Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
181 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#213 |
| commutator, cosocle | C32×C8 | C23 |
| Frattini | C8 | C2×D32 |
| Fitting | C32×Q16 | C22 |
| socle | C2×C32 | C2×D8 |
| hypercenter | Q16 | D32 |
| hypocenter | C32 | #508 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#919 | C2 | semidirect |
| 2. #63808 | C2 | semidirect |
| 3. #63655 | C2 | semidirect |
| 4. C3×#829 | C2 | semidirect |
| 5. #63547 | C2 | semidirect |
| 6. #63546 | C2 | semidirect |
| 7. D3×C3⋊C16 | C2 | semidirect |
| 23. #919 | D3 | semidirect |
| 26. #829 | D3 | semidirect |
| 144. C32 | #508 | semidirect |
| 174. C3 | #4919 | semidirect |
| 175. C3 | #5194 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#919 | 2 |
| #63808 | 2 |
| #63655 | 2 |
| C3×#829 | 2 |
| #63547 | 2 |
| #63546 | 2 |
| D3×C3⋊C16 | 2 |
| #5194 | 3 |
| #4919 | 3 |