Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
325 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #10624590 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | C3×#6913 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624886 | C3 | semidirect |
| 3. #10624774 | C3 | semidirect |
| 4. #10624771 | C3 | semidirect |
| 5. #10624487 | C3 | semidirect |
| 6. #10624487 | C3 | semidirect |
| 7. #10624487 | C3 | semidirect |
| 8. #10624487 | C3 | semidirect |
| 9. #10624487 | C3 | semidirect |
| 10. #10624486 | C3 | semidirect |
| 11. #10624485 | C3 | semidirect |
| 12. #10624485 | C3 | semidirect |
| 13. #10624484 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6913 | 3 |
| #10624886 | 3 |
| #10624774 | 3 |
| #10624771 | 3 |
| #10624487 | 3 |
| #10624486 | 3 |
| #10624485 | 3 |
| #10624484 | 3 |