Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
4390 conjugacy classes of subgroups; only the 651 normal subgroups are shown.
This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 651 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 | C23×C3 | C23×C4 |
| Fitting, prime radical | C23×C3×#41 | C2 |
| socle, cosocle | C24×C3 | C24 |
| hypercenter | C23×#41 | D3 |
| hypocenter | C3 | #14337 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#5212 | C2 | semidirect |
| 3. #72134 | C2 | semidirect |
| 4. #72131 | C2 | semidirect |
| 7. C22×C3×#41 | C4 | semidirect |
| 38. C22×#41 | Dic3 | semidirect |
| 61. C22×C3×C4 | C22⋊C4 | semidirect |
| 132. C22×C4 | #217 | semidirect |
| 197. C3 | #14337 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×#41 | 2 |
| C2×#5212 | 2 |
| #72134 | 2 |
| #72131 | 2 |
| #14337 | 3 |