Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
296 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C93 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10624463 | C3 | 1 | 12 | yes | maximal |
| 3 | C32×C27 | C32 | 1 | 9 | yes | — |
| 4 | C3×C92 | C9 | 1 | 8 | yes | — |
| 5 | C3×C92 | C9 | 1 | 1 | yes | characteristic |
| 6 | C3×C92 | C32 | 1 | 4 | yes | — |
| 7 | C32×C9 | C3×C9 | 1 | 4 | yes | — |
| 8 | C32×C9 | C3×C9 | 1 | 8 | yes | — |
| 9 | C32×C9 | C33 | 1 | 1 | yes | characteristic |
| 10 | C3×C27 | — | 3 | 36 | no | — |
| 11 | C92 | — | 3 | 24 | no | — |
| 12 | C92 | — | 3 | 12 | no | — |
| 13 | C92 | C27 | 1 | 9 | yes | complemented |
| 14 | C33 | C32×C9 | 1 | 1 | yes | characteristic |
| 15 | C3×C9 | — | 3 | 4 | no | — |
| 16 | C3×C9 | — | 3 | 8 | no | — |
| 17 | C3×C9 | — | 3 | 8 | no | — |
| 18 | C3×C9 | — | 3 | 16 | no | — |
| 19 | C3×C9 | C3×C27 | 1 | 12 | yes | — |
| 20 | C3×C9 | C27⋊C3 | 1 | 24 | yes | — |
| 21 | C3×C9 | #737 | 1 | 12 | yes | — |
| 22 | C27 | — | 9 | 9 | no | — |
| 23 | C32 | #6887 | 1 | 4 | yes | — |
| 24 | C32 | C32×C27 | 1 | 1 | yes | characteristic |
| 25 | C32 | C3×C27⋊C3 | 1 | 8 | yes | — |
| 26 | C9 | — | 3 | 12 | no | — |
| 27 | C9 | — | 3 | 24 | no | — |
| 28 | C9 | #6898 | 1 | 9 | yes | — |
| 29 | C3 | #10624552 | 1 | 1 | yes | characteristic |
| 30 | C3 | #10624710 | 1 | 4 | yes | — |
| 31 | C3 | #10624711 | 1 | 8 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C32×C9 | C33 |
| commutator | C32 | C32×C27 |
| socle, prime radical | C33 | C32×C9 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. C92 | C27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C93 | 3 |
| #10624463 | 3 |