Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
65 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#733 | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C22×3−1+2 | C3 | 1 | 3 | yes | maximal |
| 3 | C22×C3×C9 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | #733 | C22 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C2×3−1+2 | C2×C3 | 1 | 9 | yes | — |
| 6 | C2×C3×C9 | C2×C3 | 1 | 3 | yes | — |
| 7 | C22×C32 | C32 | 1 | 1 | yes | characteristic |
| 8 | C22×C9 | — | 3 | 1 | no | — |
| 9 | C22×C9 | — | 3 | 3 | no | — |
| 10 | 3−1+2 | C22×C3 | 1 | 3 | yes | — |
| 11 | C3×C9 | C22×C3 | 1 | 1 | yes | characteristic |
| 12 | C2×C32 | C2×C32 | 1 | 3 | yes | — |
| 13 | C2×C9 | — | 3 | 3 | no | — |
| 14 | C2×C9 | — | 3 | 9 | no | — |
| 15 | C22×C3 | — | 3 | 1 | no | — |
| 16 | C22×C3 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 17 | C32 | C22×C32 | 1 | 1 | yes | characteristic |
| 18 | C9 | — | 3 | 1 | no | — |
| 19 | C9 | — | 3 | 3 | no | — |
| 20 | C2×C3 | — | 3 | 3 | no | — |
| 21 | C2×C3 | C2×H(3,𝔽3) | 1 | 3 | yes | — |
| 22 | C22 | #733 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 23 | C3 | — | 3 | 1 | no | — |
| 24 | C3 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 25 | C2 | C2×#733 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C22×C32 |
| prime radical | C22×C32 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#733 | C2 | direct (N × H) |
| 4. #733 | C22 | direct (N × H) |
| 22. C22 | #733 | direct (N × H) |
| 25. C2 | C2×#733 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#733 | 2 |
| C22×3−1+2 | 3 |
| C22×C3×C9 | 3 |