Proving G is Solvable or Finding a Counter-Example

[burritoloco]

Homework Statement

Suppose G has a solvable maximal normal subgroup. Either prove G is solvable, or give a counter-example.

Homework Equations

The Attempt at a Solution

We have that G/H is simple, not necessarily abelian, so my guess is there could be a counter-example. However, I'm not sure what it is.
I know the symmetric group S_5 is not solvable and it has the unique proper normal subgroup A_5, but A_5 is not solvable, so this doesn't work as a counter.
Moreover, the quaternion group, dihedral groups are solvable too. Any help please?
My exam is tomorrow... Thanks for the help!

 

[jbunniii]

What if G is the direct product of a solvable group with something suitable? Can this give you a counterexample?