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The modularity between aggregation functions is an important property which has close links with both distributivity condition requisite in fuzzy connectives and associative law that is rather useful within fuzzy set theory. Moreover, overlap function and grouping function as two special aggregation functions which need not be associative or have the neutral element are applied to many realms such as image processing, classification. In Zong et al. (Inf Sci 477:506–527, 2018), authors discussed most general classes of 2-uninorms, some of whose structures are not indefinite. And we note that until now all studies on 2-uninorm such as Drygaś and Rak (Fuzy Sets Syst 29:82–97, 2016), Yager and Rybalov (Fuzzy Sets Syst 80:111–120, 1996) were only confined to quite special classes of 2-uninorms mentioned in Akella (Fuzzy Sts Syst 158:1631–1651, 2007), but there are not any relevant research on the general 2-uninorm. The present paper makes an attempt to explore the modularity properties of overlap (grouping) function and all general classes of 2-uninorms. During the process, we get the equivalent conditions and full structures of related aggregation operators on modularity equation, which generalize and extend the existing relevant results. [ABSTRACT FROM AUTHOR]