Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
4400 conjugacy classes of subgroups; only the 2924 normal subgroups are shown.
This group has 4400 conjugacy classes of subgroups — too many to draw. Only its 2924 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 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| socle | C33 | C34 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624879 | C3 | direct (N × H) |
| 3. C3×#6900 | C3 | semidirect |
| 4. C3×#6899 | C3 | semidirect |
| 5. C3×#6896 | C3 | semidirect |
| 6. C3×#6894 | C3 | semidirect |
| 8. C3×C9×3−1+2 | C3 | semidirect |
| 9. C3×#6887 | C3 | semidirect |
| 10. C32×C92 | C3 | semidirect |
| 21. #6900 | C32 | semidirect |
| 22. #6899 | C32 | semidirect |
| 23. #6896 | C32 | semidirect |
| 24. #6894 | C32 | semidirect |
| 26. C9×3−1+2 | C32 | semidirect |
| 27. #6887 | C32 | semidirect |
| 39. C3×C92 | C32 | semidirect |
| 55. C32×C9 | 3−1+2 | semidirect |
| 97. C3×C9 | C3×3−1+2 | semidirect |
| 98. C3×C9 | #737 | semidirect |
| 121. C9 | C3×#737 | semidirect |
| 124. C3 | #10624879 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| #10624879 | 3 |
| C3×#6900 | 3 |
| C3×#6899 | 3 |
| C3×#6896 | 3 |
| C3×#6894 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C3×C9×3−1+2 | 3 |
| C3×#6887 | 3 |
| C32×C92 | 3 |