Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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. #82031 | C2 | semidirect |
| 2. #79894 | C2 | semidirect |
| 3. C3×#2178 | C2 | semidirect |
| 4. #70926 | C2 | semidirect |
| 5. #69361 | C2 | semidirect |
| 6. #69242 | C2 | semidirect |
| 7. #69060 | C2 | semidirect |
| 8. #69056 | C2 | semidirect |
| 10. #5769 | C22 | semidirect |
| 13. C3×#443 | C22 | semidirect |
| 14. #5193 | C22 | semidirect |
| 19. C2×D48 | C22 | semidirect |
| 20. C2×D48 | C22 | semidirect |
| 21. #4901 | C22 | semidirect |
| 22. #4897 | C22 | semidirect |
| 25. #4695 | C22 | semidirect |
| 26. C3×#368 | C22 | semidirect |
| 27. C3×#359 | C22 | semidirect |
| 28. C3×C4×C16 | C22 | semidirect |
| 29. #4508 | C22 | semidirect |
| 30. #4495 | C22 | semidirect |
| 31. #4463 | C22 | semidirect |
| 38. #2178 | D3 | semidirect |
| 56. C3×Q32 | D4 | semidirect |
| 57. C3×SD32 | D4 | semidirect |
| 58. C3×D16 | D4 | semidirect |
| 90. #443 | C2×D3 | semidirect |
| 91. #368 | C2×D3 | semidirect |
| 94. #359 | C2×D3 | semidirect |
| 96. C4×C16 | C2×D3 | semidirect |
| 149. Q32 | D12 | semidirect |
| 151. SD32 | D12 | semidirect |
| 156. D16 | D12 | semidirect |
| 279. C3 | #33858 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #82031 | 2 |
| #79894 | 2 |
| C3×#2178 | 2 |
| #70926 | 2 |
| #69361 | 2 |
| #69242 | 2 |
| #69060 | 2 |
| #69056 | 2 |
| #33858 | 3 |