Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
| direct product | C3⋊D3×#123 |
| Fitting | C32×#123 ⋊ C2 |
| Sylow | C32 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#123 | C2 |
| prime radical | C2×D4×C3⋊D3 | C2 |
| hypercenter | #123 | C3⋊D3 |
| hypocenter | C32 | C2×#123 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #123 | |
1026 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #11787968, C3⋊D3×#352, #11796171, … |
| maximal normal subgroup of | #11787968, C3⋊D3×#352, #11796171, … |
| maximal quotient of | C3⋊D3×#381, C3⋊D3×#385, C3⋊D3×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8005);