Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×#60401⋊C3 |
| Fitting, Sylow | C2×#60401 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C24×C3 |
| Frattini | C23 | C24×A4 |
| Fitting | C2×#60401 | C3 |
| hypercenter | C2×D4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
7902 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 7902 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236802, C2×#72236841, C2×#72236832, … |
| maximal normal subgroup of | C2×#72236802, C2×#72236841, C2×#72236832, … |
Past the SmallGroups library coverage — no SmallGroups id.