The group of orthogonal transformations of $\mathbb{R}^3$ is direct product of the group of rotations and the group $Z=\langle x\mapsto -x\rangle\cong \mathbb{Z}/2$. The finite subgroups of group of rotations are well known: rotational symmetries of regular polyhedrons. 

These subgroups, along with their direct product with $Z$ will give some (families) of finite groups of orthogonal transformations.

It looks that some finite groups of orthogonal transformations need not be of this form (see Shafarevich- Algebra I; I couldn't understand arguments there).

How do we cover all finite subgroups of orthogonal transformations?

A more general question in this context may be: **How to get subgroups of $H\times K$ knowing subgroups of $H$ and $K$?**