(International Tables) Maximal subgroups of space group 69

Fmmm	No. 69	F2/m2/m2/m	D_2h²³

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 1/2, 1/2); t(1/2, 0, 1/2); (2); (3); (5)

General position

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

(0, 0, 0)+ (0, 1/2, 1/2)+ (1/2, 0, 1/2)+ (1/2, 1/2, 0)+

(1) x, y, z	(2) -x, -y, z	(3) -x, y, -z	(4) x, -y, -z
(5) -x, -y, -z	(6) x, y, -z	(7) x, -y, z	(8) -x, y, z

I Maximal translationengleiche subgroups

[2] Fmm2 (42)	(1; 2; 7; 8)+
[2] Fm2m (42, Fmm2)	(1; 3; 6; 8)+	c, a, b
[2] F2mm (42, Fmm2)	(1; 4; 6; 7)+	b, c, a
[2] F222 (22)	(1; 2; 3; 4)+
[2] F112/m (12, A112/m)	(1; 2; 5; 6)+	1/2(a - b), b, c
[2] F12/m1 (12, C12/m1)	(1; 3; 5; 7)+	a, b, 1/2(-a + c)
[2] F2/m11 (12, C12/m1)	(1; 4; 5; 8)+	-b, a, 1/2(b + c)

II Maximal klassengleiche subgroups

Loss of centring translations

[2] Aeaa (68, Ccce)	1; 2; 3; 4; (1; 2; 3; 4) + (0, 1/2, 1/2); (5; 6; 7; 8) + (1/2, 0, 1/2); (5; 6; 7; 8) + (1/2, 1/2, 0)	c, b, -a	1/4, 1/4, 0
[2] Bbeb (68, Ccce)	1; 2; 3; 4; (1; 2; 3; 4) + (1/2, 0, 1/2); (5; 6; 7; 8) + (0, 1/2, 1/2); (5; 6; 7; 8) + (1/2, 1/2, 0)	c, a, b	1/4, 1/4, 0
[2] Ccce (68)	1; 2; 3; 4; (1; 2; 3; 4) + (1/2, 1/2, 0); (5; 6; 7; 8) + (0, 1/2, 1/2); (5; 6; 7; 8) + (1/2, 0, 1/2)		0, 1/4, 1/4
[2] Cmme (67)	1; 2; 7; 8; (1; 2; 7; 8) + (1/2, 1/2, 0); (3; 4; 5; 6) + (0, 1/2, 1/2); (3; 4; 5; 6) + (1/2, 0, 1/2)		0, 1/4, 1/4
[2] Bmem (67, Cmme)	1; 3; 6; 8; (1; 3; 6; 8) + (1/2, 0, 1/2); (2; 4; 5; 7) + (0, 1/2, 1/2); (2; 4; 5; 7) + (1/2, 1/2, 0)	c, a, b	1/4, 1/4, 0
[2] Aemm (67, Cmme)	1; 4; 6; 7; (1; 4; 6; 7) + (0, 1/2, 1/2); (2; 3; 5; 8) + (1/2, 0, 1/2); (2; 3; 5; 8) + (1/2, 1/2, 0)	b, c, a	1/4, 0, 1/4
[2] Cccm (66)	1; 2; 5; 6; (1; 2; 5; 6) + (1/2, 1/2, 0); (3; 4; 7; 8) + (0, 1/2, 1/2); (3; 4; 7; 8) + (1/2, 0, 1/2)		1/4, 1/4, 0
[2] Bbmb (66, Cccm)	1; 3; 5; 7; (1; 3; 5; 7) + (1/2, 0, 1/2); (2; 4; 6; 8) + (0, 1/2, 1/2); (2; 4; 6; 8) + (1/2, 1/2, 0)	c, a, b	1/4, 0, 1/4
[2] Amaa (66, Cccm)	1; 4; 5; 8; (1; 4; 5; 8) + (0, 1/2, 1/2); (2; 3; 6; 7) + (1/2, 0, 1/2); (2; 3; 6; 7) + (1/2, 1/2, 0)	b, c, a	0, 1/4, 1/4
[2] Ammm (65, Cmmm)	1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + (0, 1/2, 1/2)	b, c, a
[2] Bmmm (65, Cmmm)	1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + (1/2, 0, 1/2)	c, a, b
[2] Cmmm (65)	1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + (1/2, 1/2, 0)
[2] Aeam (64, Cmce)	1; 2; 5; 6; (1; 2; 5; 6) + (0, 1/2, 1/2); (3; 4; 7; 8) + (1/2, 0, 1/2); (3; 4; 7; 8) + (1/2, 1/2, 0)	c, b, -a
[2] Bbem (64, Cmce)	1; 2; 5; 6; (1; 2; 5; 6) + (1/2, 0, 1/2); (3; 4; 7; 8) + (0, 1/2, 1/2); (3; 4; 7; 8) + (1/2, 1/2, 0)	c, a, b
[2] Aema (64, Cmce)	1; 3; 5; 7; (1; 3; 5; 7) + (0, 1/2, 1/2); (2; 4; 6; 8) + (1/2, 0, 1/2); (2; 4; 6; 8) + (1/2, 1/2, 0)	b, c, a
[2] Ccme (64, Cmce)	1; 3; 5; 7; (1; 3; 5; 7) + (1/2, 1/2, 0); (2; 4; 6; 8) + (0, 1/2, 1/2); (2; 4; 6; 8) + (1/2, 0, 1/2)	-b, a, c
[2] Bmeb (64, Cmce)	1; 4; 5; 8; (1; 4; 5; 8) + (1/2, 0, 1/2); (2; 3; 6; 7) + (0, 1/2, 1/2); (2; 3; 6; 7) + (1/2, 1/2, 0)	a, -c, b
[2] Cmce (64)	1; 4; 5; 8; (1; 4; 5; 8) + (1/2, 1/2, 0); (2; 3; 6; 7) + (0, 1/2, 1/2); (2; 3; 6; 7) + (1/2, 0, 1/2)
[2] Amam (63, Cmcm)	1; 3; 6; 8; (1; 3; 6; 8) + (0, 1/2, 1/2); (2; 4; 5; 7) + (1/2, 0, 1/2); (2; 4; 5; 7) + (1/2, 1/2, 0)	c, b, -a	1/4, 1/4, 0
[2] Amma (63, Cmcm)	1; 2; 7; 8; (1; 2; 7; 8) + (0, 1/2, 1/2); (3; 4; 5; 6) + (1/2, 0, 1/2); (3; 4; 5; 6) + (1/2, 1/2, 0)	b, c, a	1/4, 0, 1/4
[2] Bmmb (63, Cmcm)	1; 2; 7; 8; (1; 2; 7; 8) + (1/2, 0, 1/2); (3; 4; 5; 6) + (0, 1/2, 1/2); (3; 4; 5; 6) + (1/2, 1/2, 0)	a, -c, b	0, 1/4, 1/4
[2] Bbmm (63, Cmcm)	1; 4; 6; 7; (1; 4; 6; 7) + (1/2, 0, 1/2); (2; 3; 5; 8) + (0, 1/2, 1/2); (2; 3; 5; 8) + (1/2, 1/2, 0)	c, a, b	1/4, 1/4, 0
[2] Cmcm (63)	1; 3; 6; 8; (1; 3; 6; 8) + (1/2, 1/2, 0); (2; 4; 5; 7) + (0, 1/2, 1/2); (2; 4; 5; 7) + (1/2, 0, 1/2)		0, 1/4, 1/4
[2] Ccmm (63, Cmcm)	1; 4; 6; 7; (1; 4; 6; 7) + (1/2, 1/2, 0); (2; 3; 5; 8) + (0, 1/2, 1/2); (2; 3; 5; 8) + (1/2, 0, 1/2)	-b, a, c	1/4, 0, 1/4

