Which groups $G$ has the property that for all subgroups $H$ , there is a surjective map from $G$ to $H$? I tried many examples , but i can't find any counterexample .
But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal subgroups).
Thus I have two question:  


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*Is there any counter example of group $G$ and its subgroup $H$ s.t there is no surjective homomorphism from $G$ to $H$ ?



  
*If exists some counterexamples , which sort of groups or subgroups have this property?
  

I would also appreciate any reference . 

Take $G = F(\{x_1,\dots, x_n\})$, the free group on $n$ generators. The commutator subgroup $G' = [G,G]$ is a free group of infinite rank and thus $G$ cannot surject onto $G'$, as it simply doesn't have enough generators (it has finite rank).
