Group theory Finite group that has no decomposition of a given cardinality.

I'm looking for an example of a finite group. $ G $ whose order can be written as a product $ | G | = ab $ of two numbers $ a, b $ such that $ AB ne G $ for any subset $ A, B G $ subset of cardinality $ | A | = a $ Y $ | B | = b $.

**Observation. ** Such a group $ G $ it can not be abelian and also $ G $ they do not contain cardinality subgroups $ a $ or $ b $.