Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
524 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator | C33 | C32×C9 |
| Frattini, cosocle | C34 | C33 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10624699 | C3 | semidirect |
| 5. #10624691 | C3 | semidirect |
| 6. C3×#6868 | C3 | semidirect |
| 7. C3×#6867 | C3 | semidirect |
| 13. C3×#731 | C9 | semidirect |
| 33. C3×3−1+2 | C3×C9 | semidirect |
| 52. C32×C9 | C3×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#731 | 3 |
| #10624862 | 3 |
| #10624699 | 3 |
| #10624698 | 3 |
| #10624691 | 3 |
| C3×#6868 | 3 |
| C3×#6867 | 3 |