Enlarged unit cell

[3] a' = 3a

Fmmm (69)	<2; 3; 5>	3a, b, c
Fmmm (69)	<(2; 3; 5) + (2, 0, 0)>	3a, b, c	1, 0, 0
Fmmm (69)	<(2; 3; 5) + (4, 0, 0)>	3a, b, c	2, 0, 0

[3] b' = 3b

Fmmm (69)	<2; 3; 5>	a, 3b, c
Fmmm (69)	<3; (2; 5) + (0, 2, 0)>	a, 3b, c	0, 1, 0
Fmmm (69)	<3; (2; 5) + (0, 4, 0)>	a, 3b, c	0, 2, 0

[3] c' = 3c

Fmmm (69)	<2; 3; 5>	a, b, 3c
Fmmm (69)	<2; (3; 5) + (0, 0, 2)>	a, b, 3c	0, 0, 1
Fmmm (69)	<2; (3; 5) + (0, 0, 4)>	a, b, 3c	0, 0, 2

Series of maximal isomorphic subgroups

[p] a' = pa

Fmmm (69)	<(2; 3; 5) + (2u, 0, 0)>	pa, b, c	u, 0, 0
	p > 2; 0 ≤ u < p p conjugate subgroups for the prime p

[p] b' = pb

Fmmm (69)	<3; (2; 5) + (0, 2u, 0)>	a, pb, c	0, u, 0
	p > 2; 0 ≤ u < p p conjugate subgroups for the prime p

[p] c' = pc

Fmmm (69)	<2; (3; 5) + (0, 0, 2u)>	a, b, pc	0, 0, u
	p > 2; 0 ≤ u < p p conjugate subgroups for the prime p

I Minimal translationengleiche supergroups

[2] I4/mmm (139); [2] I4/mcm (140); [3] Fm-3 (202)

II Minimal non-isomorphic klassengleiche supergroups

Additional centring translations

none

Decreased unit cell

[2] a' = 1/2a, b' = 1/2b, c' = 1/2c Pmmm (47)

International Tables for Crystallography (2006). Vol. A1, ch. 2.3, Maximal subgroups of space group 69.