# Are there non-triangle Fuchsian groups?

The only Fuchsian groups I know of (apart from silly ones like finite cyclic groups) are the subgroups of triangle groups. For instance, the modular group $\text{PSL}(2,\mathbb{Z})$.

I've only ever seen hard-to-construct examples of others.

1. Can anyone give some easier-to-construct examples?
2. Is there some sort of classification of Fuchsian groups, or of a large subclass of them?

There must exist a lot since, as the link says, almost all Fuchsian groups are non-triangular.