Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1434 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 314 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 | C33 | C34 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624885 | C3 | semidirect |
| 3. C3×#6896 | C3 | semidirect |
| 4. #10624783 | C3 | semidirect |
| 5. #10624782 | C3 | semidirect |
| 6. #10624781 | C3 | semidirect |
| 7. #10624780 | C3 | semidirect |
| 8. #10624780 | C3 | semidirect |
| 9. #10624780 | C3 | semidirect |
| 13. #10624700 | C3 | semidirect |
| 14. #10624697 | C3 | semidirect |
| 29. #6895 | C32 | semidirect |
| 33. #6887 | C32 | semidirect |
| 34. C3×C9⋊C9 | C32 | semidirect |
| 35. C3×C9⋊C9 | C32 | semidirect |
| 36. C3×C9⋊C9 | C32 | semidirect |
| 37. C3×C9⋊C9 | C32 | semidirect |
| 40. C3×C92 | C32 | semidirect |
| 41. 3−1+2⋊C9 | C32 | semidirect |
| 42. 3−1+2⋊C9 | C32 | semidirect |
| 43. 3−1+2⋊C9 | C32 | semidirect |
| 44. 3−1+2⋊C9 | C32 | semidirect |
| 45. 3−1+2⋊C9 | C32 | semidirect |
| 46. 3−1+2⋊C9 | C32 | semidirect |
| 59. #737 | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×3−1+4 | 3 |
| #10624885 | 3 |
| C3×#6896 | 3 |
| #10624783 | 3 |
| #10624782 | 3 |
| #10624781 | 3 |
| #10624780 | 3 |
| #10624779 | 3 |
| #10624778 | 3 |
| #10624777 | 3 |
| #10624700 | 3 |
| #10624697 | 3 |