Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
689 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 689 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 | C3×#6856 | C3 |
| 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. #10624906 | C3 | semidirect |
| 2. #10624894 | C3 | semidirect |
| 3. #10624883 | C3 | semidirect |
| 4. #10624677 | C3 | semidirect |
| 5. #10624677 | C3 | semidirect |
| 6. #10624651 | C3 | semidirect |
| 7. #10624634 | C3 | semidirect |
| 8. #10624630 | C3 | semidirect |
| 9. #10624627 | C3 | semidirect |
| 10. #10624616 | C3 | semidirect |
| 11. #10624615 | C3 | semidirect |
| 12. #10624613 | C3 | semidirect |
| 13. #10624593 | C3 | semidirect |
| 14. #10624591 | C3 | semidirect |
| 16. C3×#737 | C32 | semidirect |
| 23. #6894 | C32 | semidirect |
| 24. #6894 | C32 | semidirect |
| 27. #6893 | C32 | semidirect |
| 28. #6893 | C32 | semidirect |
| 31. C9⋊H(3,𝔽3) | C32 | semidirect |
| 32. C9⋊H(3,𝔽3) | C32 | semidirect |
| 40. C9×3−1+2 | C32 | semidirect |
| 46. #6860 | C32 | semidirect |
| 47. #6860 | C32 | semidirect |
| 48. #6860 | C32 | semidirect |
| 49. #6859 | C32 | semidirect |
| 50. #6859 | C32 | semidirect |
| 51. #6859 | C32 | semidirect |
| 52. #6858 | C32 | semidirect |
| 53. #6858 | C32 | semidirect |
| 54. #6858 | C32 | semidirect |
| 55. #6857 | C32 | semidirect |
| 56. #6857 | C32 | semidirect |
| 57. #6857 | C32 | semidirect |
| 58. #6857 | C32 | semidirect |
| 59. #6857 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624906 | 3 |
| #10624894 | 3 |
| #10624883 | 3 |
| #10624677 | 3 |
| #10624651 | 3 |
| #10624634 | 3 |
| #10624630 | 3 |
| #10624627 | 3 |
| #10624616 | 3 |
| #10624615 | 3 |
| #10624613 | 3 |
| #10624593 | 3 |
| #10624591 | 3 |
| C3×#6856 | 3 |