The converse is true: [[nilpotent implies solvable]].
=Sources of examples ===
Note that [[prime power order implies nilpotent]], so any finite example must be of order not a prime power order. Further, [[equivalence of definitions of finite nilpotent group]] tells us that any nilpotent group must be the direct product of its Sylow subgroups. Hence, we must look for groups whose order is not a prime power, and that are not direct products of their Sylow subgroups, or equivalently, that have non-normal Sylow subgroups. We also need to make sure that the group remains solvable. It turns out that [[order has only two prime factors implies solvable]], so '' any'' non-nilpotent group whose order has only two prime factors must give an example. Further, since [[nilpotency is subgroup-closed]] and [[solvability is finite direct product-closed]], we can take the direct product of any non-nilpotent solvable group with any (nilpotent or not) solvable group and get more examples of non-nilpotent solvable groups.
==Related specific information==