Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 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. ΓL(1,𝔽8)×#135 | C2 | semidirect |
| 2. C2×D56⋊C3 | C2 | semidirect |
| 3. #6402⋊C3 | C2 | semidirect |
| 4. #6400⋊C3 | C2 | semidirect |
| 5. #6386⋊C3 | C2 | semidirect |
| 6. #89965 | C3 | semidirect |
| 9. Q16×ΓL(1,𝔽8) | C22 | semidirect |
| 11. D8×ΓL(1,𝔽8) | C22 | semidirect |
| 15. ΓL(1,𝔽8)⋊C16 | C22 | semidirect |
| 16. C7×#135 | C2×C3 | semidirect |
| 17. C2×D56 | C2×C3 | semidirect |
| 18. #6402 | C2×C3 | semidirect |
| 19. #6400 | C2×C3 | semidirect |
| 20. #6386 | C2×C3 | semidirect |
| 34. C7×Q16 | C22×C3 | semidirect |
| 36. C7×D8 | C22×C3 | semidirect |
| 40. C7⋊C16 | C22×C3 | semidirect |
| 77. #135 | AGL(1,𝔽7) | semidirect |
| 99. ΓL(1,𝔽8) | #509 | semidirect |
| 102. Q16 | C2×AGL(1,𝔽7) | semidirect |
| 106. D8 | C2×AGL(1,𝔽7) | semidirect |
| 130. C7 | C3×#509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#135 | 2 |
| C2×D56⋊C3 | 2 |
| #6402⋊C3 | 2 |
| #6400⋊C3 | 2 |
| #6386⋊C3 | 2 |
| #89965 | 3 |
| C3×#509 | 7 |