Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
15 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C81 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Dic27 | — | 3 | 1 | no | maximal |
| 3 | C81 | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C2×C27 | D3 | 1 | 1 | yes | characteristic |
| 5 | Dic9 | — | 9 | 1 | no | — |
| 6 | C27 | Dic3 | 1 | 1 | yes | characteristic |
| 7 | C2×C9 | D9 | 1 | 1 | yes | characteristic |
| 8 | Dic3 | — | 27 | 1 | no | — |
| 9 | C9 | Dic9 | 1 | 1 | yes | characteristic |
| 10 | C2×C3 | D27 | 1 | 1 | yes | characteristic |
| 11 | C4 | — | 81 | 1 | no | Sylow |
| 12 | C3 | Dic27 | 1 | 1 | yes | characteristic |
| 13 | C2 | D81 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D81 |
| commutator, hypocenter | C81 | C4 |
| Frattini | C2×C27 | D3 |
| Fitting, cosocle | C2×C81 | C2 |
| socle, prime radical | C2×C3 | D27 |
| LCS | G ▷ C81 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C81 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C81 | 2 |
| Dic27 | 3 |