Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
359 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C3×H(3,𝔽3) |
| commutator | C32 | C3×C92 |
| Frattini, cosocle | C32×C9 | C33 |
| socle | C32 | C9×H(3,𝔽3) |
| prime radical | C3×H(3,𝔽3) | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C27 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #10624681 | C3 | semidirect |
| 6. C3×#6864 | C3 | semidirect |
| 7. #10624469 | C3 | semidirect |
| 8. #10624468 | C3 | semidirect |
| 9. #10624467 | C3 | semidirect |
| 12. #6903 | C9 | semidirect |
| 23. #6873 | C9 | semidirect |
| 24. #6872 | C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6903 | 3 |
| C3×#6873 | 3 |
| C3×#6872 | 3 |
| #10624697 | 3 |
| #10624681 | 3 |
| C3×#6864 | 3 |
| #10624469 | 3 |
| #10624468 | 3 |
| #10624467 | 3 |