Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
676 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 267 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 |
|---|---|---|
| 1. C22×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×D8×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 3. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C22×C7×D8 | C3 | semidirect |
| 7. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 9. D8×ΓL(1,𝔽8) | C22 | direct (N × H) |
| 10. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 11. C22×C7×D4 | C2×C3 | semidirect |
| 12. C2×C7×D8 | C2×C3 | semidirect |
| 13. C22×C7×C8 | C2×C3 | semidirect |
| 19. D4×ΓL(1,𝔽8) | C23 | semidirect |
| 22. C8×ΓL(1,𝔽8) | C23 | semidirect |
| 25. C2×C7×D4 | C22×C3 | semidirect |
| 27. C7×D8 | C22×C3 | semidirect |
| 28. C2×C7×C8 | C22×C3 | semidirect |
| 34. C22×ΓL(1,𝔽8) | D8 | direct (N × H) |
| 38. C22×D8 | ΓL(1,𝔽8) | direct (N × H) |
| 43. C7×D4 | C23×C3 | semidirect |
| 46. C7×C8 | C23×C3 | semidirect |
| 54. C2×ΓL(1,𝔽8) | C2×D8 | direct (N × H) |
| 56. C22×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 57. C2×D8 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 58. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 61. C22×C7 | C3×D8 | semidirect |
| 73. ΓL(1,𝔽8) | C22×D8 | direct (N × H) |
| 76. C2×D4 | C22×ΓL(1,𝔽8) | semidirect |
| 78. D8 | C22×ΓL(1,𝔽8) | direct (N × H) |
| 79. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 81. C2×C7 | C2×C3×D8 | semidirect |
| 93. D4 | C23×ΓL(1,𝔽8) | semidirect |
| 96. C8 | C23×ΓL(1,𝔽8) | semidirect |
| 97. C7 | C22×C3×D8 | semidirect |
| 103. C22 | D8×ΓL(1,𝔽8) | direct (N × H) |
| 109. C2 | C2×D8×ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×ΓL(1,𝔽8) | 2 |
| C2×D8×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| C22×C7×D8 | 3 |
| C22×C3×D8 | 7 |