Finite Group DatabasebetaFinite Group Databasebeta← C22×C32×PSU(3,𝔽2)⋊C3 #72228430
Order 7776.
2642 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 2642 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | PSU(3,𝔽2) | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | SL(2,𝔽3) |
| LCS | G ▷ PSU(3,𝔽2) | |
| UCS | C1 ◁ C22×C32 | |
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 |
|---|---|
| C22×C32×PSU(3,𝔽2) | 3 |
| C2×C32×PSU(3,𝔽2)⋊C3 | 2 |
| C22×C3×PSU(3,𝔽2)⋊C3 | 3 |
| C22×C32×(C3⋊D3)⋊C3 | 4 |
| C22×C32×SL(2,𝔽3) | 9 |