Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
39 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×H(3,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1094 | C3 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×C22⋊C9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #731 | — | 4 | 1 | no | maximal, Sylow |
| 5 | C2×H(3,𝔽3) | — | 3 | 1 | no | — |
| 6 | C22×C32 | — | 3 | 1 | no | — |
| 7 | C22×C32 | C32 | 1 | 1 | yes | characteristic |
| 8 | C22⋊C9 | — | 3 | 1 | no | — |
| 9 | C22⋊C9 | — | 3 | 2 | no | — |
| 10 | 3−1+2 | — | 4 | 2 | no | — |
| 11 | H(3,𝔽3) | A4 | 1 | 1 | yes | characteristic |
| 12 | C3×C9 | — | 4 | 1 | no | — |
| 13 | C2×C32 | — | 3 | 3 | no | — |
| 14 | C2×C32 | — | 3 | 1 | no | — |
| 15 | C22×C3 | — | 3 | 1 | no | — |
| 16 | C22×C3 | — | 9 | 1 | no | — |
| 17 | C22×C3 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 18 | C32 | — | 3 | 1 | no | — |
| 19 | C32 | C3×A4 | 1 | 1 | yes | characteristic |
| 20 | C9 | — | 12 | 1 | no | — |
| 21 | C9 | — | 12 | 2 | no | — |
| 22 | C2×C3 | — | 3 | 1 | no | — |
| 23 | C2×C3 | — | 9 | 1 | no | — |
| 24 | C2×C3 | — | 9 | 3 | no | — |
| 25 | C22 | #731 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 26 | C3 | — | 3 | 1 | no | — |
| 27 | C3 | — | 9 | 1 | no | — |
| 28 | C3 | C22⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 29 | C2 | — | 3 | 1 | no | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1094 | C3 | semidirect |
| 3. C3×C22⋊C9 | C3 | semidirect |
| 25. C22 | #731 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(3,𝔽3) | 3 |
| #1094 | 3 |
| C3×C22⋊C9 | 3 |
| #731 | 4 |