Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×#625 |
| Sylow | C32 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C2×C32×Q8 | C22 |
| prime radical | C2×#625 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | D4×Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
595 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | #11779238, #11779648, #11780217, … |
| maximal normal subgroup of | #11779238, #11779648, #11780217, … |
| maximal quotient of | Q8×(C2×D3)⋊Dic3, C4⋊C4×#625, Q8×#3724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7288);