Group in which order of commutator divides order of element

Definition
A group in which order of commutator divides order of element is a group where, for any elements $$x$$ and $$y$$ such that $$x$$ has finite order, the commutator $$[x,y]$$ also has finite order and the order of $$[x,y]$$ divides the order of $$x$$.

Note that the definition does not depend on whether we use the left definition of commutator ($$[x,y] = xyx^{-1}y^{-1}$$) or the right definition ($$[x,y] = x^{-1}y^{-1}xy$$).

Examples among finite 2-groups
Any group of nilpotency class two satisfies this property, because class two implies commutator map is endomorphism. We list here the finite p-groups of small order that have class greater than two but satisfy the property:

Examples among finite p-groups for other primes p
Any Lazard Lie group automatically satisfies this property. In particular, any p-group of nilpotency class less than p automatically satisfies this property.