Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
202 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×A6 | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | A6 | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 3 | C22×A5 | — | 6 | 2 | no | maximal |
| 4 | C22×#157 | — | 10 | 1 | no | maximal |
| 5 | C2×A5 | — | 6 | 6 | no | — |
| 6 | C22×S4 | — | 15 | 2 | no | maximal |
| 7 | C22×C3⋊D3 | — | 10 | 1 | no | — |
| 8 | C2×#157 | — | 10 | 3 | no | — |
| 9 | C2×#157 | — | 10 | 3 | no | — |
| 10 | A5 | — | 6 | 2 | no | — |
| 11 | C22×A4 | — | 15 | 2 | no | — |
| 12 | C2×S4 | — | 15 | 6 | no | — |
| 13 | C2×S4 | — | 15 | 6 | no | — |
| 14 | C22×D5 | — | 36 | 1 | no | — |
| 15 | C22×C32 | — | 10 | 1 | no | — |
| 16 | C2×C3⋊D3 | — | 10 | 3 | no | — |
| 17 | C2×C3⋊D3 | — | 10 | 3 | no | — |
| 18 | #157 | — | 10 | 1 | no | — |
| 19 | #157 | — | 10 | 3 | no | — |
| 20 | C22×D4 | — | 45 | 1 | no | Sylow |
| 21 | C22×D3 | — | 60 | 2 | no | — |
| 22 | C2×A4 | — | 15 | 6 | no | — |
| 23 | S4 | — | 15 | 2 | no | — |
| 24 | S4 | — | 15 | 6 | no | — |
| 25 | C22×C5 | — | 36 | 1 | no | — |
| 26 | C2×D5 | — | 36 | 3 | no | — |
| 27 | C2×D5 | — | 36 | 3 | no | — |
| 28 | C2×C32 | — | 10 | 3 | no | — |
| 29 | C3⋊D3 | — | 10 | 1 | no | — |
| 30 | C3⋊D3 | — | 10 | 3 | no | — |
| 31 | C24 | — | 15 | 2 | no | — |
| 32 | C2×D4 | — | 45 | 3 | no | — |
| 33 | C2×D4 | — | 45 | 6 | no | — |
| 34 | C2×D4 | — | 45 | 3 | no | — |
| 35 | C22×C4 | — | 45 | 1 | no | — |
| 36 | C22×C3 | — | 20 | 2 | no | — |
| 37 | C2×D3 | — | 60 | 6 | no | — |
| 38 | C2×D3 | — | 60 | 6 | no | — |
| 39 | A4 | — | 15 | 2 | no | — |
| 40 | C2×C5 | — | 36 | 3 | no | — |
| 41 | D5 | — | 36 | 1 | no | — |
| 42 | D5 | — | 36 | 3 | no | — |
| 43 | C32 | — | 10 | 1 | no | Sylow |
| 44 | C23 | — | 15 | 6 | no | — |
| 45 | C23 | — | 45 | 1 | no | — |
| 46 | C23 | — | 45 | 6 | no | — |
| 47 | D4 | — | 45 | 1 | no | — |
| 48 | D4 | — | 45 | 6 | no | — |
| 49 | D4 | — | 45 | 3 | no | — |
| 50 | D4 | — | 45 | 6 | no | — |
| 51 | C2×C4 | — | 45 | 3 | no | — |
| 52 | C2×C4 | — | 45 | 3 | no | — |
| 53 | C2×C3 | — | 20 | 6 | no | — |
| 54 | D3 | — | 60 | 2 | no | — |
| 55 | D3 | — | 60 | 6 | no | — |
| 56 | C5 | — | 36 | 1 | no | Sylow |
| 57 | C22 | — | 15 | 2 | no | — |
| 58 | C22 | — | 45 | 6 | no | — |
| 59 | C22 | — | 45 | 3 | no | — |
| 60 | C22 | — | 45 | 3 | no | — |
| 61 | C22 | — | 90 | 2 | no | — |
| 62 | C22 | A6 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 63 | C4 | — | 45 | 1 | no | — |
| 64 | C4 | — | 45 | 3 | no | — |
| 65 | C3 | — | 20 | 2 | no | — |
| 66 | C2 | — | 45 | 1 | no | — |
| 67 | C2 | — | 45 | 3 | no | — |
| 68 | C2 | C2×A6 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C22 | A6 |
| commutator, hypocenter | A6 | C22 |
| Frattini | C1 | G |
| LCS | G ▷ A6 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×A6 | C2 | direct (N × H) |
| 2. A6 | C22 | direct (N × H) |
| 62. C22 | A6 | direct (N × H) |
| 68. C2 | C2×A6 | 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 |
| C22×A5 | 6 |
| C22×#157 | 10 |
| C22×S4 | 15 |