Are subgroups of order $p^{n-1}$ maximal?

Question

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$
Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups?
If not, what property $G$ must have to it became true?

Answer 1

By Lagrange's theorem, for a finite group $G$ the order of a subgroup $H\subset G$ divides the order of $G$. Given that $G$ is a group of order $p^n$ and $H$ is a group of order $p^{n-1}$, if $H$ is not maximal there exists a subgroup $M\subsetneq G$ with $H\subsetneq M$. Then by Lagrange's theorem $|M|$ divides $p^n$ and $p^{n-1}$ divides $|M|$. It follows that $|M|=p^{n-1}$ or $|M|=p^n$, and hence $M=H$ or $M=G$, a contradiction. This shows that $H$ is indeed maximal.