Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
176 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×H(3,𝔽3) | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×C32 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C2×H(3,𝔽3) | C22 | 1 | 7 | yes | direct product factor |
| 4 | C22×C32 | C2×C3 | 1 | 28 | yes | complemented |
| 5 | H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 6 | C23×C3 | — | 3 | 4 | no | — |
| 7 | C23×C3 | C32 | 1 | 1 | yes | characteristic |
| 8 | C2×C32 | C22×C3 | 1 | 28 | yes | complemented |
| 9 | C22×C3 | — | 3 | 28 | no | — |
| 10 | C22×C3 | C2×C32 | 1 | 7 | yes | — |
| 11 | C32 | C23×C3 | 1 | 4 | yes | complemented |
| 12 | C23 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C2×C3 | — | 3 | 28 | no | — |
| 14 | C2×C3 | C22×C32 | 1 | 7 | yes | — |
| 15 | C22 | C2×H(3,𝔽3) | 1 | 7 | yes | direct product factor |
| 16 | C3 | — | 3 | 4 | no | — |
| 17 | C3 | C23×C32 | 1 | 1 | yes | characteristic |
| 18 | C2 | C22×H(3,𝔽3) | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C23×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×H(3,𝔽3) | C2 | direct (N × H) |
| 2. C23×C32 | C3 | semidirect |
| 3. C2×H(3,𝔽3) | C22 | direct (N × H) |
| 4. C22×C32 | C2×C3 | semidirect |
| 5. H(3,𝔽3) | C23 | direct (N × H) |
| 8. C2×C32 | C22×C3 | semidirect |
| 11. C32 | C23×C3 | semidirect |
| 12. C23 | H(3,𝔽3) | direct (N × H) |
| 15. C22 | C2×H(3,𝔽3) | direct (N × H) |
| 18. C2 | C22×H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(3,𝔽3) | 2 |
| C23×C32 | 3 |