Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
49 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#1088 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#1088 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#272 | C22 | 1 | 1 | yes | characteristic |
| 4 | #1088 | — | 2 | 2 | no | — |
| 5 | #1088 | C4 | 1 | 2 | yes | complemented |
| 6 | C2×H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic |
| 7 | #272 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | #272 | D4 | 1 | 1 | yes | characteristic |
| 9 | C3×C4⋊C4 | — | 9 | 1 | no | maximal |
| 10 | C2×C3×D3 | — | 12 | 1 | no | — |
| 11 | H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3×C4 | — | 9 | 1 | no | — |
| 13 | C2×C3×C4 | — | 9 | 2 | no | — |
| 14 | C2×C32 | — | 4 | 1 | no | — |
| 15 | C3×D3 | — | 12 | 1 | no | — |
| 16 | C3×D3 | — | 12 | 1 | no | — |
| 17 | C4⋊C4 | — | 9 | 1 | no | Sylow |
| 18 | C22×C3 | — | 9 | 1 | no | — |
| 19 | C2×D3 | — | 12 | 1 | no | — |
| 20 | C3×C4 | — | 9 | 2 | no | — |
| 21 | C3×C4 | — | 18 | 2 | no | — |
| 22 | C32 | — | 4 | 1 | no | — |
| 23 | C2×C4 | — | 9 | 1 | no | — |
| 24 | C2×C4 | — | 9 | 2 | no | — |
| 25 | C2×C3 | — | 9 | 1 | no | — |
| 26 | C2×C3 | — | 9 | 1 | no | — |
| 27 | C2×C3 | — | 24 | 1 | no | — |
| 28 | C2×C3 | PSU(3,𝔽2) | 1 | 1 | yes | characteristic |
| 29 | D3 | — | 12 | 1 | no | — |
| 30 | D3 | — | 12 | 1 | no | — |
| 31 | C22 | — | 9 | 1 | no | — |
| 32 | C4 | — | 9 | 2 | no | — |
| 33 | C4 | — | 18 | 2 | no | — |
| 34 | C3 | — | 24 | 1 | no | — |
| 35 | C3 | #3777 | 1 | 1 | yes | characteristic |
| 36 | C2 | — | 9 | 1 | no | — |
| 37 | C2 | — | 9 | 1 | no | — |
| 38 | C2 | SU(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | PSU(3,𝔽2) |
| commutator | #272 | C2×C4 |
| Fitting | C2×H(3,𝔽3) | Q8 |
| prime radical, cosocle | C2×#272 | C22 |
| hypocenter | H(3,𝔽3) | C4⋊C4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #1088 | C4 | semidirect |
| 11. H(3,𝔽3) | C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1088 | 2 |
| C3×C4⋊C4 | 9 |