Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
252 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 90 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)×#498 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#365 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#364 | C2 | semidirect |
| 6. C7×#2230 | C3 | semidirect |
| 16. C7×#498 | C2×C3 | semidirect |
| 18. C7×#365 | C2×C3 | semidirect |
| 19. C7×#364 | C2×C3 | semidirect |
| 24. Q16×ΓL(1,𝔽8) | Q8 | semidirect |
| 26. D8×ΓL(1,𝔽8) | Q8 | semidirect |
| 63. #2230 | ΓL(1,𝔽8) | direct (N × H) |
| 66. C7×Q16 | C3×Q8 | semidirect |
| 68. C7×D8 | C3×Q8 | semidirect |
| 94. #498 | C2×ΓL(1,𝔽8) | semidirect |
| 96. #365 | C2×ΓL(1,𝔽8) | semidirect |
| 97. #364 | C2×ΓL(1,𝔽8) | semidirect |
| 158. ΓL(1,𝔽8) | #2230 | direct (N × H) |
| 161. Q16 | Q8×ΓL(1,𝔽8) | semidirect |
| 163. D8 | Q8×ΓL(1,𝔽8) | semidirect |
| 196. C7 | C3×#2230 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#498 | 2 |
| ΓL(1,𝔽8)×#443 | 2 |
| ΓL(1,𝔽8)×#365 | 2 |
| ΓL(1,𝔽8)×#364 | 2 |
| ΓL(1,𝔽8)×#360 | 2 |
| C7×#2230 | 3 |
| C3×#2230 | 7 |