Minimal non-Iwasawa finite groups

In this note, we describe first the structure of minimal non-Iwasawa finite groups. Then we determine the minimal non-Iwasawa finite groups which are modular. Also, we find connections between minimal non-Iwasawa finite groups and the subgroup commutativity degree, and we give an example of a family of non-nilpotent modular finite groups $G_n$, $n\in\mathbb{N}$, whose subgroup commutativity degree tends to $1$ as $n$ tends to infinity.