Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
631 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 109 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 | C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C23×C3 | C3⋊S4 |
| prime radical | C22×C3×SL(2,𝔽3) | D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9⋊#226 | C2 | direct (N × H) |
| 8. C9⋊#226 | C22 | direct (N × H) |
| 84. C22×C9 | #226 | semidirect |
| 113. C2×C9 | C2×#226 | semidirect |
| 134. C9 | C22×#226 | semidirect |
| 149. C22 | C9⋊#226 | direct (N × H) |
| 156. C2 | C2×C9⋊#226 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×SL(2,𝔽3) | 2 |
| C2×C9⋊#226 | 2 |
| C22×C3⋊#226 | 3 |
| C22×#3688 | 3 |
| C22×#3674 | 3 |
| C22×C3⋊Dic9 | 4 |