Search using subgoal graphs is a recent preprocessing-based path-planning algorithm that can find shortest paths on 8-neighbor grids several orders of magnitude faster than A*, while requiring little preprocessing time and memory overhead. In this paper, we first generalize the ideas behind subgoal graphs to a framework that can be specialized to different types of environments (represented as weighted directed graphs) through the choice of a reachability relation. Intuitively, a reachability relation identifies pairs of vertices for which a shortest path can be found quickly. A subgoal graph can then be constructed as an overlay graph that is guaranteed to have edges only between vertices that satisfy the reachability relation, which allows one to find shortest paths on the original graph quickly. In the context of this general framework, subgoal graphs on grids use freespace-reachability (originally called h-reachability) as the reachability relation, which holds for pairs of vertices if and only if their distance on the grid with blocked cells is equal to their distance on the grid without blocked cells (freespace assumption). We apply this framework to state lattices by using variants of freespace-reachability as the reachability relation. We provide preliminary results on (x,y,theta)-state lattices, which shows that subgoal graphs can be used to speed up path planning on state lattices as well, although the speed-up is not as significant as it is on grids.