Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
160 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C3×C9 | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×C32 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×C9 | C3 | 1 | 3 | yes | maximal, direct product factor |
| 4 | C2×C3×C9 | C22 | 1 | 7 | yes | direct product factor |
| 5 | C22×C32 | C2×C3 | 1 | 7 | yes | — |
| 6 | C22×C9 | C2×C3 | 1 | 21 | yes | direct product factor |
| 7 | C3×C9 | C23 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C23×C3 | C9 | 1 | 3 | yes | direct product factor |
| 9 | C23×C3 | C32 | 1 | 1 | yes | characteristic |
| 10 | C2×C32 | C22×C3 | 1 | 7 | yes | — |
| 11 | C2×C9 | C22×C3 | 1 | 21 | yes | direct product factor |
| 12 | C22×C3 | C2×C9 | 1 | 21 | yes | direct product factor |
| 13 | C22×C3 | C2×C32 | 1 | 7 | yes | — |
| 14 | C32 | C23×C3 | 1 | 1 | yes | characteristic |
| 15 | C9 | C23×C3 | 1 | 3 | yes | direct product factor |
| 16 | C23 | C3×C9 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 17 | C2×C3 | C22×C9 | 1 | 21 | yes | direct product factor |
| 18 | C2×C3 | C22×C32 | 1 | 7 | yes | — |
| 19 | C22 | C2×C3×C9 | 1 | 7 | yes | direct product factor |
| 20 | C3 | C23×C9 | 1 | 3 | yes | direct product factor |
| 21 | C3 | C23×C32 | 1 | 1 | yes | characteristic |
| 22 | C2 | C22×C3×C9 | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C32 |
| socle, prime radical | C23×C32 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C9 | C2 | direct (N × H) |
| 3. C23×C9 | C3 | direct (N × H) |
| 4. C2×C3×C9 | C22 | direct (N × H) |
| 6. C22×C9 | C2×C3 | direct (N × H) |
| 7. C3×C9 | C23 | direct (N × H) |
| 8. C23×C3 | C9 | direct (N × H) |
| 11. C2×C9 | C22×C3 | direct (N × H) |
| 12. C22×C3 | C2×C9 | direct (N × H) |
| 15. C9 | C23×C3 | direct (N × H) |
| 16. C23 | C3×C9 | direct (N × H) |
| 17. C2×C3 | C22×C9 | direct (N × H) |
| 19. C22 | C2×C3×C9 | direct (N × H) |
| 20. C3 | C23×C9 | direct (N × H) |
| 22. C2 | C22×C3×C9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C9 | 2 |
| C23×C32 | 3 |
| C23×C9 | 3 |