Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C22×((C3⋊S4)⋊C3)⋊D3 |
| Fitting | C24×C33 ⋊ C2×D3 |
| derived | C3×C22⋊H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C3⋊S4)⋊C3)⋊D3 |
| commutator, cosocle, hypocenter | C3×C22⋊H(3,𝔽3) | C24 |
| Frattini | C3 | C22×C3⋊D3×S4 |
| Fitting | C24×C33 | C2×D3 |
| socle | C24×C32 | D32 |
| LCS | G ▷ C3×C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
9310 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 9310 conjugacy classes of subgroups — too many to draw. Only its 249 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
156 conjugacy classes — view the full character table.
| maximal subgroup of | #72222936 |
| maximal normal subgroup of | #72222936, #72222993 |
Past the SmallGroups library coverage — no SmallGroups id.