Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
873 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 278 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 | C3×#6911 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C3×C32⋊C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624883 | C3 | semidirect |
| 3. C3×#6894 | C3 | semidirect |
| 4. #10624862 | C3 | semidirect |
| 5. #10624795 | C3 | semidirect |
| 6. #10624795 | C3 | semidirect |
| 7. #10624792 | C3 | semidirect |
| 8. #10624698 | C3 | semidirect |
| 9. #10624698 | C3 | semidirect |
| 10. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 11. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 12. #10624691 | C3 | semidirect |
| 23. C9⋊H(3,𝔽3) | C32 | semidirect |
| 26. C9×H(3,𝔽3) | C32 | semidirect |
| 27. C3×C9⋊C9 | C32 | semidirect |
| 28. C3×C9⋊C9 | C32 | semidirect |
| 29. C3×C9⋊C9 | C32 | semidirect |
| 36. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 37. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 38. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 39. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 40. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 41. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 42. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 43. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 44. #6868 | C32 | semidirect |
| 59. C3×H(3,𝔽3) | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6911 | 3 |
| #10624883 | 3 |
| C3×#6894 | 3 |
| #10624862 | 3 |
| #10624795 | 3 |
| #10624792 | 3 |
| #10624698 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| #10624691 | 3 |