Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
776 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 269 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C33 | C34 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624569 | C3 | semidirect |
| 2. #10624566 | C3 | semidirect |
| 3. #10624566 | C3 | semidirect |
| 4. #10624566 | C3 | semidirect |
| 5. #10624557 | C3 | semidirect |
| 6. #10624557 | C3 | semidirect |
| 7. #10624556 | C3 | semidirect |
| 8. #10624556 | C3 | semidirect |
| 9. #10624556 | C3 | semidirect |
| 13. #10624548 | C3 | semidirect |
| 14. C9×C32⋊C9 | C3 | semidirect |
| 15. C3×#6855 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624569 | 3 |
| #10624566 | 3 |
| #10624557 | 3 |
| #10624556 | 3 |
| #10624555 | 3 |
| #10624548 | 3 |
| C9×C32⋊C9 | 3 |
| C3×#6855 | 3 |