In mathematics a group is sometimes called an Iwasawa group or M-group or modular group if its lattice of subgroups is modular.
Finite modular groups are also called Iwasawa groups, after (Iwasawa 1941) where they were classified. Both finite and infinite M-groups are presented in textbook form in (Schmidt 1994, Ch. 2.4). Modern study includes (Zimmerman 1989). A finite p-group is a modular group if and only if every subgroup is permutable, by (Schmidt 1994, Lemma 2.3.2, p. 55). Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other words, a finite p-group is an Iwasawa group if and only if it is a PT-group.