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 |
|---|---|---|
| 1. C24×C3×Q8 | C2 | semidirect |
| 3. C23×#215 | C2 | direct (N × H) |
| 4. C24×C3⋊C8 | C2 | semidirect |
| 8. C23×C3×Q8 | C22 | semidirect |
| 13. C22×#215 | C22 | direct (N × H) |
| 14. C23×C3⋊C8 | C22 | semidirect |
| 15. C24×Q8 | D3 | semidirect |
| 23. C22×C3×Q8 | C23 | semidirect |
| 29. C2×#215 | C23 | direct (N × H) |
| 30. C22×C3⋊C8 | C23 | semidirect |
| 33. C23×Q8 | C2×D3 | semidirect |
| 40. C24×C3 | SD16 | semidirect |
| 45. C2×C3×Q8 | C24 | semidirect |
| 51. #215 | C24 | direct (N × H) |
| 52. C2×C3⋊C8 | C24 | semidirect |
| 57. C22×Q8 | C22×D3 | semidirect |
| 65. C23×C3 | C2×SD16 | semidirect |
| 70. C3×Q8 | C25 | semidirect |
| 76. C3⋊C8 | C25 | semidirect |
| 79. C24 | #215 | direct (N × H) |
| 82. C2×Q8 | C23×D3 | semidirect |
| 90. C22×C3 | C22×SD16 | semidirect |
| 99. C23 | C2×#215 | direct (N × H) |
| 102. Q8 | C24×D3 | semidirect |
| 109. C2×C3 | C23×SD16 | semidirect |
| 114. C22 | C22×#215 | direct (N × H) |
| 119. C3 | C24×SD16 | semidirect |
| 121. C2 | C23×#215 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×Q8 | 2 |
| C24×D12 | 2 |
| C23×#215 | 2 |
| C24×C3⋊C8 | 2 |
| C24×SD16 | 3 |