Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
3113 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 3113 conjugacy classes of subgroups — too many to draw. Only its 314 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 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×H(3,𝔽3) | C3 | semidirect |
| 2. C3×#6909 | C3 | semidirect |
| 3. C32×AΣL(1,𝔽27) | C3 | semidirect |
| 4. #10624881 | C3 | semidirect |
| 5. #10624880 | C3 | semidirect |
| 6. #10624735 | C3 | semidirect |
| 7. #10624730 | C3 | semidirect |
| 9. C35 | C32 | semidirect |
| 10. C32×3−1+2 | C32 | semidirect |
| 16. C32×H(3,𝔽3) | C32 | semidirect |
| 17. C32×H(3,𝔽3) | C32 | semidirect |
| 20. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 21. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 22. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 24. #6891 | C32 | semidirect |
| 26. C32⋊H(3,𝔽3) | C32 | semidirect |
| 35. C34 | H(3,𝔽3) | semidirect |
| 59. C3×H(3,𝔽3) | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×H(3,𝔽3) | 3 |
| C3×#6909 | 3 |
| C32×AΣL(1,𝔽27) | 3 |
| #10624881 | 3 |
| #10624880 | 3 |
| #10624735 | 3 |
| #10624730 | 3 |