Modular not implies permutable
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., modular subgroup) need not satisfy the second subgroup property (i.e., permutable subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about modular subgroup|Get more facts about permutable subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property modular subgroup but not permutable subgroup|View examples of subgroups satisfying property modular subgroup and permutable subgroup
By fact (1), it suffices to construct an example of a group with a maximal subgroup that is not permutable. Indeed, in the symmetric group:S3, the subgroup of order two generated by a transposition is maximal but not permutable.