Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1220 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 296 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 |
|---|---|---|
| 2. C3×#6894 | C3 | semidirect |
| 3. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 4. C32×C32⋊C9 | C3 | semidirect |
| 5. #10624797 | C3 | direct (N × H) |
| 6. C3×3−1+2⋊C9 | C3 | semidirect |
| 7. C3×3−1+2⋊C9 | C3 | semidirect |
| 8. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 9. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 10. C3×#6866 | C3 | semidirect |
| 21. #6894 | C32 | semidirect |
| 23. C9×H(3,𝔽3) | C32 | semidirect |
| 33. C3×C32⋊C9 | C32 | semidirect |
| 34. 3−1+2⋊C9 | C32 | semidirect |
| 35. 3−1+2⋊C9 | C32 | semidirect |
| 36. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 37. H(3,𝔽3)⋊C9 | C32 | semidirect |
| 38. #6866 | C32 | semidirect |
| 188. C3 | #10624797 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×AΣL(1,𝔽27) | 3 |
| C3×#6894 | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| C32×C32⋊C9 | 3 |
| #10624797 | 3 |
| C3×3−1+2⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| C3×#6866 | 3 |