Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#90632⋊C3 |
| Fitting, socle, Sylow | C27 ⋊ ΓL(1,𝔽8) |
| derived | #90632 ⋊ C2×C3 |
| Hall | C2×#90632 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #90632⋊C3 |
| commutator, cosocle, hypocenter | #90632 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C27 | ΓL(1,𝔽8) |
| LCS | G ▷ #90632 | |
| UCS | C1 ◁ C2 | |
1596 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209839, #72209843, #72209846, … |
| maximal normal subgroup of | C22×#90632⋊C3, C4×#90632⋊C3 |
| maximal quotient of | C2×#72209829, C22×#90632⋊C3, C4×#90632⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id.