Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2158 conjugacy classes of subgroups; only the 499 normal subgroups are shown.
This group has 2158 conjugacy classes of subgroups — too many to draw. Only its 499 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, cosocle | C24×C3 | C24 |
| Fitting | C3×C4×#95 | C2 |
| prime radical | C23×D12 | C22 |
| hypercenter | C4×#95 | D3 |
| hypocenter | C3 | #15155 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C4×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4935 | C2 | semidirect |
| 3. C2×#4677 | C2 | semidirect |
| 4. C3×C4×#95 | C2 | semidirect |
| 18. C4×#95 | D3 | semidirect |
| 34. C3×#95 | D4 | semidirect |
| 82. #95 | D12 | semidirect |
| 142. C3×C4 | #399 | semidirect |
| 187. C4 | #4677 | semidirect |
| 188. C3 | #15155 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4939 | 2 |
| C2×#4935 | 2 |
| C2×#4677 | 2 |
| C3×C4×#95 | 2 |
| #15155 | 3 |