Finite Group DatabasebetaFinite Group Databasebeta| order | 8112 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#72220622 |
| Fitting, socle | C2×C132 ⋊ C2×C3×C4 |
| derived, Sylow | C132 ⋊ C22×C3×C4 |
| Hall | C2×D13⋊#261 ⋊ C3 |
| Hall | #92789 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72220622 |
| commutator, hypocenter | C132 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C132 | C2×C3×C4 |
| prime radical | C2×#11759818⋊C2 | C2 |
| cosocle | C13⋊D13 | C23×C3 |
| LCS | G ▷ C132 | |
| UCS | C1 ◁ C2 | |
232 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 156 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | #72241924, C2×#72220623 |
| maximal normal subgroup of | #72241924, C2×#72220623 |
| maximal quotient of | #72241924 |
Past the SmallGroups library coverage — no SmallGroups id.