Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
| direct product | Q8×#638 |
| Fitting | C22×C32×Q8 ⋊ C2 |
| Sylow | C32 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C22×C32×Q8 | C2 |
| prime radical | C2×#638 | C22 |
| hypercenter | C22×Q8 | C3⋊D3 |
| hypocenter | C32 | D4×Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
840 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | #11884046, #11884049, #11884067, … |
| maximal normal subgroup of | #11884046, #11884049, #11884067, … |
| maximal quotient of | Q8×#3752, Q8×#3757, C4⋊C4×#638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8168);