Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1722 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 1722 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 | C32 | C32×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| prime radical | C3×H(5,𝔽3) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624883 | C3 | semidirect |
| 3. C3×C9⋊H(3,𝔽3) | C3 | semidirect |
| 4. #10624779 | C3 | semidirect |
| 5. #10624778 | C3 | semidirect |
| 6. #10624777 | C3 | semidirect |
| 7. #10624698 | C3 | semidirect |
| 8. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 10. C32×3−1+2 | C32 | semidirect |
| 15. C33×C9 | C32 | semidirect |
| 18. #6891 | C32 | semidirect |
| 19. C3×C32⋊C9 | C32 | semidirect |
| 20. C3×C32⋊C9 | C32 | semidirect |
| 21. C3×C32⋊C9 | C32 | semidirect |
| 22. C3×C32⋊C9 | C32 | semidirect |
| 23. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 24. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 25. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 26. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 41. C3×H(3,𝔽3) | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×H(5,𝔽3) | 3 |
| #10624883 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| #10624779 | 3 |
| #10624778 | 3 |
| #10624777 | 3 |
| #10624698 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |