Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
591 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C33 | C32×C9 |
| Frattini, cosocle | C34 | C33 |
| prime radical | C3×#6909 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624862 | C3 | semidirect |
| 3. #10624699 | C3 | semidirect |
| 4. #10624699 | C3 | semidirect |
| 5. #10624699 | C3 | semidirect |
| 6. #10624698 | C3 | semidirect |
| 7. C3×3−1+2⋊C9 | C3 | semidirect |
| 8. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 9. #10624690 | C3 | semidirect |
| 10. #10624689 | C3 | semidirect |
| 11. #10624689 | C3 | semidirect |
| 12. #10624558 | C3 | semidirect |
| 13. #10624555 | C3 | semidirect |
| 20. #6909 | C9 | semidirect |
| 62. C3×3−1+2 | C3×C9 | semidirect |
| 63. C3×3−1+2 | C3×C9 | semidirect |
| 78. C3×H(3,𝔽3) | C3×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6909 | 3 |
| #10624862 | 3 |
| #10624699 | 3 |
| #10624698 | 3 |
| C3×3−1+2⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| #10624690 | 3 |
| #10624689 | 3 |
| #10624558 | 3 |
| #10624555 | 3 |