Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
252 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 63 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. C2×C3×D32 | C2 | semidirect |
| 2. #71465 | C2 | semidirect |
| 3. #71464 | C2 | semidirect |
| 4. C2×#4520 | C2 | semidirect |
| 5. C2×#4519 | C2 | semidirect |
| 6. C2×#4449 | C2 | semidirect |
| 7. #67589 | C2 | semidirect |
| 27. C2×D32 | D3 | semidirect |
| 52. C3×C32 | D4 | semidirect |
| 53. C3⋊C32 | D4 | semidirect |
| 113. C32 | #67 | semidirect |
| 199. C3 | #13729 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D32 | 2 |
| #71465 | 2 |
| #71464 | 2 |
| C2×#4520 | 2 |
| C2×#4519 | 2 |
| C2×#4449 | 2 |
| #67589 | 2 |
| #13729 | 3 |