Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
866 conjugacy classes of subgroups; only the 458 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 458 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C33 | C34 |
| commutator | C32 | C33×C9 |
| prime radical | C34 | C33 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10624561 | C3 | semidirect |
| 6. #10624559 | C3 | semidirect |
| 7. #10624559 | C3 | semidirect |
| 9. #10624553 | C3 | semidirect |
| 10. C9×C9⋊C9 | C3 | semidirect |
| 11. #10624544 | C3 | semidirect |
| 15. #6900 | C9 | semidirect |
| 62. C9⋊C9 | C3×C9 | semidirect |
| 63. C9⋊C9 | C3×C9 | semidirect |
| 69. C92 | C3×C9 | semidirect |
| 70. C92 | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6900 | 3 |
| C3×C9×3−1+2 | 3 |
| #10624561 | 3 |
| #10624560 | 3 |
| #10624559 | 3 |
| #10624558 | 3 |
| #10624553 | 3 |
| C9×C9⋊C9 | 3 |
| #10624544 | 3 |