Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×Q8×Dic67 |
| Sylow | C67 ⋊ C42×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D67 |
| commutator | C2×C67 | C22×C42 |
| Frattini | C23 | C23×D67 |
| Fitting | C2×C4×Q8×C67 | C2 |
| socle | C23×C67 | C24 |
| hypercenter | C2×C4×Q8 | D67 |
| hypocenter | C67 | C42×Q8 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×Q8 | |
684 conjugacy classes of subgroups; only the 425 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 425 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1400 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Q8⋊C8×Dic67, #30746440, #30746450, … |
| maximal normal subgroup of | Q8⋊C8×Dic67, #30746440, #30746450, … |
| maximal quotient of | C4×C4⋊C4×Dic67, C4×#16863282, #387×Dic67, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6861);