Finite Group DatabasebetaFinite Group DatabasebetaOrder 729.
136 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 43 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 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| prime radical | C3×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #6894 | C3 | semidirect |
| 3. C9⋊H(3,𝔽3) | C3 | semidirect |
| 4. C9×3−1+2 | C3 | semidirect |
| 5. C3×C32⋊C9 | C3 | semidirect |
| 6. 3−1+2⋊C9 | C3 | semidirect |
| 7. 3−1+2⋊C9 | C3 | semidirect |
| 8. 3−1+2⋊C9 | C3 | semidirect |
| 9. 3−1+2⋊C9 | C3 | semidirect |
| 10. H(3,𝔽3)⋊C9 | C3 | semidirect |
| 11. H(3,𝔽3)⋊C9 | C3 | semidirect |
| 12. #6866 | C3 | semidirect |
| 13. #6866 | C3 | semidirect |
| 52. 3−1+2 | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×AΣL(1,𝔽27) | 3 |
| #6894 | 3 |
| C9⋊H(3,𝔽3) | 3 |
| C9×3−1+2 | 3 |
| C3×C32⋊C9 | 3 |
| 3−1+2⋊C9 | 3 |
| H(3,𝔽3)⋊C9 | 3 |
| #6866 | 3 |