Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1217 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×#737 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×C32⋊C27 | C3 | 1 | 12 | yes | maximal, complemented |
| 3 | C3×#737 | — | 3 | 12 | no | — |
| 4 | C3×#737 | — | 3 | 24 | no | — |
| 5 | C32×3−1+2 | C9 | 1 | 8 | yes | — |
| 6 | C32×H(3,𝔽3) | C9 | 1 | 1 | yes | characteristic |
| 7 | C33×C9 | C32 | 1 | 4 | yes | — |
| 8 | C32×C27 | C32 | 1 | 9 | yes | — |
| 9 | C32⋊C27 | — | 3 | 36 | no | — |
| 10 | C34 | C3×C9 | 1 | 4 | yes | — |
| 11 | #737 | — | 3 | 27 | no | — |
| 12 | #737 | — | 3 | 216 | no | — |
| 13 | C3×3−1+2 | — | 3 | 24 | no | — |
| 14 | C3×3−1+2 | — | 3 | 72 | no | — |
| 15 | C3×H(3,𝔽3) | — | 3 | 12 | no | — |
| 16 | C32×C9 | — | 3 | 12 | no | — |
| 17 | C32×C9 | — | 3 | 24 | no | — |
| 18 | C32×C9 | C3×C9 | 1 | 8 | yes | — |
| 19 | C32×C9 | H(3,𝔽3) | 1 | 12 | yes | — |
| 20 | C32×C9 | 3−1+2 | 1 | 24 | yes | — |
| 21 | C32×C9 | C33 | 1 | 1 | yes | characteristic |
| 22 | C3×C27 | — | 3 | 36 | no | — |
| 23 | C33 | — | 3 | 12 | no | — |
| 24 | C33 | — | 3 | 36 | no | — |
| 25 | C33 | C32⋊C9 | 1 | 12 | yes | — |
| 26 | C33 | C32×C9 | 1 | 1 | yes | characteristic |
| 27 | 3−1+2 | — | 3 | 216 | no | — |
| 28 | H(3,𝔽3) | — | 3 | 27 | no | — |
| 29 | C3×C9 | — | 3 | 24 | no | — |
| 30 | C3×C9 | — | 3 | 12 | no | — |
| 31 | C3×C9 | — | 3 | 24 | no | — |
| 32 | C3×C9 | — | 3 | 72 | no | — |
| 33 | C3×C9 | C32⋊C9 | 1 | 24 | yes | — |
| 34 | C3×C9 | C3×H(3,𝔽3) | 1 | 4 | yes | — |
| 35 | C3×C9 | C3×3−1+2 | 1 | 8 | yes | — |
| 36 | C27 | — | 9 | 9 | no | — |
| 37 | C32 | — | 3 | 12 | no | — |
| 38 | C32 | — | 3 | 36 | no | — |
| 39 | C32 | — | 9 | 36 | no | — |
| 40 | C32 | C3×C32⋊C9 | 1 | 4 | yes | — |
| 41 | C32 | #6903 | 1 | 9 | yes | — |
| 42 | C9 | — | 3 | 24 | no | — |
| 43 | C9 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 44 | C9 | #6891 | 1 | 8 | yes | — |
| 45 | C3 | — | 9 | 12 | no | — |
| 46 | C3 | #10624547 | 1 | 1 | yes | characteristic |
| 47 | C3 | #10624684 | 1 | 12 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C32×C9 | C33 |
| commutator, socle | C33 | C32×C9 |
| prime radical | C32×H(3,𝔽3) | C9 |
| LCS | G ▷ C33 ▷ 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 |
|---|---|---|
| 2. C3×C32⋊C27 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| C3×C32⋊C27 | 3 |