Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1535 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 1535 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| socle | C32 | C32×H(3,𝔽3) |
| prime radical | #10624596 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#737 | C3 | semidirect |
| 2. #10624872 | C3 | semidirect |
| 3. #10624869 | C3 | semidirect |
| 4. C3×C9×3−1+2 | C3 | semidirect |
| 5. #10624613 | C3 | semidirect |
| 6. #10624612 | C3 | semidirect |
| 7. #10624611 | C3 | semidirect |
| 8. #10624610 | C3 | semidirect |
| 9. #10624609 | C3 | semidirect |
| 10. #10624608 | C3 | semidirect |
| 11. #10624607 | C3 | semidirect |
| 12. #10624602 | C3 | semidirect |
| 14. #10624590 | C3 | semidirect |
| 15. #10624583 | C3 | semidirect |
| 34. #6894 | C32 | semidirect |
| 38. C9⋊H(3,𝔽3) | C32 | semidirect |
| 51. C9×3−1+2 | C32 | semidirect |
| 52. C9×3−1+2 | C32 | semidirect |
| 53. C9×3−1+2 | C32 | semidirect |
| 56. C9×H(3,𝔽3) | C32 | semidirect |
| 62. #6860 | C32 | semidirect |
| 63. #6860 | C32 | semidirect |
| 65. #6858 | C32 | semidirect |
| 66. #6858 | C32 | semidirect |
| 68. #6857 | C32 | semidirect |
| 69. #6857 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| #10624872 | 3 |
| #10624869 | 3 |
| C3×C9×3−1+2 | 3 |
| #10624613 | 3 |
| #10624612 | 3 |
| #10624611 | 3 |
| #10624610 | 3 |
| #10624609 | 3 |
| #10624608 | 3 |
| #10624607 | 3 |
| #10624602 | 3 |
| #10624596 | 3 |
| #10624590 | 3 |
| #10624583 | 3 |