Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
391 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 53 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 |
| 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. #10624883 | C3 | semidirect |
| 2. #10624818 | C3 | semidirect |
| 3. #10624806 | C3 | semidirect |
| 4. #10624735 | C3 | semidirect |
| 5. #10624600 | C3 | semidirect |
| 6. #10624599 | C3 | semidirect |
| 7. #10624598 | C3 | semidirect |
| 8. #10624597 | C3 | semidirect |
| 9. #10624564 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624883 | 3 |
| #10624818 | 3 |
| #10624806 | 3 |
| #10624735 | 3 |
| #10624600 | 3 |
| #10624599 | 3 |
| #10624598 | 3 |
| #10624597 | 3 |
| #10624564 | 3 |