Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
43 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#1088 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #6249 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×#272 | C4 | 1 | 1 | yes | characteristic |
| 4 | #1088 | C4 | 1 | 1 | yes | characteristic |
| 5 | #1088 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×H(3,𝔽3) | C8 | 1 | 1 | yes | characteristic, complemented |
| 7 | #272 | C8 | 1 | 1 | yes | characteristic, complemented |
| 8 | #272 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C3⋊C8 | — | 9 | 1 | no | maximal |
| 10 | C2×C3×D3 | — | 12 | 1 | no | — |
| 11 | H(3,𝔽3) | C2×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3×C4 | — | 9 | 1 | no | — |
| 13 | C3⋊C8 | — | 9 | 2 | no | — |
| 14 | C2×C32 | — | 4 | 1 | no | — |
| 15 | C3×D3 | — | 12 | 1 | no | — |
| 16 | C3×D3 | — | 12 | 1 | no | — |
| 17 | C2×C8 | — | 27 | 1 | no | Sylow |
| 18 | C22×C3 | — | 9 | 1 | no | — |
| 19 | C2×D3 | — | 12 | 1 | no | — |
| 20 | C3×C4 | — | 9 | 1 | no | — |
| 21 | C3×C4 | — | 9 | 1 | no | — |
| 22 | C32 | — | 4 | 1 | no | — |
| 23 | C2×C4 | — | 9 | 1 | no | — |
| 24 | C8 | — | 27 | 2 | no | — |
| 25 | C2×C3 | — | 9 | 1 | no | — |
| 26 | C2×C3 | — | 9 | 1 | no | — |
| 27 | C2×C3 | — | 12 | 1 | no | — |
| 28 | C2×C3 | AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 29 | D3 | — | 12 | 1 | no | — |
| 30 | D3 | — | 12 | 1 | no | — |
| 31 | C22 | — | 9 | 1 | no | — |
| 32 | C4 | — | 9 | 1 | no | — |
| 33 | C4 | — | 9 | 1 | no | — |
| 34 | C3 | — | 12 | 1 | no | — |
| 35 | C3 | C2×AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 36 | C2 | — | 9 | 1 | no | — |
| 37 | C2 | — | 9 | 1 | no | — |
| 38 | C2 | #6249 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #6249 |
| commutator, hypocenter | H(3,𝔽3) | C2×C8 |
| Frattini | C3 | C2×AGL(1,𝔽9) |
| Fitting | C2×H(3,𝔽3) | C8 |
| socle | C2×C3 | AGL(1,𝔽9) |
| prime radical | C2×#272 | C4 |
| cosocle | #1088 | C22 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #6249 | C2 | direct (N × H) |
| 6. C2×H(3,𝔽3) | C8 | semidirect |
| 7. #272 | C8 | semidirect |
| 11. H(3,𝔽3) | C2×C8 | semidirect |
| 38. C2 | #6249 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1088 | 2 |
| #6249 | 2 |
| C2×C3⋊C8 | 9 |