Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
24030 conjugacy classes of subgroups; only the 4939 normal subgroups are shown.
This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 4939 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C24×C3×D4 | C2 |
| socle | C25×C3 | C23 |
| hypercenter | C24×D4 | D3 |
| hypocenter | C3 | C23×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C24×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×D4 | C2 | semidirect |
| 2. C24×#67 | C2 | semidirect |
| 3. C24×D12 | C2 | semidirect |
| 4. C22×#940 | C2 | direct (N × H) |
| 5. C23×C4×Dic3 | C2 | semidirect |
| 10. C23×C3×D4 | C22 | semidirect |
| 13. C23×#67 | C22 | semidirect |
| 16. C23×D12 | C22 | semidirect |
| 17. C2×#940 | C22 | direct (N × H) |
| 18. C22×C4×Dic3 | C22 | semidirect |
| 21. C24×D4 | D3 | semidirect |
| 32. C22×C3×D4 | C23 | semidirect |
| 33. C23×C3×C4 | D4 | semidirect |
| 36. C22×#67 | C23 | semidirect |
| 37. C23×Dic3 | D4 | semidirect |
| 40. C22×D12 | C23 | semidirect |
| 41. #940 | C23 | direct (N × H) |
| 42. C2×C4×Dic3 | C23 | semidirect |
| 50. C23×D4 | C2×D3 | semidirect |
| 67. C2×C3×D4 | C24 | semidirect |
| 68. C22×C3×C4 | C2×D4 | semidirect |
| 71. C2×#67 | C24 | semidirect |
| 72. C22×Dic3 | C2×D4 | semidirect |
| 75. C2×D12 | C24 | semidirect |
| 76. C4×Dic3 | C24 | semidirect |
| 89. C22×D4 | C22×D3 | semidirect |
| 92. C23×C4 | #67 | semidirect |
| 99. C23×C3 | #127 | semidirect |
| 109. C2×C3×C4 | C22×D4 | semidirect |
| 112. C2×Dic3 | C22×D4 | semidirect |
| 131. C2×D4 | C23×D3 | semidirect |
| 134. C22×C4 | C2×#67 | semidirect |
| 140. C22×C3 | C2×#127 | semidirect |
| 147. C3×C4 | C23×D4 | semidirect |
| 148. Dic3 | C23×D4 | semidirect |
| 156. C23 | #940 | direct (N × H) |
| 166. C2×C4 | C22×#67 | semidirect |
| 169. C2×C3 | C22×#127 | semidirect |
| 179. C22 | C2×#940 | direct (N × H) |
| 184. C4 | C23×#67 | semidirect |
| 185. C3 | C23×#127 | semidirect |
| 188. C2 | C22×#940 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×D4 | 2 |
| C24×#67 | 2 |
| C24×D12 | 2 |
| C22×#940 | 2 |
| C23×C4×Dic3 | 2 |
| C23×#127 | 3 |