Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
878 conjugacy classes of subgroups; only the 260 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 260 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| prime radical | C3×#6856 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6894 | C3 | semidirect |
| 2. C3×#6893 | C3 | semidirect |
| 3. C3×C9×3−1+2 | C3 | semidirect |
| 4. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 5. #10624623 | C3 | direct (N × H) |
| 6. C3×#6860 | C3 | semidirect |
| 7. C3×#6860 | C3 | semidirect |
| 8. C3×#6859 | C3 | semidirect |
| 9. C3×#6858 | C3 | semidirect |
| 10. C3×#6858 | C3 | semidirect |
| 11. C3×#6857 | C3 | semidirect |
| 12. C3×#6857 | C3 | semidirect |
| 13. C3×#6857 | C3 | semidirect |
| 24. #6894 | C32 | semidirect |
| 26. #6893 | C32 | semidirect |
| 28. C9×3−1+2 | C32 | semidirect |
| 30. C9×H(3,𝔽3) | C32 | semidirect |
| 48. #6860 | C32 | semidirect |
| 49. #6860 | C32 | semidirect |
| 50. #6859 | C32 | semidirect |
| 51. #6858 | C32 | semidirect |
| 52. #6858 | C32 | semidirect |
| 53. #6857 | C32 | semidirect |
| 54. #6857 | C32 | semidirect |
| 55. #6857 | C32 | semidirect |
| 218. C3 | #10624623 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6894 | 3 |
| C3×#6893 | 3 |
| C3×C9×3−1+2 | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| #10624623 | 3 |
| C3×#6860 | 3 |
| C3×#6859 | 3 |
| C3×#6858 | 3 |
| C3×#6857 | 3 |
| C3×#6856 | 3 |