Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
78 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×A6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | PGL(2,𝔽9) | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | A6 | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C2×AGL(1,𝔽9) | — | 10 | 1 | no | maximal |
| 5 | C2×A5 | — | 12 | 1 | no | — |
| 6 | C2×#157 | — | 10 | 1 | no | — |
| 7 | AGL(1,𝔽9) | — | 10 | 2 | no | — |
| 8 | A5 | — | 12 | 1 | no | — |
| 9 | C2×S4 | — | 30 | 1 | no | — |
| 10 | C22×D5 | — | 36 | 1 | no | maximal |
| 11 | C2×C3⋊D3 | — | 10 | 1 | no | — |
| 12 | #157 | — | 10 | 1 | no | — |
| 13 | #157 | — | 10 | 1 | no | — |
| 14 | C2×D8 | — | 45 | 1 | no | maximal, Sylow |
| 15 | C2×A4 | — | 30 | 1 | no | — |
| 16 | S4 | — | 30 | 1 | no | — |
| 17 | S4 | — | 30 | 1 | no | — |
| 18 | C22×C5 | — | 36 | 1 | no | — |
| 19 | C2×D5 | — | 36 | 1 | no | — |
| 20 | C2×D5 | — | 36 | 1 | no | — |
| 21 | C2×D5 | — | 36 | 2 | no | — |
| 22 | C2×D5 | — | 36 | 2 | no | — |
| 23 | C2×C32 | — | 10 | 1 | no | — |
| 24 | C3⋊D3 | — | 10 | 1 | no | — |
| 25 | C3⋊D3 | — | 10 | 1 | no | — |
| 26 | C2×D4 | — | 45 | 1 | no | — |
| 27 | C2×D4 | — | 45 | 1 | no | — |
| 28 | D8 | — | 45 | 2 | no | — |
| 29 | D8 | — | 45 | 2 | no | — |
| 30 | C2×C8 | — | 45 | 1 | no | — |
| 31 | C2×D3 | — | 120 | 1 | no | — |
| 32 | A4 | — | 30 | 1 | no | — |
| 33 | C2×C5 | — | 36 | 2 | no | — |
| 34 | C2×C5 | — | 36 | 1 | no | — |
| 35 | D5 | — | 36 | 2 | no | — |
| 36 | D5 | — | 36 | 1 | no | — |
| 37 | D5 | — | 36 | 1 | no | — |
| 38 | C32 | — | 10 | 1 | no | Sylow |
| 39 | C23 | — | 30 | 1 | no | — |
| 40 | C23 | — | 90 | 1 | no | — |
| 41 | D4 | — | 45 | 1 | no | — |
| 42 | D4 | — | 45 | 2 | no | — |
| 43 | D4 | — | 45 | 1 | no | — |
| 44 | D4 | — | 90 | 1 | no | — |
| 45 | D4 | — | 90 | 1 | no | — |
| 46 | C2×C4 | — | 45 | 1 | no | — |
| 47 | C8 | — | 45 | 2 | no | — |
| 48 | C2×C3 | — | 40 | 1 | no | — |
| 49 | D3 | — | 120 | 1 | no | — |
| 50 | D3 | — | 120 | 1 | no | — |
| 51 | C5 | — | 36 | 1 | no | Sylow |
| 52 | C22 | — | 30 | 1 | no | — |
| 53 | C22 | — | 36 | 1 | no | — |
| 54 | C22 | — | 45 | 1 | no | — |
| 55 | C22 | — | 90 | 2 | no | — |
| 56 | C22 | — | 90 | 1 | no | — |
| 57 | C22 | — | 180 | 1 | no | — |
| 58 | C4 | — | 45 | 1 | no | — |
| 59 | C4 | — | 45 | 1 | no | — |
| 60 | C3 | — | 40 | 1 | no | — |
| 61 | C2 | — | 36 | 2 | no | — |
| 62 | C2 | — | 45 | 1 | no | — |
| 63 | C2 | — | 45 | 1 | no | — |
| 64 | C2 | PGL(2,𝔽9) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | PGL(2,𝔽9) |
| commutator, cosocle, hypocenter | A6 | C22 |
| Frattini | C1 | G |
| socle | C2×A6 | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×A6 | C2 | semidirect |
| 2. PGL(2,𝔽9) | C2 | direct (N × H) |
| 3. A6 | C22 | semidirect |
| 64. C2 | PGL(2,𝔽9) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×A6 | 2 |
| PGL(2,𝔽9) | 2 |
| C2×AGL(1,𝔽9) | 10 |
| C22×D5 | 36 |
| C2×D8 | 45 |