Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
| direct product | C3⋊D3×C4⋊Q8 |
| Fitting | C32×C4⋊Q8 ⋊ C2 |
| Sylow | C32 ⋊ C2×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C4⋊Q8 | C2 |
| prime radical | C22×C3⋊D3 | C23 |
| hypercenter | C4⋊Q8 | C3⋊D3 |
| hypocenter | C32 | C2×C4⋊Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
870 conjugacy classes of subgroups; only the 235 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 235 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#332, #11776101, #11776105, … |
| maximal normal subgroup of | C3⋊D3×#332, #11776101, #11776105, … |
| maximal quotient of | C3⋊D3×#384, C3⋊D3×#389, C3⋊D3×#391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8076);