Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
755 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 755 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 | C3×#6908 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C3×C9⋊C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624887 | C3 | semidirect |
| 3. C3×#6887 | C3 | semidirect |
| 4. #10624789 | C3 | semidirect |
| 5. #10624788 | C3 | semidirect |
| 6. #10624787 | C3 | semidirect |
| 7. #10624786 | C3 | semidirect |
| 8. #10624700 | C3 | semidirect |
| 9. #10624688 | C3 | semidirect |
| 10. #10624688 | C3 | semidirect |
| 56. 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 |
|---|---|
| C3×#6908 | 3 |
| #10624887 | 3 |
| C3×#6887 | 3 |
| #10624789 | 3 |
| #10624788 | 3 |
| #10624787 | 3 |
| #10624786 | 3 |
| #10624700 | 3 |
| #10624688 | 3 |