Triadic concept analysis is a new approach for data analysis and information processing. One of the important problems of triadic concept analysis is the construction of triadic concepts. Firstly, properties of a triadic concept in which one of its extent, intent and modus is an empty set are studied, and the judgment method of this kind of triadic concept is also obtained. Secondly, another kind of special triadic concept, called object-conditional triadic concept, is researched in detail. An operation is defined on the set of conditional triadic concepts, and a construction method of triadic concepts is then proposed based on object-conditional triadic concepts using the operation. It is shown that every triadic concept can be generated by some object-conditional triadic concepts if its extent and modus are both non-empty sets. Thirdly, a triadic context is decomposed into a series of dyadic contexts according to its conditions. The relationship between object-conditional triadic concepts of the triadic context and object dyadic concepts of those decomposed dyadic contexts is then studied. Furthermore, steps are given to generate all triadic concepts from object-conditional triadic concepts. Finally, the detailed process of constructing the triadic concepts is demonstrated by an example, and the triadic diagram is given to describe all generated triadic concepts more clearly.