Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 135 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×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 5. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×C8×D4 | C3 | semidirect |
| 16. C22×C7×C8 | C2×C3 | semidirect |
| 18. C7×C4⋊C8 | C2×C3 | semidirect |
| 19. C7×C22⋊C8 | C2×C3 | semidirect |
| 20. C4×C7×C8 | C2×C3 | semidirect |
| 23. D4×ΓL(1,𝔽8) | C8 | direct (N × H) |
| 32. C8×ΓL(1,𝔽8) | D4 | direct (N × H) |
| 49. C22×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 53. C4×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 56. C8×D4 | ΓL(1,𝔽8) | direct (N × H) |
| 58. C7×D4 | C3×C8 | semidirect |
| 67. C7×C8 | C3×D4 | semidirect |
| 81. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 83. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 84. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 85. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 88. C22×C7 | C2×C3×C8 | semidirect |
| 92. C4×C7 | C2×C3×C8 | semidirect |
| 106. ΓL(1,𝔽8) | C8×D4 | direct (N × H) |
| 130. D4 | C8×ΓL(1,𝔽8) | direct (N × H) |
| 139. C8 | D4×ΓL(1,𝔽8) | direct (N × H) |
| 140. C7 | C3×C8×D4 | semidirect |
| 147. C22 | C2×C8×ΓL(1,𝔽8) | semidirect |
| 151. C4 | C2×C8×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×ΓL(1,𝔽8) | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C7×C8×D4 | 3 |
| C3×C8×D4 | 7 |