| P-43m | No. 215 | Td1 |
| Axes | Coordinates | Wyckoff positions | ||||||||
| 1a | 1b | 3c | 3d | 4e | 6f | 6g | 12h | |||
| 12i | 24j | |||||||||
| I Maximal translationengleiche subgroups | ||||||||||
| [2] P23 | 1a | 1b | 3c | 3d | 4e | 6f | 6i | 6g; 6h | ||
| (195) | 12j | 2 × 12j | ||||||||
| [4] R3m | (rhombohedral axes) | 1a | 1a | 3b | 3b | 1a; 3b | 2 × 3b | 2 × 3b | 2 × 6c | |
| (160) | 2 × 3b; 6c | 4 × 6c | ||||||||
| a - b, b - c, | (1/3)(2x - y - z), (1/3)(x + y - 2z), | 3a | 3a | 9b | 9b | 3a; 9b | 2 × 9b | 2 × 9b | 2 × 18c | |
| a + b + c | (1/3)(x + y + z) | 2 × 9b; 18c | 4 × 18c | |||||||
| (hexagonal axes) | ||||||||||
| conjugate: | -a - b, b + c, | (1/3)(-2x - y + z), (1/3)(-x + y + 2z), | ||||||||
| -a + b - c | (1/3)(-x + y - z) | |||||||||
| (hexagonal axes) | ||||||||||
| conjugate: | a + b, -b + c, | (1/3)(2x + y + z), (1/3)(x - y + 2z), | ||||||||
| a - b - c | (1/3)(x - y - z) | |||||||||
| (hexagonal axes) | ||||||||||
| conjugate: | -a + b, -b - c, | (1/3)(-2x + y - z), (1/3)(-x - y - 2z), | ||||||||
| -a - b + c | (1/3)(-x - y + z) | |||||||||
| (hexagonal axes) | ||||||||||
| [3] P-42m | 1a | 1b | 1d; 2f | 1c; 2e | 4n | 2g; 4i | 2h; 4j | 4k; 4l; 4m | ||
| (111) | 4n; 8o | 3 × 8o | ||||||||
| conjugate: | b, c, a | y, z, x | ||||||||
| conjugate: | c, a, b | z, x, y | ||||||||
| II Maximal klassengleiche subgroups | ||||||||||
| Enlarged unit cell, non-isomorphic | ||||||||||
| [2] F-43c | 2a, 2b, 2c | (1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) | 8a | 8b | 24c | 24d | 32e | 48f | 48g | 96h |
| (219) | 96h | 2 × 96h | ||||||||
| [2] F-43m | 2a, 2b, 2c | (1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) | 4a; 4b | 4c; 4d | 24g | 24f | 2 × 16e | 2 × 24f | 2 × 24g | 96i |
| (216) | 2 × 48h | 2 × 96i | ||||||||
| [4] I-43m | 2a, 2b, 2c | (1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0); | 2a; 6b | 8c | 24g | 12d; 12e | 8c; 24g | 2 × 12e; 24f | 2 × 24g | 2 × 48h |
| (217) | +(0, (1/2), 0); +(0, 0, (1/2)) | 2 × 24g; 48h | 4 × 48h | |||||||
| [4] I-43m | 2a, 2b, 2c | (1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); | 8c | 2a; 6b | 12d; 12e | 24g | 8c; 24g | 2 × 24g | 2 × 12e; 24f | 2 × 48h |
| (217) | +((1/2), 0, 0); +(0, (1/2), 0); | 2 × 24g; 48h | 4 × 48h | |||||||
| +(0, 0, (1/2)) | ||||||||||
| Axes | Coordinates | Wyckoff positions | ||||||
| 1a | 1b | 3c | 3d | 4e | 6f | |||
| 6g | 12h | 12i | 24j | |||||
| Enlarged unit cell, isomorphic | ||||||||
| [27] P-43m | 3a, 3b, 3c | (1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); | 1a; 2 × 4e; 6f; | 1b; 2 × 4e; 6g; | 3c; 6g; 12h; | 3d; 6f; 12h; | 3 × 4e; | 3 × 6f; 6 × 12i; |
| ±(0, (1/3), 0); ±(0, 0, (1/3)); | 12i | 12i | 3 × 12i; 24j | 3 × 12i; 24j | 6 × 12i; 24j | 3 × 24j | ||
| ±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); | ||||||||
| ±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); | 3 × 6g; 6 × 12i; | 3 × 12h; | 9 × 12i; | 27 × 24j | ||||
| ±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); | 3 × 24j | 12 × 24j | 9 × 24j | |||||
| ±((1/3), (1/3), (1/3)); ±((1/3), (1/3), (2/3)); | ||||||||
| ±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) | ||||||||
| [p3] P-43m | pa, pb, pc | (1/p)x, (1/p)y, (1/p)z; | 1a; (p - 1) × 4e; | 1b; (p - 1) × 4e; | 3c; ((p - 1)/2) × 6g; | 3d; ((p - 1)/2) × 6f; | p × 4e; | p × 6f; |
| +((u/p), (v/p), (w/p)) | ((p - 1)/2) × 6f; | ((p - 1)/2) × 6g; | ((p - 1)/2) × 12h; | ((p - 1)/2) × 12h; | p(p - 1) × 12i; | p(p - 1) × 12i; | ||
| p = prime > 2; | (((p - 1)(p - 2))/2) × 12i; | (((p - 1)(p - 2))/2) × 12i; | ((p(p - 1))/2) × 12i; | ((p(p - 1))/2) × 12i; | ((p(p - 1)(p - 2))/6) × | ((p(p - 1)2)/4) × 24j | ||
| u, v, w = 1, . . ., p - 1 | (((p - 1)(p - 2)(p - 3))/24) | (((p - 1)(p - 2)(p - 3))/24) | (((p2 - 1)(p - 2))/8) × | (((p2 - 1)(p - 2))/8) × | 24j | |||
| × 24j | × 24j | 24j | 24j | |||||
| p × 6g; | p × 12h; | p2 × 12i; | p3 × 24j | |||||
| p(p - 1) × 12i; | ((p(p2 - 1))/2) × 24j | ((p2(p - 1))/2) × 24j | ||||||
| ((p(p - 1)2)/4) × 24j | ||||||||
