Bread wheat is an allohexaploid species originating from two successive and recent rounds of hybridization between three diploid species that were very similar in terms of chromosome number, genome size, TE content, gene content and synteny. As a result, it has long been considered that most of the genes were in three pairs of homoeologous copies. However, these so‐called triads represent only one half of wheat genes, while the remaining half belong to homoeologous groups with various number of copies across subgenomes. In this study, we examined and compared the distribution, conservation, function, expression and epigenetic profiles of triads with homoeologous groups having undergone a deletion (dyads) or a duplication (tetrads) in one subgenome. We show that dyads and tetrads are mostly located in distal regions and have lower expression level and breadth than triads. Moreover, they are enriched in functions related to adaptation and more associated with the repressive H3K27me3 modification. Altogether, these results suggest that triads mainly correspond to housekeeping genes and are part of the core genome, while dyads and tetrads belong to the Triticeae dispensable genome. In addition, by comparing the different categories of dyads and tetrads, we hypothesize that, unlike most of the allopolyploid species, subgenome dominance and biased fractionation are absent in hexaploid wheat. Differences observed between the three subgenomes are more likely related to two successive and ongoing waves of post‐polyploid diploidization, that had impacted A and B more significantly than D, as a result of the evolutionary history of hexaploid wheat.