Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
297 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #10624596 |
| commutator, Frattini, cosocle | C34 | C33 |
| prime radical | #10624474 | C3 |
| LCS | G ▷ C34 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C34 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624882 | C3 | semidirect |
| 2. #10624774 | C3 | semidirect |
| 3. #10624736 | C3 | semidirect |
| 4. #10624692 | C3 | semidirect |
| 5. #10624476 | C3 | semidirect |
| 6. #10624476 | C3 | semidirect |
| 7. #10624475 | C3 | semidirect |
| 8. #10624475 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624882 | 3 |
| #10624774 | 3 |
| #10624736 | 3 |
| #10624692 | 3 |
| #10624476 | 3 |
| #10624475 | 3 |
| #10624474 | 3 |