Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
90 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×Q8 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×SL(2,𝔽3) | C3 | 1 | 12 | yes | maximal, direct product factor |
| 3 | C2×C33 | — | 4 | 1 | no | maximal |
| 4 | C32×C4 | — | 3 | 1 | no | — |
| 5 | C33 | — | 4 | 1 | no | Sylow |
| 6 | C3×Q8 | C32 | 1 | 4 | yes | complemented |
| 7 | SL(2,𝔽3) | C32 | 1 | 9 | yes | direct product factor |
| 8 | C2×C32 | — | 4 | 12 | no | — |
| 9 | C2×C32 | A4 | 1 | 1 | yes | characteristic |
| 10 | C3×C4 | — | 3 | 4 | no | — |
| 11 | C32 | — | 4 | 12 | no | — |
| 12 | C32 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, direct product factor |
| 13 | Q8 | C33 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2×C3 | — | 4 | 9 | no | — |
| 15 | C2×C3 | C3×A4 | 1 | 4 | yes | — |
| 16 | C4 | — | 3 | 1 | no | — |
| 17 | C3 | — | 4 | 9 | no | — |
| 18 | C3 | C3×SL(2,𝔽3) | 1 | 4 | yes | direct product factor |
| 19 | C2 | C32×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | A4 |
| commutator, cosocle, hypocenter | Q8 | C33 |
| Frattini | C2 | C32×A4 |
| Fitting | C32×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×Q8 | C3 | semidirect |
| 2. C3×SL(2,𝔽3) | C3 | direct (N × H) |
| 6. C3×Q8 | C32 | semidirect |
| 7. SL(2,𝔽3) | C32 | direct (N × H) |
| 12. C32 | SL(2,𝔽3) | direct (N × H) |
| 13. Q8 | C33 | semidirect |
| 18. C3 | C3×SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q8 | 3 |
| C3×SL(2,𝔽3) | 3 |
| C2×C33 | 4 |