Finite Group DatabasebetaFinite Group Databasebeta← C22×C3×D3×SL(2,𝔽3) #13181442
Order 1728.
1639 conjugacy classes of subgroups; only the 292 normal subgroups are shown.
This group has 1639 conjugacy classes of subgroups — too many to draw. Only its 292 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 | D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C32 |
| Frattini | C2 | C22×C3×D3×A4 |
| Fitting | C22×C32×Q8 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| LCS | G ▷ C3×Q8 | |
| 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×D3×SL(2,𝔽3) | 2 |
| C22×C3×D3×Q8 | 3 |
| C23×C3×SL(2,𝔽3) | 3 |
| C22×D3×SL(2,𝔽3) | 3 |
| C23×C32×D3 | 4 |