Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
785 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 785 conjugacy classes of subgroups — too many to draw. Only its 278 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 ◁ C3×C32⋊C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624886 | C3 | semidirect |
| 4. C3×#6887 | C3 | semidirect |
| 5. #10624789 | C3 | semidirect |
| 6. #10624785 | C3 | semidirect |
| 7. #10624699 | C3 | semidirect |
| 9. C3×#6867 | C3 | semidirect |
| 18. #6894 | C32 | semidirect |
| 24. C3×C9⋊C9 | C32 | semidirect |
| 29. C3×C92 | C32 | semidirect |
| 30. 3−1+2⋊C9 | C32 | semidirect |
| 32. #6867 | C32 | semidirect |
| 33. #6867 | C32 | semidirect |
| 51. C32×C9 | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#731 | 3 |
| #10624886 | 3 |
| #10624883 | 3 |
| C3×#6887 | 3 |
| #10624789 | 3 |
| #10624785 | 3 |
| #10624699 | 3 |
| #10624698 | 3 |
| C3×#6867 | 3 |