Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
212 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 71 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 |
|---|---|---|
| 2. #68089 | C2 | semidirect |
| 4. C3×#1313 | C2 | semidirect |
| 5. #67517 | C2 | semidirect |
| 7. #67513 | C2 | semidirect |
| 30. #1313 | D3 | semidirect |
| 89. C3×C16 | M16 | semidirect |
| 153. C16 | ΓL(1,𝔽25) | semidirect |
| 200. C3 | #12087 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #69077 | 2 |
| #68089 | 2 |
| #68083 | 2 |
| C3×#1313 | 2 |
| #67517 | 2 |
| #67516 | 2 |
| #67513 | 2 |
| #12087 | 3 |