Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
428 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 228 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 |
|---|---|---|
| 2. ΓL(1,𝔽8)×#322 | C2 | semidirect |
| 3. C82×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C7×#1455 | C3 | semidirect |
| 11. C7×#322 | C2×C3 | semidirect |
| 12. C7×C82 | C2×C3 | semidirect |
| 33. C8×ΓL(1,𝔽8) | M16 | semidirect |
| 34. #1455 | ΓL(1,𝔽8) | direct (N × H) |
| 51. #322 | C2×ΓL(1,𝔽8) | semidirect |
| 52. C82 | C2×ΓL(1,𝔽8) | semidirect |
| 58. C7×C8 | C3×M16 | semidirect |
| 82. ΓL(1,𝔽8) | #1455 | direct (N × H) |
| 100. C8 | M16×ΓL(1,𝔽8) | semidirect |
| 101. C7 | C3×#1455 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#322 | 2 |
| C82×ΓL(1,𝔽8) | 2 |
| C7×#1455 | 3 |
| C3×#1455 | 7 |