Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10624840 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10624830 | C3 | 1 | 2 | yes | maximal |
| 3 | #10624457 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | C3×#733 | — | 3 | 1 | no | — |
| 5 | #6887 | C32 | 1 | 1 | yes | characteristic |
| 6 | #6880 | — | 3 | 3 | no | — |
| 7 | #6875 | — | 3 | 6 | no | — |
| 8 | #6869 | — | 3 | 2 | no | — |
| 9 | C32⋊C27 | — | 3 | 1 | no | — |
| 10 | C3×3−1+2 | — | 9 | 1 | no | — |
| 11 | C32×C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 12 | #733 | — | 3 | 3 | no | — |
| 13 | #733 | — | 9 | 2 | no | — |
| 14 | C27⋊C3 | — | 3 | 6 | no | — |
| 15 | C3×C27 | — | 9 | 1 | no | — |
| 16 | C9⋊C9 | — | 3 | 2 | no | — |
| 17 | C32⋊C9 | — | 3 | 1 | no | — |
| 18 | C92 | — | 3 | 1 | no | — |
| 19 | C33 | #731 | 1 | 1 | yes | characteristic |
| 20 | 3−1+2 | — | 9 | 3 | no | — |
| 21 | 3−1+2 | — | 27 | 2 | no | — |
| 22 | C3×C9 | — | 9 | 1 | no | — |
| 23 | C3×C9 | — | 9 | 1 | no | — |
| 24 | C3×C9 | — | 27 | 1 | no | — |
| 25 | C3×C9 | AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 26 | C3×C9 | AΣL(1,𝔽27) | 1 | 2 | yes | — |
| 27 | C27 | — | 9 | 6 | no | — |
| 28 | C27 | — | 27 | 1 | no | — |
| 29 | C32 | — | 3 | 1 | no | — |
| 30 | C32 | — | 9 | 1 | no | — |
| 31 | C32 | #6857 | 1 | 1 | yes | characteristic |
| 32 | C9 | — | 3 | 2 | no | — |
| 33 | C9 | — | 3 | 1 | no | — |
| 34 | C9 | — | 27 | 1 | no | — |
| 35 | C9 | — | 27 | 3 | no | — |
| 36 | C3 | — | 3 | 1 | no | — |
| 37 | C3 | — | 9 | 1 | no | — |
| 38 | C3 | #10624484 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #10624484 |
| commutator, Frattini, cosocle | #6887 | C32 |
| prime radical | C33 | #731 |
| LCS | G ▷ #6887 ▷ C32×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C32×C9 ◁ #6887 ◁ G | |
| LECS | G ▷ #6887 ▷ C32×C9 ▷ C32 ▷ C3 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624840 | 3 |
| #10624830 | 3 |
| #10624457 | 3 |