Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
752 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 752 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, 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 |
|---|---|---|
| 1. #10624872 | C3 | semidirect |
| 3. C3×#6887 | C3 | semidirect |
| 4. #10624810 | C3 | semidirect |
| 5. #10624809 | C3 | semidirect |
| 6. #10624808 | C3 | semidirect |
| 7. #10624807 | C3 | semidirect |
| 8. #10624806 | C3 | semidirect |
| 11. #10624692 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624872 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C3×#6887 | 3 |
| #10624810 | 3 |
| #10624809 | 3 |
| #10624808 | 3 |
| #10624807 | 3 |
| #10624806 | 3 |
| #10624805 | 3 |
| #10624804 | 3 |
| #10624692 | 3 |