Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
748 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 748 conjugacy classes of subgroups — too many to draw. Only its 163 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. #83906 | C2 | semidirect |
| 2. C2×#5193 | C2 | semidirect |
| 3. C22×D48 | C2 | semidirect |
| 4. #79167 | C2 | semidirect |
| 5. C3×#2138 | C2 | semidirect |
| 6. #71401 | C2 | semidirect |
| 7. #70652 | C2 | semidirect |
| 8. #69235 | C2 | semidirect |
| 9. #69202 | C2 | semidirect |
| 20. #5191 | C22 | semidirect |
| 21. C2×#828 | C22 | semidirect |
| 23. #5151 | C22 | semidirect |
| 24. C2×#826 | C22 | semidirect |
| 30. #4872 | C22 | semidirect |
| 32. C3×#358 | C22 | semidirect |
| 33. C3×#357 | C22 | semidirect |
| 34. Dic3⋊C16 | C22 | semidirect |
| 35. #4491 | C22 | semidirect |
| 44. #2138 | D3 | semidirect |
| 69. C2×D24 | D4 | semidirect |
| 109. #358 | C2×D3 | semidirect |
| 111. #357 | C2×D3 | semidirect |
| 356. C3 | #33585 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #83906 | 2 |
| C2×#5193 | 2 |
| C22×D48 | 2 |
| #79167 | 2 |
| C3×#2138 | 2 |
| #71401 | 2 |
| #70652 | 2 |
| #69235 | 2 |
| #69202 | 2 |
| #33585 | 3 |