Finite Group DatabasebetaFinite Group DatabasebetaOrder 2016.
262 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 63 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)×#217 | C2 | semidirect |
| 3. C7×#834 | C3 | semidirect |
| 5. C23×C3×ΓL(1,𝔽8) | C4 | semidirect |
| 8. C2×C3×C4×ΓL(1,𝔽8) | C4 | semidirect |
| 13. C7×#217 | C2×C3 | semidirect |
| 24. C23×C3×C7 | C3×C4 | semidirect |
| 28. C23×ΓL(1,𝔽8) | Dic3 | semidirect |
| 31. C2×C3×C4×C7 | C3×C4 | semidirect |
| 37. C2×C4×ΓL(1,𝔽8) | Dic3 | semidirect |
| 47. #834 | ΓL(1,𝔽8) | direct (N × H) |
| 73. C3×ΓL(1,𝔽8) | #99 | semidirect |
| 74. C23×C7 | C3×Dic3 | semidirect |
| 78. C2×C4×C7 | C3×Dic3 | semidirect |
| 83. #217 | C2×ΓL(1,𝔽8) | semidirect |
| 115. C23×C3 | C4×ΓL(1,𝔽8) | semidirect |
| 123. C2×C3×C4 | C4×ΓL(1,𝔽8) | semidirect |
| 125. C3×C7 | C3×#99 | semidirect |
| 127. ΓL(1,𝔽8) | #834 | direct (N × H) |
| 159. C23 | Dic3×ΓL(1,𝔽8) | semidirect |
| 163. C2×C4 | Dic3×ΓL(1,𝔽8) | semidirect |
| 164. C7 | C3×#834 | semidirect |
| 186. C3 | ΓL(1,𝔽8)×#99 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#217 | 2 |
| C7×#834 | 3 |
| ΓL(1,𝔽8)×#99 | 3 |
| C3×#834 | 7 |