Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
168 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C33×C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊(C3⋊Dic3) | C2 | 1 | 2 | yes | maximal |
| 3 | #634 | — | 3 | 13 | no | maximal |
| 4 | C2×C33 | C22 | 1 | 1 | yes | characteristic |
| 5 | C32×C4 | D3 | 1 | 13 | yes | — |
| 6 | C3⋊Dic3 | — | 3 | 26 | no | — |
| 7 | C33 | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | Dic6 | — | 9 | 13 | no | — |
| 9 | C2×C32 | C2×D3 | 1 | 13 | yes | — |
| 10 | C3×C4 | C3⋊D3 | 1 | 13 | yes | — |
| 11 | Dic3 | — | 9 | 26 | no | — |
| 12 | C32 | Dic6 | 1 | 13 | yes | complemented |
| 13 | Q8 | — | 27 | 1 | no | Sylow |
| 14 | C2×C3 | C2×C3⋊D3 | 1 | 13 | yes | — |
| 15 | C4 | — | 27 | 2 | no | — |
| 16 | C4 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 17 | C3 | #634 | 1 | 13 | yes | complemented |
| 18 | C2 | C2×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊D3) |
| commutator, socle, prime radical, cosocle | C2×C33 | C22 |
| Fitting | C33×C4 | C2 |
| hypercenter | C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C33 | Q8 | semidirect |
| 12. C32 | Dic6 | semidirect |
| 17. C3 | #634 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C4 | 2 |
| C3⋊(C3⋊Dic3) | 2 |
| #634 | 3 |