Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2801 conjugacy classes of subgroups; only the 323 normal subgroups are shown.
This group has 2801 conjugacy classes of subgroups — too many to draw. Only its 323 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, Frattini, cosocle | C33 | C34 |
| socle | C32 | C32×H(3,𝔽3) |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C34×C9 | C3 | semidirect |
| 2. C3×#6908 | C3 | semidirect |
| 3. #10624869 | C3 | semidirect |
| 4. #10624734 | C3 | semidirect |
| 5. #10624733 | C3 | semidirect |
| 6. #10624732 | C3 | semidirect |
| 7. #10624731 | C3 | semidirect |
| 8. #10624730 | C3 | semidirect |
| 10. C3×#737 | C32 | semidirect |
| 20. C3×#733 | C32 | semidirect |
| 21. C3×#732 | C32 | semidirect |
| 22. C3×#731 | C32 | semidirect |
| 24. #6893 | C32 | semidirect |
| 56. C32×C9 | H(3,𝔽3) | semidirect |
| 115. C32 | #6908 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C34×C9 | 3 |
| C3×#6908 | 3 |
| #10624869 | 3 |
| #10624734 | 3 |
| #10624733 | 3 |
| #10624732 | 3 |
| #10624731 | 3 |
| #10624730 | 3 |