Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
570 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 242 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 |
| prime radical | C34 | C33 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10624674 | C3 | semidirect |
| 5. #10624655 | C3 | semidirect |
| 6. #10624649 | C3 | semidirect |
| 8. #10624626 | C3 | semidirect |
| 9. #10624618 | C3 | semidirect |
| 11. #10624588 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624899 | 3 |
| #10624885 | 3 |
| #10624674 | 3 |
| #10624659 | 3 |
| #10624655 | 3 |
| #10624649 | 3 |
| #10624646 | 3 |
| #10624626 | 3 |
| #10624618 | 3 |
| #10624600 | 3 |
| #10624588 | 3 |