Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
653 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 653 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 |
| 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. #10624905 | C3 | semidirect |
| 2. #10624902 | C3 | semidirect |
| 3. #10624883 | C3 | semidirect |
| 4. #10624669 | C3 | semidirect |
| 5. #10624669 | C3 | semidirect |
| 6. #10624654 | C3 | semidirect |
| 7. #10624638 | C3 | semidirect |
| 8. #10624628 | C3 | semidirect |
| 9. #10624622 | C3 | semidirect |
| 10. #10624620 | C3 | semidirect |
| 11. #10624614 | C3 | semidirect |
| 12. #10624601 | C3 | semidirect |
| 13. #10624599 | C3 | semidirect |
| 14. #10624597 | C3 | semidirect |
| 15. C3×#6859 | C3 | semidirect |
| 20. #6895 | C32 | semidirect |
| 21. #6895 | C32 | semidirect |
| 30. C9⋊H(3,𝔽3) | C32 | semidirect |
| 31. C9⋊H(3,𝔽3) | C32 | semidirect |
| 38. C9×H(3,𝔽3) | C32 | semidirect |
| 40. #6887 | C32 | semidirect |
| 46. #6861 | C32 | semidirect |
| 47. #6861 | C32 | semidirect |
| 48. #6861 | C32 | semidirect |
| 49. #6861 | C32 | semidirect |
| 50. #6861 | C32 | semidirect |
| 51. #6861 | C32 | semidirect |
| 58. #6859 | C32 | semidirect |
| 59. #6859 | C32 | semidirect |
| 60. #6859 | C32 | semidirect |
| 63. #6856 | C32 | semidirect |
| 64. #6856 | C32 | semidirect |
| 65. #6856 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624905 | 3 |
| #10624902 | 3 |
| #10624883 | 3 |
| #10624669 | 3 |
| #10624654 | 3 |
| #10624638 | 3 |
| #10624628 | 3 |
| #10624622 | 3 |
| #10624620 | 3 |
| #10624614 | 3 |
| #10624601 | 3 |
| #10624599 | 3 |
| #10624597 | 3 |
| C3×#6859 | 3 |