Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
252 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 75 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. C3×D4×C16 | C2 | semidirect |
| 2. #70850 | C2 | semidirect |
| 4. #69061 | C2 | semidirect |
| 5. #67561 | C2 | semidirect |
| 6. #67560 | C2 | semidirect |
| 7. #67501 | C2 | semidirect |
| 25. D4×C16 | D3 | semidirect |
| 96. C3×C16 | D8 | semidirect |
| 123. C3×D4 | SD32 | semidirect |
| 171. C16 | #213 | semidirect |
| 196. D4 | #800 | semidirect |
| 229. C3 | #12432 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×C16 | 2 |
| #70850 | 2 |
| #70847 | 2 |
| #69061 | 2 |
| #67561 | 2 |
| #67560 | 2 |
| #67501 | 2 |
| #12432 | 3 |