Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#5925 |
| Fitting | C3×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C3 |
| Frattini | C2 | C22×C3×S4 |
| Fitting | C3×H(5,𝔽2) | D3 |
| hypercenter | C3×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
303 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
69 conjugacy classes — view the full character table.
| maximal subgroup of | #11922460, #11922463, #11922468, … |
| maximal normal subgroup of | #11922460, #11922463, #11922468, … |
| maximal quotient of | C3×#85363, C3×#85365, C3×#85366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8449);