Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
264 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 6. C7×#442 | C3 | semidirect |
| 9. D8×ΓL(1,𝔽8) | C4 | semidirect |
| 16. C4×C7×D4 | C2×C3 | semidirect |
| 17. C7×#106 | C2×C3 | semidirect |
| 18. C7×#102 | C2×C3 | semidirect |
| 19. C7×#97 | C2×C3 | semidirect |
| 23. D4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 29. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 32. C7×D8 | C3×C4 | semidirect |
| 51. #442 | ΓL(1,𝔽8) | direct (N × H) |
| 54. C7×D4 | C2×C3×C4 | semidirect |
| 60. C7×C8 | C2×C3×C4 | semidirect |
| 73. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 74. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 75. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 76. #97 | C2×ΓL(1,𝔽8) | semidirect |
| 94. ΓL(1,𝔽8) | #442 | direct (N × H) |
| 97. D8 | C4×ΓL(1,𝔽8) | semidirect |
| 116. D4 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 122. C8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 123. C7 | C3×#442 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D8×ΓL(1,𝔽8) | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| C7×#442 | 3 |
| C3×#442 | 7 |