Considered are subgraphs of oriented graphs in the context of oriented colouring that are analogous to cliques in traditional vertex colouring. Bounds on the sizes of these subgraphs are given for planar, outerplaner and series-parallel graphs. The authors show, among other things, that a planar graph cannot contain an induced subgraph \(D\) with more than 36 vertices such that each pair of vertices in \(D\) are joined by a directed path of length at most two. The following conjecture is stated. Let \(D\) be an orientation of a planar graph such that for any two vertices \(x\), \(y\in V(G)\) there exists an \(xy/yx\) directed path of length at most two. Then \(D\) has at most fifteen vertices.