Triadic configurations are fundamental to many social structural processes and provide the basis for a variety of social network theories and methodologies. This paper addresses the question of how much of the patterning of triads is accounted for by lower-order properties pertaining to nodes and dyads. The empirical base is a collection of 82 social networks representing a number of different species (humans, baboons, macaques, bison, cattle, goats, sparrows, caribou, and more) and an assortment of social relations (friendship, negative sentiments, choice of work partners, advice seeking, reported social interactions, victories in agonistic encounters, dominance, and co-observation). Methodology uses low dimensional representations of triad censuses for these social networks, as compared to censuses expected given four lower-order social network properties. Results show that triadic structure is largely accounted for by properties more local than triads: network density, nodal indegree and outdegree distributions, and the dyad census. These findings reinforce the observation that structural configurations that can be realized in empirical social networks are severely constrained by very local network properties, making some configurations extremely improbable.