Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
566 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 50 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, Frattini, cosocle | C34 | C33 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3)2 | C3 | semidirect |
| 2. #10624745 | C3 | semidirect |
| 3. #10624737 | C3 | semidirect |
| 4. #10624667 | C3 | semidirect |
| 5. #10624656 | C3 | semidirect |
| 6. #10624639 | C3 | semidirect |
| 7. #10624628 | C3 | semidirect |
| 8. #10624596 | C3 | semidirect |
| 9. #10624590 | C3 | semidirect |
| 10. C32×3−1+2 | C32 | semidirect |
| 15. #6891 | C32 | semidirect |
| 16. #6891 | C32 | semidirect |
| 17. #6891 | C32 | semidirect |
| 23. C32⋊H(3,𝔽3) | C32 | semidirect |
| 24. C3×C32⋊C9 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)2 | 3 |
| #10624745 | 3 |
| #10624737 | 3 |
| #10624667 | 3 |
| #10624656 | 3 |
| #10624639 | 3 |
| #10624628 | 3 |
| #10624596 | 3 |
| #10624590 | 3 |