Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
350 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 47 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 | #6911 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6911 | C3 | semidirect |
| 2. #10624739 | C3 | semidirect |
| 3. #10624692 | C3 | semidirect |
| 4. #10624570 | C3 | semidirect |
| 5. #10624481 | C3 | semidirect |
| 6. #10624480 | C3 | semidirect |
| 10. C3×#733 | C32 | semidirect |
| 14. C3×#732 | C32 | semidirect |
| 20. #6868 | C32 | semidirect |
| 25. #6855 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6911 | 3 |
| #10624739 | 3 |
| #10624692 | 3 |
| #10624570 | 3 |
| #10624481 | 3 |
| #10624480 | 3 |