Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×S4×#127 |
| Fitting | C23×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×S4 |
| commutator, cosocle | C22×A4 | C25 |
| Frattini | C22 | C24×S4 |
| Fitting | C23×#127 | D3 |
| socle | C25 | C23×D3 |
| hypercenter | C2×#127 | S4 |
| hypocenter | A4 | C22×#127 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
22435 conjugacy classes of subgroups; only the 766 normal subgroups are shown.
This group has 22435 conjugacy classes of subgroups — too many to draw. Only its 766 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72246076, C2×D42×S4 |
| maximal normal subgroup of | C2×D42×S4 |
Past the SmallGroups library coverage — no SmallGroups id.