Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 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)×#443 | C2 | semidirect |
| 2. #89965⋊C3 | C2 | semidirect |
| 3. #89467⋊C3 | C2 | semidirect |
| 4. #89290⋊C3 | C2 | semidirect |
| 5. #89258⋊C3 | C2 | semidirect |
| 6. #11728680 | C3 | semidirect |
| 18. C7×#443 | C2×C3 | semidirect |
| 19. #89965 | C2×C3 | semidirect |
| 20. #89467 | C2×C3 | semidirect |
| 21. #89290 | C2×C3 | semidirect |
| 22. #89258 | C2×C3 | semidirect |
| 27. Q16×ΓL(1,𝔽8) | D4 | semidirect |
| 29. D8×ΓL(1,𝔽8) | D4 | semidirect |
| 76. C7×Q16 | C3×D4 | semidirect |
| 78. C7×D8 | C3×D4 | semidirect |
| 111. #443 | AGL(1,𝔽7) | semidirect |
| 184. ΓL(1,𝔽8) | #2213 | semidirect |
| 188. Q16 | C4⋊AGL(1,𝔽7) | semidirect |
| 194. D8 | C4⋊AGL(1,𝔽7) | semidirect |
| 229. C7 | C3×#2213 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#443 | 2 |
| #89965⋊C3 | 2 |
| #89467⋊C3 | 2 |
| #89290⋊C3 | 2 |
| #89258⋊C3 | 2 |
| #11728680 | 3 |
| C3×#2213 | 7 |