Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
13310 conjugacy classes of subgroups; only the 4427 normal subgroups are shown.
This group has 13310 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 |
|---|---|---|
| 2. C24×Dic6 | C2 | semidirect |
| 3. C24×C3×C8 | C2 | semidirect |
| 4. C23×#204 | C2 | direct (N × H) |
| 12. C23×Dic6 | C22 | semidirect |
| 13. C23×C3×C8 | C22 | semidirect |
| 14. C22×#204 | C22 | direct (N × H) |
| 18. C24×C8 | D3 | semidirect |
| 28. C22×Dic6 | C23 | semidirect |
| 29. C22×C3×C8 | C23 | semidirect |
| 30. C2×#204 | C23 | direct (N × H) |
| 39. C23×C8 | C2×D3 | semidirect |
| 40. C24×C3 | SD16 | semidirect |
| 50. C2×Dic6 | C24 | semidirect |
| 51. C2×C3×C8 | C24 | semidirect |
| 52. #204 | C24 | direct (N × H) |
| 64. C22×C8 | C22×D3 | semidirect |
| 65. C23×C3 | C2×SD16 | semidirect |
| 75. Dic6 | C25 | semidirect |
| 76. C3×C8 | C25 | semidirect |
| 79. C24 | #204 | direct (N × H) |
| 89. C2×C8 | C23×D3 | semidirect |
| 90. C22×C3 | C22×SD16 | semidirect |
| 99. C23 | C2×#204 | direct (N × H) |
| 108. C8 | C24×D3 | semidirect |
| 109. C2×C3 | C23×SD16 | semidirect |
| 114. C22 | C22×#204 | direct (N × H) |
| 119. C3 | C24×SD16 | semidirect |
| 121. C2 | C23×#204 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D12 | 2 |
| C24×Dic6 | 2 |
| C24×C3×C8 | 2 |
| C23×#204 | 2 |
| C24×SD16 | 3 |