Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
15150 conjugacy classes of subgroups; only the 4427 normal subgroups are shown.
This group has 15150 conjugacy classes of subgroups — too many to draw. Only its 4427 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. C25×C3×C4 | C2 | semidirect |
| 3. C23×#217 | C2 | semidirect |
| 4. C22×#929 | C2 | direct (N × H) |
| 5. C23×(C2×D3)⋊C4 | C2 | semidirect |
| 6. C23×Dic3⋊C4 | C2 | semidirect |
| 12. C24×C3×C4 | C22 | semidirect |
| 17. C22×#217 | C22 | semidirect |
| 18. C2×#929 | C22 | direct (N × H) |
| 19. C22×(C2×D3)⋊C4 | C22 | semidirect |
| 20. C22×Dic3⋊C4 | C22 | semidirect |
| 22. C25×C4 | D3 | semidirect |
| 37. C23×C3×C4 | C23 | semidirect |
| 44. C2×#217 | C23 | semidirect |
| 45. #929 | C23 | direct (N × H) |
| 46. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 47. C2×Dic3⋊C4 | C23 | semidirect |
| 56. C24×C4 | C2×D3 | semidirect |
| 75. C22×C3×C4 | C24 | semidirect |
| 82. #217 | C24 | semidirect |
| 83. (C2×D3)⋊C4 | C24 | semidirect |
| 84. Dic3⋊C4 | C24 | semidirect |
| 101. C23×C4 | C22×D3 | semidirect |
| 108. C23×C3 | #123 | semidirect |
| 146. C22×C4 | C23×D3 | semidirect |
| 152. C22×C3 | C2×#123 | semidirect |
| 169. C23 | #929 | direct (N × H) |
| 184. C2×C3 | C22×#123 | semidirect |
| 193. C22 | C2×#929 | direct (N × H) |
| 201. C3 | C23×#123 | semidirect |
| 204. C2 | C22×#929 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C3×C4 | 2 |
| C24×#67 | 2 |
| C23×#217 | 2 |
| C22×#929 | 2 |
| C23×(C2×D3)⋊C4 | 2 |
| C23×Dic3⋊C4 | 2 |
| C23×#123 | 3 |