Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1166 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 229 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 | D3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C33 | C24×C3 |
| Frattini | C3 | C22×C3×D32 |
| Fitting | C22×C3×H(3,𝔽3) | C22 |
| socle | C22×C32 | C2×C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
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×C3⋊((C3⋊D3)⋊C3) | 2 |
| C22×D3×H(3,𝔽3) | 2 |
| C22×C3×(C3⋊D3)⋊C3 | 2 |
| C2×D3×(C3⋊D3)⋊C3 | 2 |
| C22×D3×C3⋊D3 | 3 |
| C22×C3×D32 | 3 |
| C23×(C3⋊D3)⋊C3 | 3 |