Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1088 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #272 | C4 | 1 | 1 | yes | characteristic |
| 3 | H(3,𝔽3) | C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C3⋊C8 | — | 9 | 1 | no | maximal |
| 5 | C3×D3 | — | 12 | 1 | no | — |
| 6 | C3×C4 | — | 9 | 1 | no | — |
| 7 | C32 | — | 4 | 1 | no | — |
| 8 | C8 | — | 27 | 1 | no | Sylow |
| 9 | C2×C3 | — | 9 | 1 | no | — |
| 10 | D3 | — | 12 | 1 | no | — |
| 11 | C4 | — | 9 | 1 | no | — |
| 12 | C3 | — | 12 | 1 | no | — |
| 13 | C3 | AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 14 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | H(3,𝔽3) | C8 |
| Frattini, socle | C3 | AGL(1,𝔽9) |
| prime radical | #272 | C4 |
| cosocle | #1088 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. H(3,𝔽3) | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1088 | 2 |
| C3⋊C8 | 9 |