Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3×C4×A4 |
| Fitting | C24×C3×C4 ⋊ C3 |
| derived | C22 ⋊ C22×C32×C4 |
| Sylow | C24×C4 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C4 | A4 |
| commutator, hypocenter | C22 | C22×C32×C4 |
| Frattini | C2 | C23×C3×A4 |
| Fitting | C24×C3×C4 | C3 |
| socle | C25×C3 | C2×C3 |
| prime radical | C23×C3×A4 | C2 |
| cosocle | C23 | C23×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C22×C3×C4 | |
742 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | C3×A4×#95, C3×A4×C22⋊C8, C2×C3×C42×A4, … |
| maximal normal subgroup of | C3×A4×#95, C3×A4×C22⋊C8, C2×C3×C42×A4, … |
| maximal quotient of | C2×C3×C42×A4, C2×C3×A4×C22⋊C4, C2×C3×A4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8591);