Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C3 |
| direct product | C3⋊D3×H(5,𝔽2) |
| Fitting | C32×H(5,𝔽2) ⋊ C2 |
| Sylow | C32 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C25 |
| Fitting | C32×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | C3⋊D3 |
| hypocenter | C32 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
2694 conjugacy classes of subgroups; only the 717 normal subgroups are shown.
This group has 2694 conjugacy classes of subgroups — too many to draw. Only its 717 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
102 conjugacy classes — view the full character table.
| maximal subgroup of | #11784820, #11784864, #11858011, … |
| maximal normal subgroup of | #11784820, #11784864, #11858011, … |
| maximal quotient of | C3⋊D3×#518, C3⋊D3×#520, #11900158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8639);