Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
199 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×D32 |
| commutator, socle | C2×C32 | C22×C8 |
| Fitting | C2×C32×C8 | C22 |
| prime radical | #3828 | C4 |
| cosocle | C32×C8 | C23 |
| hypercenter | C2×C8 | D32 |
| hypocenter | C32 | #504 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#812 | C2 | semidirect |
| 3. C2×C3×C3⋊C16 | C2 | semidirect |
| 4. #63544 | C2 | semidirect |
| 5. #63542 | C2 | semidirect |
| 21. #812 | D3 | semidirect |
| 22. C2×C3⋊C16 | D3 | semidirect |
| 117. C32 | #504 | semidirect |
| 141. C3 | #4901 | semidirect |
| 142. C3 | #4907 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64110 | 2 |
| C3×#812 | 2 |
| C2×C3×C3⋊C16 | 2 |
| #63544 | 2 |
| #63542 | 2 |
| #4907 | 3 |
| #4901 | 3 |