Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
182 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 88 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 | C3×#235 | 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. C2×C3⋊(C3⋊C16) | C2 | semidirect |
| 3. C3×#812 | C2 | semidirect |
| 4. #63543 | C2 | semidirect |
| 5. D3×C3⋊C16 | C2 | semidirect |
| 20. #812 | D3 | semidirect |
| 108. C32 | #504 | semidirect |
| 131. C3 | #4907 | semidirect |
| 132. C3 | #5140 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#901 | 2 |
| C2×C3⋊(C3⋊C16) | 2 |
| C3×#812 | 2 |
| #63543 | 2 |
| D3×C3⋊C16 | 2 |
| #5140 | 3 |
| #4907 | 3 |