Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2741 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 2741 conjugacy classes of subgroups — too many to draw. Only its 368 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 | C3×C32⋊H(3,𝔽3) | C3 |
| 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. C33×3−1+2 | C3 | semidirect |
| 2. C3×C9⋊H(3,𝔽3) | C3 | semidirect |
| 3. C3×C9⋊H(3,𝔽3) | C3 | semidirect |
| 5. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 6. #10624563 | C3 | semidirect |
| 7. #10624555 | C3 | semidirect |
| 8. #10624555 | C3 | semidirect |
| 9. #10624547 | C3 | semidirect |
| 15. C32×3−1+2 | C32 | semidirect |
| 21. C33×C9 | C32 | semidirect |
| 29. C3×C32⋊C9 | C32 | semidirect |
| 30. C3×C32⋊C9 | C32 | semidirect |
| 31. C3×C32⋊C9 | C32 | semidirect |
| 32. C3×C32⋊C9 | C32 | semidirect |
| 50. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 59. C3×H(3,𝔽3) | 3−1+2 | semidirect |
| 65. C32×C9 | H(3,𝔽3) | semidirect |
| 103. C33 | C3×3−1+2 | semidirect |
| 148. C32 | C9⋊H(3,𝔽3) | semidirect |
| 158. C9 | C32⋊H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×3−1+2 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C3×C32⋊H(3,𝔽3) | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| #10624563 | 3 |
| #10624555 | 3 |
| #10624547 | 3 |