Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
936 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 936 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#127 | C2 |
| hypercenter | #127 | C3⋊D3 |
| hypocenter | C32 | #550 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3830 | C2 | semidirect |
| 2. C32×#127 | C2 | semidirect |
| 3. #64149 | C2 | semidirect |
| 4. #64148 | C2 | semidirect |
| 5. #64147 | C2 | semidirect |
| 6. D4×C3⋊Dic3 | C2 | semidirect |
| 7. C4×C4⋊(C3⋊D3) | C2 | semidirect |
| 8. C4×#634 | C2 | semidirect |
| 28. C3×#127 | D3 | semidirect |
| 51. #634 | D4 | semidirect |
| 79. C32×C4 | #41 | semidirect |
| 85. #127 | C3⋊D3 | semidirect |
| 139. C3×C4 | #237 | semidirect |
| 143. C32 | #550 | semidirect |
| 178. C4 | #3830 | semidirect |
| 179. C3 | #5718 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3830 | 2 |
| C32×#127 | 2 |
| #64149 | 2 |
| #64148 | 2 |
| #64147 | 2 |
| D4×C3⋊Dic3 | 2 |
| C4×C4⋊(C3⋊D3) | 2 |
| C4×#634 | 2 |
| #5718 | 3 |