Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
684 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 243 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×D4 | C2 |
| prime radical | C2×C32×D4 | C22 |
| hypercenter | C4×D4 | C3⋊D3 |
| hypocenter | C32 | #554 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4⋊(C3⋊Dic3) | C2 | semidirect |
| 4. #64133 | C2 | semidirect |
| 5. #64099 | C2 | semidirect |
| 6. #64087 | C2 | semidirect |
| 7. #64078 | C2 | semidirect |
| 8. C4×#634 | C2 | semidirect |
| 35. C32×D4 | Q8 | semidirect |
| 84. C3×D4 | Dic6 | semidirect |
| 127. C32 | #554 | semidirect |
| 131. D4 | #634 | semidirect |
| 156. C3 | #5543 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C4×D4 | 2 |
| D4×C3⋊Dic3 | 2 |
| C2×C4⋊(C3⋊Dic3) | 2 |
| #64133 | 2 |
| #64099 | 2 |
| #64087 | 2 |
| #64078 | 2 |
| C4×#634 | 2 |
| #5543 | 3 |