Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C32×C8 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C2×H(3,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 4 | C32×C4 | C2×C3 | 1 | 4 | yes | — |
| 5 | H(3,𝔽3) | C8 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 6 | C3×C8 | — | 3 | 4 | no | — |
| 7 | C3×C8 | C32 | 1 | 1 | yes | characteristic |
| 8 | C2×C32 | C3×C4 | 1 | 4 | yes | — |
| 9 | C3×C4 | — | 3 | 4 | no | — |
| 10 | C3×C4 | C2×C32 | 1 | 1 | yes | characteristic |
| 11 | C32 | C3×C8 | 1 | 4 | yes | complemented |
| 12 | C8 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C2×C3 | — | 3 | 4 | no | — |
| 14 | C2×C3 | C32×C4 | 1 | 1 | yes | characteristic |
| 15 | C4 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 16 | C3 | — | 3 | 4 | no | — |
| 17 | C3 | C32×C8 | 1 | 1 | yes | characteristic |
| 18 | C2 | C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C32 |
| commutator | C3 | C32×C8 |
| Frattini, cosocle | C3×C4 | C2×C32 |
| socle | C2×C3 | C32×C4 |
| prime radical | C2×H(3,𝔽3) | C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×C8 | C3 | semidirect |
| 5. H(3,𝔽3) | C8 | direct (N × H) |
| 11. C32 | C3×C8 | semidirect |
| 12. C8 | 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 |
|---|---|
| C4×H(3,𝔽3) | 2 |
| C32×C8 | 3 |