Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
804 conjugacy classes of subgroups; only the 332 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 332 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C33 | C34 |
| prime radical | C34 | C33 |
| LCS | G ▷ C33 ▷ 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. #10624575 | C3 | semidirect |
| 4. #10624572 | C3 | semidirect |
| 7. #10624561 | C3 | semidirect |
| 8. #10624561 | C3 | semidirect |
| 10. #10624550 | C3 | semidirect |
| 11. #10624550 | C3 | semidirect |
| 12. C9×C9⋊C9 | C3 | semidirect |
| 75. C9⋊C9 | 3−1+2 | semidirect |
| 80. C92 | 3−1+2 | semidirect |
| 144. C9 | #6896 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6900 | 3 |
| C3×#6896 | 3 |
| #10624575 | 3 |
| #10624572 | 3 |
| #10624567 | 3 |
| #10624565 | 3 |
| #10624561 | 3 |
| #10624560 | 3 |
| #10624550 | 3 |
| C9×C9⋊C9 | 3 |