A while back I asked this question about classifying all noncommutative $p^3$ groups (with $p \ge 3$). The book (see this) I am using gave the following hint
...there is a normal subgroup $N$ of order $p^2$, which is commutative. Now show that $G$ has an element $c$ of order $p$ not in $N$, and deduce that $G = N \rtimes \langle c \rangle$, etc.
I couldn't figure out how to show that such a $c$ exists. When I spoke with some people in the chatroom, they said that the hint is based upon a false claim---that it isn't always true that such a $c$ exists. One of them offered the following as a counterexample:
$$G= \left\{ \begin{pmatrix}1+pa & b \\ 0 & 1 \end{pmatrix} \mid a,b \in \Bbb Z/p^2 \Bbb Z \right\}$$
Is this in fact a counterexample? How does one show this?
Yes, there exist normal subgroups of order $p^2$ for which there is no $c$ of order $p$ outside of $N$. However there exists such an $N$ for which such a $c$ exists.