Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
2000 conjugacy classes of subgroups; only the 336 normal subgroups are shown.
This group has 2000 conjugacy classes of subgroups — too many to draw. Only its 336 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C23×C7 | C24×C3 |
| Fitting | C7×#2679 | C3 |
| hypercenter | #2679 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#2679 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #2679 | |
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×ΓL(1,𝔽8)×#120 | 2 |
| ΓL(1,𝔽8)×#394 | 2 |
| ΓL(1,𝔽8)×#390 | 2 |
| C7×#2679 | 3 |
| C3×#2679 | 7 |