Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
342 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 50 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, Frattini, cosocle | C34 | C33 |
| prime radical | C32⋊H(3,𝔽3) | C32 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624824 | C3 | semidirect |
| 3. #10624824 | C3 | semidirect |
| 4. #10624824 | C3 | semidirect |
| 5. #10624816 | C3 | semidirect |
| 6. #10624815 | C3 | semidirect |
| 8. #10624675 | C3 | semidirect |
| 9. #10624671 | C3 | semidirect |
| 12. #10624617 | C3 | semidirect |
| 13. #10624614 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624882 | 3 |
| #10624824 | 3 |
| #10624816 | 3 |
| #10624815 | 3 |
| #10624814 | 3 |
| #10624675 | 3 |
| #10624671 | 3 |
| #10624661 | 3 |
| #10624620 | 3 |
| #10624617 | 3 |
| #10624614 | 3 |