Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
260 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 171 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 |
|---|---|---|
| 4. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 5. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×#405 | C3 | semidirect |
| 8. M16×ΓL(1,𝔽8) | C4 | semidirect |
| 9. C2×C8×ΓL(1,𝔽8) | C4 | semidirect |
| 19. C7×#97 | C2×C3 | semidirect |
| 20. C4×C7×C8 | C2×C3 | semidirect |
| 32. C7×M16 | C3×C4 | semidirect |
| 33. C2×C7×C8 | C3×C4 | semidirect |
| 52. #405 | ΓL(1,𝔽8) | direct (N × H) |
| 77. #97 | C2×ΓL(1,𝔽8) | semidirect |
| 78. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 95. ΓL(1,𝔽8) | #405 | direct (N × H) |
| 97. M16 | C4×ΓL(1,𝔽8) | semidirect |
| 98. C2×C8 | C4×ΓL(1,𝔽8) | semidirect |
| 124. C7 | C3×#405 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C7×#405 | 3 |
| C3×#405 | 7 |