Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
526 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 62 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 | C34 | C33 |
| prime radical | #10624742 | C3 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#731 | C3 | semidirect |
| 2. H(3,𝔽3)×3−1+2 | C3 | semidirect |
| 3. #10624812 | C3 | semidirect |
| 4. #10624783 | C3 | semidirect |
| 5. #10624779 | C3 | semidirect |
| 7. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 8. #10624610 | C3 | semidirect |
| 9. #10624602 | C3 | semidirect |
| 127. 3−1+2 | AΣL(1,𝔽27) | semidirect |
| 136. H(3,𝔽3) | #731 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#731 | 3 |
| H(3,𝔽3)×3−1+2 | 3 |
| #10624812 | 3 |
| #10624783 | 3 |
| #10624779 | 3 |
| #10624742 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| #10624610 | 3 |
| #10624602 | 3 |