Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#5576 |
| Fitting | C32×C4×Q8 ⋊ C2 |
| Sylow | C32 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C3 |
| Frattini | C22 | C23×C3×D3 |
| Fitting | C32×C4×Q8 | C2 |
| socle | C22×C32 | C24 |
| hypercenter | C3×C4×Q8 | D3 |
| hypocenter | C3 | C3×#548 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 | |
615 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#80404, C3×#79794, #11883945, … |
| maximal normal subgroup of | C3×#80404, C3×#79794, #11883945, … |
| maximal quotient of | C3×Q8×(C2×D3)⋊C4, C3×#82457, C3×#82460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7555);