C12/m1

UNIQUE AXIS b

CELL CHOICE 1

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

[2] C1m1 (8)

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

[2] C121(5)

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

[2] P-1 (2)

(1/2)(a - b),

2x, x + y, z

1a

1c

1b

1g

1d; 1e

1f; 1h

2i

2i

2i

2 × 2i

b, c

II Maximal klassengleiche subgroups

Loss of centring translations

[2] P121/a1 (14)

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

[2] P12/a1 (13)

x + (1/4), y + (1/4), z

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

[2] P121/m1 (11)

x + (1/4), y + (1/4), z

2e

2e

2e

2e

2a; 2b

2c; 2d

4f

4f

2 × 2e

2 × 4f

[2] P12/m1 (10)

1a; 1e

1b; 1d

1c; 1h

1f; 1g

4o

4o

2i; 2j

2k; 2l

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

[2] C12/c1 (15)

a, b, 2c

x, y, (1/2)z; +(0, 0, (1/2))

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

[2] C12/c1 (15)

a, b, 2c

x, y, (1/2)z + (1/4); +(0, 0, (1/2))

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

[2] C12/c1 (15)

a - 2c, b, 2c

x, y, x + (1/2)z; +(0, 0, (1/2))

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

or: I12/a1

2c, -b, a

(1/2)z, -y, x; +((1/2), 0, 0)

[2] C12/c1 (15)

a - 2c, b, 2c

x, y, x + (1/2)z + (1/4); +(0, 0, (1/2))

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

or: I12/a1

2c, -b, a

(1/2)z + (1/4), -y, x; +((1/2), 0, 0)

Enlarged unit cell, isomorphic

[3] C12/m1

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

[p] C12/m1

a, pb, c

x, (1/p)y, z; +(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

[2] C12/m1

a, b, 2c

x, y, (1/2)z; +(0, 0, (1/2))

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

[2] C12/m1

a, b, 2c

x, y, (1/2)z + (1/4); +(0, 0, (1/2))

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

[2] I12/m1

a, b, 2c

x, y, (1/2)z; +(0, 0, (1/2))

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

[2] I12/m1

a, b, 2c

x, y, (1/2)z + (1/4); +(0, 0, (1/2))

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

[3] C12/m1

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

[3] C12/m1

3a, b, -a + c

(1/3)(x + z), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

[3] C12/m1

3a, b, a + c

(1/3)(x - z), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

[3] C12/m1

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

[p] C12/m1

a, b, pc

x, y, (1/p)z; +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

[p] C12/m1

pa, b, qa + c

(1/p)(x - qz), y, z; +((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

* cell choice 2 (A12/m1)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I12/m1)

‡ cell choice 3 (I12/m1) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I12/m1) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3

A12/m1

UNIQUE AXIS b

CELL CHOICE 2

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

[2] A1m1

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

[2] A121

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

[2] P-1

a, b, (1/2)(-b + c)

x, y + z, 2z

1a

1c

1e

1d

1b; 1g

1f; 1h

2i

2i

2i

2 × 2i

II Maximal klassengleiche subgroups

Loss of centring translations

[2] P121/c1

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

[2] P12/c1

x, y + (1/4), z + (1/4)

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

[2] P121/m1

x, y + (1/4), z + (1/4)

2e

2e

2e

2e

2a; 2c

2b; 2d

4f

4f

2 × 2e

2 × 4f

[2] P12/m1

1a; 1f

1b; 1c

1e; 1g

1d; 1h

4o

4o

2i; 2k

2j; 2l

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

[2] A12/n1

2a + c, b, c

(1/2)x, y, -(1/2)x + z; +((1/2), 0, (1/2))

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

[2] A12/n1

2a + c, b, c

(1/2)x - (1/4), y, -(1/2)x + z - (1/4); +((1/2), 0, (1/2))

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

[2] I12/a1

2a, b, c

(1/2)x, y, z; +((1/2), 0, 0)

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

[2] I12/a1

2a, b, c

(1/2)x - (1/4), y, z + (1/2); +((1/2), 0, 0)

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

Enlarged unit cell, isomorphic

[3] A12/m1

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

[p] A12/m1

a, pb, c

x, (1/p)y, z; +(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

[2] A12/m1

2a, b, c

(1/2)x, y, z; +((1/2), 0, 0)

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

[2] A12/m1

2a, b, c

(1/2)x - (1/4), y, z; +((1/2), 0, 0)

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

[2] C12/m1

2a + c, b, c

(1/2)x, y, -(1/2)x + z; +((1/2), 0, (1/2))

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

[2] C12/m1

2a + c, b, c

(1/2)x - (1/4), y, -(1/2)x + z - (1/4); +((1/2), 0, (1/2))

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

[3] A12/m1

a - 2c, b, 3c

x, y, (1/3)(2x + z); ±(0, 0, (1/3))

