Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
77 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 | 8 | yes | maximal, complemented |
| 3 | C3×C22⋊C9 | C3 | 1 | 4 | yes | maximal, complemented |
| 4 | #737 | — | 4 | 1 | no | maximal, Sylow |
| 5 | C2×H(3,𝔽3) | — | 3 | 1 | no | — |
| 6 | C22×C32 | C32 | 1 | 4 | yes | — |
| 7 | C22⋊C9 | C32 | 1 | 1 | yes | characteristic |
| 8 | C22⋊C9 | C32 | 1 | 8 | yes | — |
| 9 | 3−1+2 | — | 4 | 8 | no | — |
| 10 | H(3,𝔽3) | A4 | 1 | 1 | yes | characteristic |
| 11 | C3×C9 | — | 4 | 4 | no | — |
| 12 | C2×C32 | — | 3 | 4 | no | — |
| 13 | C22×C3 | — | 3 | 4 | no | — |
| 14 | C22×C3 | C33 | 1 | 1 | yes | characteristic |
| 15 | C32 | C3×A4 | 1 | 4 | yes | — |
| 16 | C9 | — | 4 | 1 | no | — |
| 17 | C9 | — | 4 | 8 | no | — |
| 18 | C2×C3 | — | 3 | 1 | no | — |
| 19 | C2×C3 | — | 9 | 4 | no | — |
| 20 | C22 | #737 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C3 | — | 3 | 4 | no | — |
| 22 | C3 | C32×A4 | 1 | 1 | yes | characteristic |
| 23 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×A4 |
| commutator, socle, cosocle | C22×C3 | C33 |
| Fitting, prime radical | C22×H(3,𝔽3) | C3 |
| hypercenter | H(3,𝔽3) | A4 |
| hypocenter | C22 | #737 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
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 |
| 20. C22 | #737 | 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 |
| #737 | 4 |