# Minimal non-solvable groups

By "minimal non-solvable group" I mean a non-solvable group whose all proper subgroups are solvable.

I need a proof or a counterexample for the following proposition:

Let $$G$$ be a finite minimal non-solvable group. Then one of the following occurs:

(a) $$G$$ is a finite non-abelian simple group;

(b) $$G$$ has a prime order normal subgroup $$N$$, such that the quotient group $$\dfrac{G}{N}$$ is a finite non-abelian simple group.

It's not true. A counterexample is a nonsplit extension $$2^3.L_3(2)$$ of an elementary abelian group of order $$8$$ by the simple group $${\rm SL}(3,2)$$.
Then $$G$$ has a normal subgroup $$N$$ with $$G/N$$ simple, but $$N$$ does not have prime order.