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

[3] A12/m1

a - c, b, 3c

x, y, (1/3)(x + z); ±(0, 0, (1/3))

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

[3] A12/m1

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

[3] A12/m1

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

[p] A12/m1

pa, b, c

(1/p)x, y, z; +((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

[p] A12/m1

a + qc, b, pc

x, y, (1/p)(-qx + z); +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

* cell choice 2 (A12/m1)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I12/m1)

‡ cell choice 3 (I12/m1) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I12/m1) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3

I12/m1

UNIQUE AXIS b

CELL CHOICE 3

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

I1m1

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

I121

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

P-1

a, (1/2)(-a + b - c), c

x + y, 2y, y + z

1a

1f

1d

1b

1c; 1h

1e; 1g

2i

2i

2i

2 × 2i

II Maximal klassengleiche subgroups

Loss of centring translations

P121/n1

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

P12/n1

x - (1/4), y + (1/4), z - (1/4)

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

P121/m1

x - (1/4), y + (1/4), z - (1/4)

2e

2e

2e

2e

2a; 2d

2b; 2c

4f

4f

2 × 2e

2 × 4f

P12/m1

1a; 1h

1b; 1g

1d; 1f

1c; 1e

4o

4o

2i; 2l

2j; 2k

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

I12/a1

2a, b, -a + c

(1/2)(x + z), y, z; +((1/2), 0, 0)

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

I12/a1

2a, b, -a + c

(1/2)(x + z) + (1/4), y, z; +((1/2), 0, 0)

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

I12/a1

2a, b, a + c

(1/2)(x - z), y, z; +((1/2), 0, 0)

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

or: C12/c1

-a + c, -b, 2a

z, -y, (1/2)(x + z); +(0, 0, (1/2))

I12/a1

2a, b, a + c

(1/2)(x - z) + (1/4), y, z; +((1/2), 0, 0)

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

or: C12/c1

-a + c, -b, 2a

z, -y, (1/2)(x + z) + (1/4); +(0, 0, (1/2))

Enlarged unit cell, isomorphic

I12/m1

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

I12/m1

a, pb, c

x, (1/p)y, z; +(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

I12/m1

2a, b, -a + c

(1/2)(x + z), y, z; +((1/2), 0, 0)

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

I12/m1

2a, b, -a + c

(1/2)(x + z) + (1/4), y, z; +((1/2), 0, 0)

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

A12/m1

2a, b, -a + c

(1/2)(x + z), y, z; +((1/2), 0, 0)

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

A12/m1

2a, b, -a + c

(1/2)(x + z) + (1/4), y, z; +((1/2), 0, 0)

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

I12/m1

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

I12/m1

3a, b, 2a + c

(1/3)(x - 2z), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

I12/m1

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

I12/m1

3a, b, -2a + c

(1/3)(x + 2z), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

I12/m1

pa, b, c

(1/p)x, y, z; ±((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

I12/m1

a + 2qc, b, pc

x, y, (1/p)(-2qx + z); +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

* cell choice 2 (A12/m1)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I12/m1)

‡ cell choice 3 (I12/m1) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I12/m1) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3

A112/m

UNIQUE AXIS c

CELL CHOICE 1

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

[2] A11m (8)

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

[2] A112(5)

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

[2] P-1 (2)

a,

x, 2y, y + z

1a

1b

1d

1f

1c; 1g

1e; 1h

2i

2i

2i

2 × 2i

(1/2)(b - c), c

II Maximal klassengleiche subgroups

Loss of centring translations

[2] P1121/b (14)

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

[2] P112/b (13)

x, y + (1/4), z + (1/4)

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

[2] P1121/m (11)

x, y + (1/4), z + (1/4)

2e

2e

2e

2e

2a; 2b

2c; 2d

4f

4f

2 × 2e

2 × 4f

[2] P112/m (10)

1a; 1e

1b; 1d

1c; 1h

1f; 1g

4o

4o

2i; 2j

2k; 2l

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

[2] A112/a (15)

2a, b, c

(1/2)x, y, z; +((1/2), 0, 0)

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

[2] A112/a (15)

2a, b, c

(1/2)x + (1/4), y, z; +((1/2), 0, 0)

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

[2] A112/a (15)

2a, -2a + b, c

(1/2)x + y, y, z; +((1/2), 0, 0)

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

or: I112/b

b, 2a, -c

y, (1/2)x, -z; +(0, (1/2), 0)

[2] A112/a (15)

2a, -2a + b, c

(1/2)x + y + (1/4), y, z; +((1/2), 0, 0)

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

or: I112/b

b, 2a, -c

y, (1/2)x + (1/4), -z; +(0, (1/2), 0)

