Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
620 conjugacy classes of subgroups; only the 332 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 332 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | H(3,𝔽3) |
| commutator | C32 | C3×C92 |
| Frattini, cosocle | C32×C9 | C33 |
| socle | C33 | C32⋊C9 |
| prime radical | C34 | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #10624469 | C3 | direct (N × H) |
| 8. C3×C27⋊C3 | C9 | semidirect |
| 14. C32×C27 | C9 | semidirect |
| 37. C27⋊C3 | C3×C9 | semidirect |
| 44. C3×C27 | C3×C9 | semidirect |
| 99. C3 | #10624469 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C27⋊C3 | 3 |
| C33×C27 | 3 |
| C3×#6887 | 3 |
| #10624469 | 3 |