Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
280 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 87 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. C2×D8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#127 | C2 | semidirect |
| 3. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 6. C7×#459 | C3 | semidirect |
| 17. C2×C7×D8 | C2×C3 | semidirect |
| 18. C7×#127 | C2×C3 | semidirect |
| 19. C4×C7×D4 | C2×C3 | semidirect |
| 20. C7×C4⋊C8 | C2×C3 | semidirect |
| 21. C7×#102 | C2×C3 | semidirect |
| 30. D4×ΓL(1,𝔽8) | D4 | semidirect |
| 61. C4×ΓL(1,𝔽8) | D8 | semidirect |
| 64. #459 | ΓL(1,𝔽8) | direct (N × H) |
| 72. C7×D4 | C3×D4 | semidirect |
| 94. C2×D8 | C2×ΓL(1,𝔽8) | semidirect |
| 95. #127 | C2×ΓL(1,𝔽8) | semidirect |
| 96. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 97. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 98. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 108. C4×C7 | C3×D8 | semidirect |
| 125. ΓL(1,𝔽8) | #459 | direct (N × H) |
| 160. D4 | D4×ΓL(1,𝔽8) | semidirect |
| 167. C7 | C3×#459 | semidirect |
| 182. C4 | D8×ΓL(1,𝔽8) | 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 |
| ΓL(1,𝔽8)×#127 | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#459 | 3 |
| C3×#459 | 7 |