Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C23×D4×S4 |
| Fitting | C25×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×S4 |
| commutator, cosocle | C2×A4 | C26 |
| Frattini | C2 | C25×S4 |
| Fitting | C25×D4 | D3 |
| socle | C26 | C22×D3 |
| hypercenter | C23×D4 | S4 |
| hypocenter | A4 | C24×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
99595 conjugacy classes of subgroups; only the 3982 normal subgroups are shown.
This group has 99595 conjugacy classes of subgroups — too many to draw. Only its 3982 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | C23×S42, #72259267, C22×D4×#1020⋊C2, … |
| maximal normal subgroup of | #72259267, C22×S4×AGL(1,ℤ/8), C2×D42×S4 |
| maximal quotient of | C22×D4×#1020⋊C2, C22×S4×AGL(1,ℤ/8), C2×D42×S4 |
Past the SmallGroups library coverage — no SmallGroups id.