Modular not implies permutable

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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
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Statement

A modular subgroup of a group need not be permutable.

Facts used

  1. Maximal implies modular

Proof

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.