Undirected co-graphs are those graphs which can be generated from the single vertex graph by disjoint union and join operations. Co-graphs are exactly the P_4-free graphs (where P_4 denotes the path on 4 vertices). Co-graphs itself and several subclasses haven been intensively studied. Among these are trivially perfect graphs, threshold graphs, weakly quasi threshold graphs, and simple co-graphs. Directed co-graphs are precisely those digraphs which can be defined from the single vertex graph by applying the disjoint union, order composition, and series composition. By omitting the series composition we obtain the subclass of oriented co-graphs which has been analyzed by Lawler in the 1970s and the restriction to linear expressions was recently studied by Boeckner. There are only a few versions of subclasses of directed co-graphs until now. By transmitting the restrictions of undirected subclasses to the directed classes, we define the corresponding subclasses for directed co-graphs. We consider directed and oriented versions of threshold graphs, simple co-graphs, co-simple co-graphs, trivially perfect graphs, co-trivially perfect graphs, weakly quasi threshold graphs and co-weakly quasi threshold graphs. For all these classes we provide characterizations by finite sets of minimal forbidden induced subdigraphs. Further we analyze relations between these graph classes.