Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C22⋊#997 |
| Fitting, Sylow | C24×H(5,𝔽2) ⋊ C3 |
| derived | C22×H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C22⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C22×C3 |
| Frattini | C2 | C22×C22⋊(C22⋊A4) |
| Fitting | C24×H(5,𝔽2) | C3 |
| socle | C25 | C22⋊A4 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
152887 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 152887 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11715850, #72233869, #72234652, … |
| maximal normal subgroup of | C22×#11715850, #72233869, #72234652, … |
| maximal quotient of | #72233395, #72255786, C2×#72260605, … |
Past the SmallGroups library coverage — no SmallGroups id.