Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
102 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#226 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 3 | C32×Q16 | — | 3 | 1 | no | maximal |
| 4 | C32×Dic3 | — | 4 | 1 | no | maximal |
| 5 | C32×Q8 | — | 3 | 1 | no | — |
| 6 | C32×Q8 | D3 | 1 | 1 | yes | characteristic |
| 7 | C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 8 | C3×SL(2,𝔽3) | C2×C3 | 1 | 4 | yes | — |
| 9 | C32×C8 | — | 3 | 1 | no | — |
| 10 | C2×C33 | — | 4 | 1 | no | — |
| 11 | #226 | C32 | 1 | 1 | yes | characteristic, direct product factor |
| 12 | C3×Q16 | — | 3 | 4 | no | — |
| 13 | C32×C4 | — | 3 | 1 | no | — |
| 14 | C32×C4 | — | 6 | 1 | no | — |
| 15 | C3×Dic3 | — | 4 | 4 | no | — |
| 16 | C33 | — | 4 | 1 | no | Sylow |
| 17 | C3×Q8 | — | 3 | 4 | no | — |
| 18 | C3×Q8 | C3×D3 | 1 | 4 | yes | — |
| 19 | SL(2,𝔽3) | — | 2 | 4 | no | — |
| 20 | SL(2,𝔽3) | C2×C32 | 1 | 1 | yes | characteristic |
| 21 | C3×C8 | — | 3 | 4 | no | — |
| 22 | C2×C32 | — | 4 | 4 | no | — |
| 23 | C2×C32 | — | 8 | 4 | no | — |
| 24 | C2×C32 | S4 | 1 | 1 | yes | characteristic |
| 25 | Q16 | — | 3 | 1 | no | Sylow |
| 26 | C3×C4 | — | 3 | 4 | no | — |
| 27 | C3×C4 | — | 6 | 4 | no | — |
| 28 | Dic3 | — | 4 | 1 | no | — |
| 29 | C32 | — | 4 | 4 | no | — |
| 30 | C32 | — | 8 | 4 | no | — |
| 31 | C32 | #226 | 1 | 1 | yes | characteristic, direct product factor |
| 32 | Q8 | — | 3 | 1 | no | — |
| 33 | Q8 | C32×D3 | 1 | 1 | yes | characteristic |
| 34 | C8 | — | 3 | 1 | no | — |
| 35 | C2×C3 | — | 4 | 1 | no | — |
| 36 | C2×C3 | — | 8 | 4 | no | — |
| 37 | C2×C3 | C3×S4 | 1 | 4 | yes | — |
| 38 | C4 | — | 3 | 1 | no | — |
| 39 | C4 | — | 6 | 1 | no | — |
| 40 | C3 | — | 4 | 1 | no | — |
| 41 | C3 | — | 8 | 4 | no | — |
| 42 | C3 | C3×#226 | 1 | 4 | yes | direct product factor |
| 43 | C2 | C32×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C2×C32 |
| Frattini | C2 | C32×S4 |
| Fitting | C32×Q8 | D3 |
| prime radical | C32×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#226 | C3 | direct (N × H) |
| 11. #226 | C32 | direct (N × H) |
| 31. C32 | #226 | direct (N × H) |
| 42. C3 | C3×#226 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×SL(2,𝔽3) | 2 |
| C3×#226 | 3 |
| C32×Q16 | 3 |
| C32×Dic3 | 4 |