TO have an instanton solution, you need to map the (euclideanized) "spacetime at infinity" to the group manifold. In the case of SU(2), both the spacetime at infinity and the group manifold are $S^3$ and instantons are characterized by the integers. I hope you understand that much, at least for SU(2).
If you're interested in 4d instantons, they are characterized by $H_3(M_G)$ where $M_G$ is the group manifold -- since the spacetime at infinity is $S^3$. So, for every (homologically distinct) non-contractible 3-cycle of the group manifold, you can find an instanton. As the Wikipedia link given by @twistor says, the gauge fields corresponding to directions on that 3-cycle will have the same profile as the SU(2) instanton and the other gauge fields will have a trivial configuration (of course, up to a gauge transformation). Essentially, you're seeking the possible embeddings of SU(2) inside your gauge group and then making instantons out of those SU(2) subgroups.
If you understand that, the generalization to arbitrary number of dimensions should be straightforward.