Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
3920 conjugacy classes of subgroups; only the 404 normal subgroups are shown.
This group has 3920 conjugacy classes of subgroups — too many to draw. Only its 404 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 | C33×H(3,𝔽3) | C3 |
| 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 |
|---|---|---|
| 1. C33×3−1+2 | C3 | semidirect |
| 3. C3×#6891 | C3 | semidirect |
| 4. C3×#6891 | C3 | semidirect |
| 5. #10624563 | C3 | semidirect |
| 6. #10624555 | C3 | semidirect |
| 13. C32×3−1+2 | C32 | semidirect |
| 18. C33×C9 | C32 | semidirect |
| 21. C3×C32⋊C9 | C32 | semidirect |
| 22. C3×C32⋊C9 | C32 | semidirect |
| 23. C3×C32⋊C9 | C32 | semidirect |
| 43. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 49. C3×H(3,𝔽3) | 3−1+2 | semidirect |
| 81. C33 | C3×3−1+2 | semidirect |
| 93. C3×C9 | C3×H(3,𝔽3) | semidirect |
| 112. C32 | #6891 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×3−1+2 | 3 |
| C33×H(3,𝔽3) | 3 |
| C3×#6891 | 3 |
| #10624563 | 3 |
| #10624555 | 3 |