Finite Group DatabasebetaFinite Group DatabasebetaOrder 6048.
824 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 824 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A42×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C24×C7 | C2×C33 |
| Frattini | C1 | G |
| Fitting, socle | C25×C7 | C33 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
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×(C22⋊A4)⋊ΓL(1,𝔽8) | 3 |
| C2×A4×C22⋊ΓL(1,𝔽8) | 3 |
| C23×A4×ΓL(1,𝔽8) | 3 |
| C2×C7×A42 | 3 |
| C2×ΓL(1,𝔽8)×C22⋊A4 | 3 |
| A42×ΓL(1,𝔽8) | 2 |
| C2×C3×A4×ΓL(1,𝔽8) | 4 |
| C2×C3×A42 | 7 |