Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
198 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 62 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 | C34 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×C9 ◁ 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 |
|---|---|---|
| 4. #10624495 | C3 | semidirect |
| 5. #10624495 | C3 | semidirect |
| 6. #10624494 | C3 | semidirect |
| 7. #10624493 | C3 | semidirect |
| 8. #10624493 | C3 | semidirect |
| 9. #10624492 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624688 | 3 |
| #10624548 | 3 |
| #10624495 | 3 |
| #10624494 | 3 |
| #10624493 | 3 |
| #10624492 | 3 |