Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2078 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 2078 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 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#737 | C3 | semidirect |
| 2. C3×#6912 | C3 | semidirect |
| 3. C3×#6911 | C3 | semidirect |
| 4. C3×#6909 | C3 | semidirect |
| 5. C32×#731 | C3 | semidirect |
| 6. #10624881 | C3 | semidirect |
| 7. #10624739 | C3 | semidirect |
| 8. #10624734 | C3 | semidirect |
| 16. C32×3−1+2 | C32 | semidirect |
| 20. C33×C9 | C32 | semidirect |
| 24. C3×#733 | C32 | semidirect |
| 27. C3×#731 | C32 | semidirect |
| 28. C3×#731 | C32 | semidirect |
| 29. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 31. #6893 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| C3×#6912 | 3 |
| C3×#6911 | 3 |
| C3×#6909 | 3 |
| C32×#731 | 3 |
| #10624881 | 3 |
| #10624739 | 3 |
| #10624734 | 3 |