Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
743 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| socle | C32 | C32×H(3,𝔽3) |
| prime radical | C32⋊H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624872 | C3 | semidirect |
| 3. C3×C9×3−1+2 | C3 | semidirect |
| 4. C3×#6887 | C3 | semidirect |
| 5. #10624621 | C3 | semidirect |
| 7. #10624619 | C3 | semidirect |
| 8. #10624615 | C3 | semidirect |
| 9. #10624614 | C3 | semidirect |
| 10. #10624610 | C3 | semidirect |
| 12. #10624604 | C3 | semidirect |
| 14. #10624592 | C3 | semidirect |
| 15. #10624587 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624872 | 3 |
| #10624869 | 3 |
| C3×C9×3−1+2 | 3 |
| C3×#6887 | 3 |
| #10624621 | 3 |
| #10624620 | 3 |
| #10624619 | 3 |
| #10624615 | 3 |
| #10624614 | 3 |
| #10624610 | 3 |
| #10624609 | 3 |
| #10624604 | 3 |
| #10624598 | 3 |
| #10624592 | 3 |
| #10624587 | 3 |