Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
636 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 195 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×C4⋊Q8 | C2 |
| prime radical | C2×C4⋊(C3⋊D3) | C22 |
| hypercenter | C4⋊Q8 | C3⋊D3 |
| hypocenter | C32 | #556 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C4⋊Q8 | C2 | semidirect |
| 3. Q8×C3⋊Dic3 | C2 | semidirect |
| 4. #64107 | C2 | semidirect |
| 5. #64106 | C2 | semidirect |
| 6. #64101 | C2 | semidirect |
| 8. C4×#634 | C2 | semidirect |
| 26. C3×C4⋊Q8 | D3 | semidirect |
| 74. C4⋊Q8 | C3⋊D3 | semidirect |
| 125. C32 | #556 | semidirect |
| 153. C3 | #5732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C4⋊Q8 | 2 |
| #64156 | 2 |
| Q8×C3⋊Dic3 | 2 |
| #64107 | 2 |
| #64106 | 2 |
| #64101 | 2 |
| C4×C4⋊(C3⋊D3) | 2 |
| C4×#634 | 2 |
| #5732 | 3 |