Finite Group DatabasebetaFinite Group DatabasebetaOrder 3024.
540 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C32×C7 | C23×C3 |
| Fitting | C32×C4×C7 | C22×C3 |
| hypercenter | C4 | D32×ΓL(1,𝔽8) |
| hypocenter | C32×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D32×ΓL(1,𝔽8) | 2 |
| C4×C3⋊D3×ΓL(1,𝔽8) | 2 |
| C3×ΓL(1,𝔽8)×D12 | 2 |
| ΓL(1,𝔽8)×#625 | 2 |
| C7×#3802 | 3 |
| D3×D4×ΓL(1,𝔽8) | 3 |
| C3×#3802 | 7 |