Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D4×GO+(4,𝔽2) |
| Fitting | C32×D4 ⋊ D4 |
| Sylow | C32 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C24 |
| Fitting | C32×D4 | D4 |
| socle | C2×C32 | C22×D4 |
| hypercenter | D4 | GO+(4,𝔽2) |
| hypocenter | C32 | D42 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
664 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | GO+(4,𝔽2)⋊SD16, #11779747, #11883885, … |
| maximal normal subgroup of | GO+(4,𝔽2)⋊SD16, #11779747, #11883885, … |
| maximal quotient of | D4×D32⋊C4, C22⋊C4×GO+(4,𝔽2), D4×#3773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8399);