Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
353 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 50 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 | C3×#732 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | C32×#732 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624768 | C3 | semidirect |
| 3. #10624768 | C3 | semidirect |
| 4. #10624548 | C3 | semidirect |
| 5. #10624488 | C3 | semidirect |
| 20. #6868 | C32 | semidirect |
| 21. #6868 | C32 | semidirect |
| 23. #6855 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#732 | 3 |
| #10624768 | 3 |
| #10624548 | 3 |
| #10624488 | 3 |