Enlarged unit cell, isomorphic

[3] A112/m

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

[p] A112/m

a, b, pc

x, y, (1/p)z; +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

[2] A112/m

2a, b, c

(1/2)x, y, z; +((1/2), 0, 0)

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

[2] A112/m

2a, b, c

(1/2)x + (1/4), y, z; +((1/2), 0, 0)

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

[2] I112/m

2a, b, c

(1/2)x, y, z; +((1/2), 0, 0)

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

[2] I112/m

2a, b, c

(1/2)x + (1/4), y, z; +((1/2), 0, 0)

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

[3] A112/m

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

[3] A112/m

a - b, 3b, c

x, (1/3)(x + y), z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

[3] A112/m

a + b, 3b, c

x, (1/3)(-x + y), z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

[3] A112/m

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

[p] A112/m

pa, b, c

(1/p)x, y, z; +((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

[p] A112/m

a + qb, pb, c

x, (1/p)( - qx + y), z; +(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

* cell choice 2 (B112/m)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I112/m)

‡ cell choice 3 (I112/m) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I112/m) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3

B112/m

UNIQUE AXIS c

CELL CHOICE 2

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

[2] B11m

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

[2] B112

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

[2] P-1

(1/2)(a - c), b, c

2x, y, x + z

1a

1b

1g

1c

1d; 1f

1e; 1h

2i

2i

2i

2 × 2i

II Maximal klassengleiche subgroups

Loss of centring translations

[2] P1121/a

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

[2] P112/a

x + (1/4), y, z + (1/4)

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

[2] P1121/m

x + (1/4), y, z + (1/4)

2e

2e

2e

2e

2a; 2c

2b; 2d

4f

4f

2 × 2e

2 × 4f

[2] P112/m

1a; 1f

1b; 1c

1e; 1g

1d; 1h

4o

4o

2i; 2k

2j; 2l

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

[2] B112/n

a, a + 2b, c

x - (1/2)y, (1/2)y, z; +((1/2), (1/2), 0)

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

[2] B112/n

a, a + 2b, c

x - (1/2)y - (1/4), (1/2)y - (1/4), z; +((1/2), (1/2), 0)

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

[2] I112/b

a, 2b, c

x, (1/2)y, z; +(0, (1/2), 0)

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

[2] I112/b

a, 2b, c

x + (1/2), (1/2)y - (1/4), z; +(0, (1/2), 0)

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

Enlarged unit cell, isomorphic

[3] B112/m

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

[p] B112/m

a, b, pc

x, y, (1/p)z; +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

[2] B112/m

a, 2b, c

x, (1/2)y, z; +(0, (1/2), 0)

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

[2] B112/m

a, 2b, c

x, (1/2)y - (1/4), z; +(0, (1/2), 0)

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

[2] A112/m

a, a + 2b, c

x - (1/2)y, (1/2)y, z; +((1/2), (1/2), 0)

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

[2] A112/m

a, a + 2b, c

x - (1/2)y - (1/4), (1/2)y - (1/4), z; +((1/2), (1/2), 0)

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

[3] B112/m

3a, -2a + b, c

(1/3)(x + 2y), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

[3] B112/m

3a, -a + b, c

(1/3)(x + y), y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

[3] B112/m

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

[3] B112/m

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

[p] B112/m

a, pb, c

x, (1/p)y, z; +(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

[p] B112/m

pa, qa + b, c

(1/p)(x - qy), y, z; +((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

* cell choice 2 (B112/m)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I112/m)

‡ cell choice 3 (I112/m) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I112/m) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3

I112/m

CELL CHOICE 3

UNIQUE AXIS c

Axes

Coordinates

Wyckoff positions

2a

2b

2c

2d

4e

4f

4g

4h

4i

8j

I Maximal translationengleiche subgroups

I11m

2a

2a

2a

2a

4b

4b

4b

4b

2 × 2a

2 × 4b

I112

2a

2a

2b

2b

4c

4c

2 × 2a

2 × 2b

4c

2 × 4c

P-1

a, b, (1/2)(-a - b + c)

x + y, y + z, 2z

1a

1e

1c

1d

1b; 1h

1f; 1g

2i

2i

2i

2 × 2i

II Maximal klassengleiche subgroups

Loss of centring translations

P1121/n

2a

2c

2d

2b

4e

4e

4e

4e

4e

2 × 4e

P112/n

x - (1/4), y - (1/4), z + (1/4)

2e

2e

2f

2f

2a; 2c

2b; 2d

2 × 2e

2 × 2 f

4g

2 × 4g

P1121/m

x - (1/4), y - (1/4), z + (1/4)

2e

2e

2e

2e

2a; 2d

2b; 2c

4f

4f

2 × 2e

2 × 4f

P112/m

1a; 1h

1b; 1g

1d; 1f

1c; 1e

4o

4o

2i; 2l

2j; 2k

2m; 2n

2 × 4o

Enlarged unit cell, non-isomorphic

I112/b

a - b, 2b, c

x, (1/2)(x + y), z; +(0, (1/2), 0)

