Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C2×#13062401 |
| Fitting, socle, Sylow | C27 ⋊ ΓL(1,𝔽8) |
| derived | #90631 ⋊ C2×C3 |
| Hall | C2×#90631 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #13062401 |
| commutator, cosocle, hypocenter | #90631 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C27 | ΓL(1,𝔽8) |
| LCS | G ▷ #90631 | |
| UCS | C1 ◁ C2 | |
1608 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 1608 conjugacy classes of subgroups — too many to draw. Only its 14 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 | #72209765, #72209775, #72209785, … |
| maximal normal subgroup of | C2×#72224489, C22×#13062401 |
| maximal quotient of | C22×#13062401, D3×#13062401 |
Past the SmallGroups library coverage — no SmallGroups id.