Summary: The achromatic index of a graph \(G\) is the maximum number of colours that can be assigned to the edges of \(G\) in such a way that the colouring is proper and any two colours meet on two adjacent edges. To compute the achromatic index of complete graphs seems to be a very difficult task. Indeed that knowledge would yield all odd projective plane orders. The aim of the paper is to establish a nontrivial lower bound of the achromatic index of \(K_{q^2+q}\) for a prime \(q\geq 7\).