Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2045 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 2045 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×#6913 | C3 | semidirect |
| 3. C3×#6908 | C3 | semidirect |
| 4. #10624882 | C3 | semidirect |
| 5. #10624739 | C3 | semidirect |
| 6. #10624739 | C3 | semidirect |
| 7. #10624738 | C3 | semidirect |
| 8. #10624737 | C3 | semidirect |
| 9. #10624735 | C3 | semidirect |
| 10. #10624734 | C3 | semidirect |
| 11. #10624731 | C3 | semidirect |
| 63. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 80. C32×C9 | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| C3×#6913 | 3 |
| C3×#6908 | 3 |
| #10624882 | 3 |
| #10624739 | 3 |
| #10624738 | 3 |
| #10624737 | 3 |
| #10624735 | 3 |
| #10624734 | 3 |
| #10624731 | 3 |