Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
236 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 123 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)×#107 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 6. C7×#428 | C3 | semidirect |
| 9. M16×ΓL(1,𝔽8) | C4 | semidirect |
| 20. C7×#107 | C2×C3 | semidirect |
| 21. C7×#106 | C2×C3 | semidirect |
| 32. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 35. C7×M16 | C3×C4 | semidirect |
| 58. #428 | ΓL(1,𝔽8) | direct (N × H) |
| 68. C7×C8 | C2×C3×C4 | semidirect |
| 86. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 87. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 108. ΓL(1,𝔽8) | #428 | direct (N × H) |
| 111. M16 | C4×ΓL(1,𝔽8) | semidirect |
| 142. C8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 143. C7 | C3×#428 | 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 |
| ΓL(1,𝔽8)×#117 | 2 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| C7×#428 | 3 |
| C3×#428 | 7 |