Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
242 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 101 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 |
|---|---|---|
| 3. C3×#1372 | C2 | semidirect |
| 4. #67501 | C2 | semidirect |
| 5. #67470 | C2 | semidirect |
| 22. #1372 | D3 | semidirect |
| 69. C3×C16 | M16 | semidirect |
| 71. C3⋊C16 | M16 | semidirect |
| 127. C16 | ΓL(1,𝔽25) | semidirect |
| 172. C3 | #12023 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D3×C16 | 2 |
| #68569 | 2 |
| C3×#1372 | 2 |
| #67501 | 2 |
| #67470 | 2 |
| #12023 | 3 |