Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
768 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 768 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 | C34 | C33 |
| 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 |
|---|---|---|
| 6. #10624802 | C3 | semidirect |
| 7. #10624802 | C3 | semidirect |
| 8. #10624799 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6912 | 3 |
| #10624886 | 3 |
| C3×#6894 | 3 |
| #10624863 | 3 |
| #10624802 | 3 |
| #10624799 | 3 |
| #10624699 | 3 |
| C3×3−1+2⋊C9 | 3 |
| #10624691 | 3 |