Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
740 conjugacy classes of subgroups; only the 422 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 422 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 |
|---|---|---|
| 2. #10624561 | C3 | semidirect |
| 4. #10624559 | C3 | semidirect |
| 5. #10624559 | C3 | semidirect |
| 6. #10624559 | C3 | semidirect |
| 7. #10624559 | C3 | semidirect |
| 8. #10624559 | C3 | semidirect |
| 12. #10624553 | C3 | semidirect |
| 13. #10624551 | C3 | semidirect |
| 16. #6896 | C9 | semidirect |
| 77. C9⋊C9 | C3×C9 | semidirect |
| 78. C9⋊C9 | C3×C9 | semidirect |
| 84. C92 | C3×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6896 | 3 |
| #10624561 | 3 |
| #10624560 | 3 |
| #10624559 | 3 |
| #10624558 | 3 |
| #10624557 | 3 |
| #10624556 | 3 |
| #10624553 | 3 |
| #10624551 | 3 |