Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
744 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#638 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C23×C3⋊D3 |
| Fitting | C32×2−1+4 | C2 |
| socle | C2×C32 | C22×D4 |
| hypercenter | 2−1+4 | C3⋊D3 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×2−1+4 | C2 | semidirect |
| 2. #64367 | C2 | semidirect |
| 3. #64160 | C2 | semidirect |
| 4. #64159 | C2 | semidirect |
| 5. #64157 | C2 | semidirect |
| 6. C2×#3756 | C2 | semidirect |
| 7. #64151 | C2 | semidirect |
| 25. C3×2−1+4 | D3 | semidirect |
| 74. 2−1+4 | C3⋊D3 | semidirect |
| 124. C32 | #578 | semidirect |
| 153. C3 | #5838 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×2−1+4 | 2 |
| #64367 | 2 |
| #64160 | 2 |
| #64159 | 2 |
| #64157 | 2 |
| C2×#3756 | 2 |
| #64151 | 2 |
| #5838 | 3 |