Converse of Lagrange's thm

Lagrange's theorem for finite groups is one of the first (and fundamental)
results one studies in group theory.
It is well known that the _converse_ of Lagrange's theorem holds in (finite)
supersolvable groups, while in solvable groups (a superclass of ssolv.
groups) it does not.
My question is: which is the largest known class of finite groups in which
the converse of L. theorem holds?
Thanks.


.

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