Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
25 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | PSU(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | GO+(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | AGL(1,𝔽9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D32 | — | 2 | 1 | no | — |
| 5 | #157 | — | 2 | 1 | no | — |
| 6 | #157 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 8 | C3×D3 | — | 4 | 1 | no | — |
| 9 | SD16 | — | 9 | 1 | no | maximal, Sylow |
| 10 | C2×D3 | — | 12 | 1 | no | — |
| 11 | C32 | SD16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | Q8 | — | 9 | 1 | no | — |
| 13 | D4 | — | 9 | 1 | no | — |
| 14 | C8 | — | 9 | 1 | no | — |
| 15 | C2×C3 | — | 12 | 1 | no | — |
| 16 | D3 | — | 4 | 1 | no | — |
| 17 | D3 | — | 12 | 1 | no | — |
| 18 | C22 | — | 18 | 1 | no | — |
| 19 | C4 | — | 9 | 1 | no | — |
| 20 | C4 | — | 18 | 1 | no | — |
| 21 | C3 | — | 4 | 1 | no | — |
| 22 | C2 | — | 9 | 1 | no | — |
| 23 | C2 | — | 12 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | #157 | C22 |
| Fitting, socle, hypocenter | C32 | SD16 |
| prime radical | GO+(4,𝔽2) | C2 |
| LCS | G ▷ #157 ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. PSU(3,𝔽2) | C2 | semidirect |
| 3. AGL(1,𝔽9) | C2 | semidirect |
| 11. C32 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| PSU(3,𝔽2) | 2 |
| GO+(4,𝔽2) | 2 |
| AGL(1,𝔽9) | 2 |
| SD16 | 9 |