Finite Group DatabasebetaFinite Group Databasebeta← C22×C3×C3⋊GL(2,𝔽3) #13181379
Order 1728.
1371 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1371 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23×C3 |
| Frattini | C2 | C22×C3×C3⋊S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle | C23×C32 | S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C23×C3 | |
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×SL(2,𝔽3) | 2 |
| C2×C3×C3⋊GL(2,𝔽3) | 2 |
| C22×C3⋊GL(2,𝔽3) | 3 |
| C22×C3×GL(2,𝔽3) | 3 |
| C22×C3×#215 | 3 |
| C23×C3×C3⋊D3 | 4 |