Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1088 | C2 | 1 | 3 | yes | maximal |
| 2 | #272 | C22 | 1 | 1 | yes | characteristic |
| 3 | H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C3×Q8 | — | 9 | 1 | no | maximal |
| 5 | C3×D3 | — | 12 | 1 | no | — |
| 6 | C3×C4 | — | 9 | 3 | no | — |
| 7 | C32 | — | 4 | 1 | no | — |
| 8 | Q8 | — | 9 | 1 | no | Sylow |
| 9 | C2×C3 | — | 9 | 1 | no | — |
| 10 | D3 | — | 12 | 1 | no | — |
| 11 | C4 | — | 9 | 3 | no | — |
| 12 | C3 | — | 12 | 1 | no | — |
| 13 | C3 | PSU(3,𝔽2) | 1 | 1 | yes | characteristic |
| 14 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3 | PSU(3,𝔽2) |
| commutator, prime radical, cosocle | #272 | C22 |
| Fitting, hypocenter | H(3,𝔽3) | Q8 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. H(3,𝔽3) | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1088 | 2 |
| C3×Q8 | 9 |