4a

4b

4e

4e

4c; 4d

8f

8f

2 × 4e

8f

2 × 8f

I112/b

a - b, 2b, c

x, (1/2)(x + y) + (1/4), z; +(0, (1/2), 0)

4e

4e

4a

4b

8f

4c; 4d

2 × 4e

8f

8f

2 × 8f

I112/b

a + b, 2b, c

x, (1/2)(-x + y), z; +(0, (1/2), 0)

4a

4b

4e

4e

8f

4c; 4d

8f

2 × 4e

8f

2 × 8f

or: A112/a

2b, a - b, -c

(1/2)(x + y), x, -z; +((1/2), 0, 0)

I112/b

a + b, 2b, c

x, (1/2)(-x + y) + (1/4), z; +(0, (1/2), 0)

4e

4e

4a

4b

4c; 4d

8f

2 × 4e

8f

8f

2 × 8f

or: A112/a

2b, a - b, -c

(1/2)(x + y) + (1/4), x, -z; +((1/2), 0, 0)

Enlarged unit cell, isomorphic

I112/m

a, b, 3c

x, y, (1/3)z; ±(0, 0, (1/3))

2a; 4g

2b; 4g

2c; 4h

2d; 4h

4e; 8j

4f; 8j

3 × 4g

3 × 4h

4i; 8j

3 × 8j

I112/m

a, b, pc

x, y, (1/p)z; +(0, 0, (u/p))

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e; ((p - 1)/2)

4f; ((p - 1)/2)

p × 4g

p × 4h

4i; ((p - 1)/2)

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4g

4g

× 4h

× 4h

× 8j

× 8j

× 8j

I112/m

a - b, 2b, c

x, (1/2)(x + y), z; +(0, (1/2), 0)

2a; 2c(d*)

2b; 2d(c*)

4i

4i

4e; 4f

8j

4g; 4h

8j

2 × 4i

2 × 8j

I112/m

a - b, 2b, c

x, (1/2)(x + y) + (1/4), z; +(0, (1/2), 0)

4i

4i

2a(b*);

2b(a*);

8j

4e; 4f

8j

4g; 4h

2 × 4i

2 × 8j

2c

2d

B112/m

a - b, 2b, c

x, (1/2)(x + y), z; +(0, (1/2), 0)

2a; 2d

2b; 2c

4i

4i

8j

4e; 4f

4g; 4h

8j

2 × 4i

2 × 8j

B112/m

a - b, 2b, c

x, (1/2)(x + y) + (1/4), z; +(0, (1/2), 0)

4i

4i

2a; 2d

2b; 2c

4e; 4f

8j

8j

4g; 4h

2 × 4i

2 × 8j

I112/m

3a, b, c

(1/3)x, y, z; ±((1/3), 0, 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

8j

8j

I112/m

a + 2b, 3b, c

x, (1/3)(-2x + y), z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2d(c† );

2c(d† );

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

I112/m

a, 3b, c

x, (1/3)y, z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2d(c† † );

2c(d† † );

4e(f† );

4f(e† );

4g; 8j

4h; 8j

3 × 4i

3 × 8j

4i

4i

8j

8j

I112/m

a - 2b, 3b, c

x, (1/3)(2x + y), z; ±(0, (1/3), 0)

2a; 4i

2b; 4i

2c; 4i

2d; 4i

4e; 8j

4f; 8j

4g; 8j

4h; 8j

3 × 4i

3 × 8j

I112/m

a, pb, c

x, (1/p)y, z; ±(0, (u/p), 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c; ((p - 1)/2)

2d; ((p - 1)/2)

4e(f‡ );

4f(e‡ );

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

p = prime > 2; u = 1, . . ., p - 1

4i

4i

× 4i

× 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

I112/m

pa, 2qa + b, c

(1/p)(x - 2qy), y, z; +((u/p), 0, 0)

2a; ((p - 1)/2) ×

2b; ((p - 1)/2) ×

2c(d§ );

2d(c§ );

4e(f);

4f(e);

4g; ((p - 1)/2)

4h; ((p - 1)/2)

p × 4i

p × 8j

-(1/2)(p - 1)≤ q≤ (1/2)(p - 1)

4i

4i

((p - 1)/2) × 4i

((p - 1)/2) × 4i

((p - 1)/2) × 8j

((p - 1)/2) × 8j

× 8j

× 8j

* cell choice 2 (B112/m)

§ cell choices 1 or 2 and q = 2n + 1

† cell choice 3 (I112/m)

‡ cell choice 3 (I112/m) and p = 4n - 1

† † cell choices 2 or 3

cell choice 3 (I112/m) and p + 2q = 4n - 1

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3