Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
876 conjugacy classes of subgroups; only the 281 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 281 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 | C33 | C34 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#6894 | C3 | semidirect |
| 3. C3×#6894 | C3 | semidirect |
| 5. #10624796 | C3 | semidirect |
| 6. #10624793 | C3 | semidirect |
| 7. #10624699 | C3 | semidirect |
| 9. #10624690 | C3 | semidirect |
| 25. #6894 | C32 | semidirect |
| 26. #6894 | C32 | semidirect |
| 33. C3×C9⋊C9 | C32 | semidirect |
| 34. C3×C9⋊C9 | C32 | semidirect |
| 39. 3−1+2⋊C9 | C32 | semidirect |
| 41. #6867 | C32 | semidirect |
| 159. C9 | #6910 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6910 | 3 |
| C3×#6894 | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| #10624796 | 3 |
| #10624793 | 3 |
| #10624699 | 3 |
| #10624698 | 3 |
| #10624690 | 3 |