Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
637 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 637 conjugacy classes of subgroups — too many to draw. Only its 242 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 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C3×C32⋊C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624902 | C3 | semidirect |
| 2. #10624899 | C3 | semidirect |
| 4. #10624676 | C3 | semidirect |
| 5. #10624676 | C3 | semidirect |
| 6. #10624653 | C3 | semidirect |
| 7. #10624645 | C3 | semidirect |
| 9. #10624625 | C3 | semidirect |
| 10. #10624621 | C3 | semidirect |
| 11. #10624611 | C3 | semidirect |
| 12. #10624610 | C3 | semidirect |
| 13. #10624603 | C3 | semidirect |
| 14. #10624597 | C3 | semidirect |
| 23. #6895 | C32 | semidirect |
| 24. #6895 | C32 | semidirect |
| 27. #6894 | C32 | semidirect |
| 28. #6894 | C32 | semidirect |
| 39. C9×3−1+2 | C32 | semidirect |
| 41. #6887 | C32 | semidirect |
| 46. #6862 | C32 | semidirect |
| 47. #6862 | C32 | semidirect |
| 48. #6862 | C32 | semidirect |
| 49. #6861 | C32 | semidirect |
| 50. #6861 | C32 | semidirect |
| 51. #6861 | C32 | semidirect |
| 54. #6858 | C32 | semidirect |
| 55. #6858 | C32 | semidirect |
| 56. #6858 | C32 | semidirect |
| 57. #6858 | C32 | semidirect |
| 58. #6858 | C32 | semidirect |
| 59. #6858 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624902 | 3 |
| #10624899 | 3 |
| #10624883 | 3 |
| #10624676 | 3 |
| #10624653 | 3 |
| #10624645 | 3 |
| #10624632 | 3 |
| #10624625 | 3 |
| #10624621 | 3 |
| #10624611 | 3 |
| #10624610 | 3 |
| #10624603 | 3 |
| #10624597 | 3 |
| C3×#6857 | 3 |