Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C3⋊Dic3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #622 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C2×C33 | C4 | 1 | 1 | yes | characteristic |
| 4 | C3⋊Dic3 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | C3×Dic3 | — | 6 | 2 | no | — |
| 6 | C33 | C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C3×C8 | — | 9 | 1 | no | maximal |
| 8 | C2×C32 | — | 2 | 2 | no | — |
| 9 | C2×C32 | — | 4 | 2 | no | — |
| 10 | C2×C32 | C3×C4 | 1 | 1 | yes | characteristic |
| 11 | C3×C4 | — | 9 | 1 | no | — |
| 12 | Dic3 | — | 6 | 2 | no | — |
| 13 | C32 | — | 2 | 2 | no | — |
| 14 | C32 | — | 4 | 2 | no | — |
| 15 | C32 | C3×C8 | 1 | 1 | yes | characteristic, complemented |
| 16 | C8 | — | 9 | 1 | no | Sylow |
| 17 | C2×C3 | — | 2 | 2 | no | — |
| 18 | C2×C3 | — | 4 | 2 | no | — |
| 19 | C2×C3 | #157 | 1 | 1 | yes | characteristic |
| 20 | C4 | — | 9 | 1 | no | — |
| 21 | C3 | — | 2 | 2 | no | — |
| 22 | C3 | — | 4 | 2 | no | — |
| 23 | C3 | #622 | 1 | 1 | yes | characteristic, direct product factor |
| 24 | C2 | C3×#157 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #157 |
| commutator, hypocenter | C32 | C3×C8 |
| Frattini | C2 | C3×#157 |
| Fitting, socle, prime radical | C2×C33 | C4 |
| cosocle | C3⋊Dic3 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #622 | C3 | direct (N × H) |
| 6. C33 | C8 | semidirect |
| 15. C32 | C3×C8 | semidirect |
| 23. C3 | #622 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C3⋊Dic3 | 2 |
| #622 | 3 |
| C3×C8 | 9 |