Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1898 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 1898 conjugacy classes of subgroups — too many to draw. Only its 314 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 | C32×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×3−1+2 | C3 | semidirect |
| 3. C3×#6894 | C3 | semidirect |
| 4. C3×C9⋊H(3,𝔽3) | C3 | semidirect |
| 5. C3×C9×3−1+2 | C3 | semidirect |
| 6. #10624798 | C3 | direct (N × H) |
| 7. C3×3−1+2⋊C9 | C3 | semidirect |
| 8. C3×3−1+2⋊C9 | C3 | semidirect |
| 9. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 10. C3×#6866 | C3 | semidirect |
| 19. C32×3−1+2 | C32 | semidirect |
| 26. #6894 | C32 | semidirect |
| 28. C9⋊H(3,𝔽3) | C32 | semidirect |
| 30. C9×3−1+2 | C32 | semidirect |
| 39. 3−1+2⋊C9 | C32 | semidirect |
| 40. 3−1+2⋊C9 | C32 | semidirect |
| 41. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 42. #6866 | C32 | semidirect |
| 65. C3×3−1+2 | 3−1+2 | semidirect |
| 130. 3−1+2 | C3×3−1+2 | semidirect |
| 151. C3×C9 | AΣL(1,𝔽27) | semidirect |
| 185. C9 | C3×AΣL(1,𝔽27) | semidirect |
| 191. C3 | #10624798 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×3−1+2 | 3 |
| C32×AΣL(1,𝔽27) | 3 |
| C3×#6894 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C3×C9×3−1+2 | 3 |
| #10624798 | 3 |
| C3×3−1+2⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| C3×#6866 | 3 |