Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
199 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #10624566 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | C3×H(3,𝔽3) | 3−1+2 |
| LCS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624709 | C3 | semidirect |
| 2. #10624709 | C3 | semidirect |
| 3. C3×#6875 | C3 | semidirect |
| 8. #10624473 | C3 | semidirect |
| 9. #10624473 | C3 | semidirect |
| 10. #10624472 | C3 | semidirect |
| 11. #10624472 | C3 | semidirect |
| 12. #10624471 | C3 | semidirect |
| 13. #10624465 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624709 | 3 |
| C3×#6875 | 3 |
| #10624704 | 3 |
| #10624702 | 3 |
| #10624700 | 3 |
| #10624694 | 3 |
| #10624473 | 3 |
| #10624472 | 3 |
| #10624471 | 3 |
| #10624465 | 3 |