I will start by the following example.
If I study a group of order 8, are their Sylow's subgroups of order 8 or 4?
For example for $\mathbf{Q}_{8}$, the maximal p-subgroups are of order 4, $\mathbf{C}_{4}$, and there are 3 of this subgroups, isomorphics between them, $3{\equiv}_{2}1$ but $\forall n \; 3{\nmid}2^n$). Too, $\forall x \in \mathbf{Q}_{8} xix^{-1}\in \langle i\rangle \Leftrightarrow \langle i\rangle\lhd\mathbf{Q}_{8}$, and by this way, $\{\langle i\rangle ,\langle j\rangle ,\langle k\rangle \}$ are not conjugate between them.
Then, my answer is that for a group of order $p^n$ its unique Sylow $p$-subgroup is itself, if assuming the previous example as counterexample, resulting that it is not the same Sillow's $p$-subgroups and maximal $p$-subgroups.
If I admit that a maximal subgroups can be the self whole group, then for a $p$-group, its Silow subgroup is the maximal subgroup and it is itself.
The problem for me is the definition of maximal subgroup, beacuse in ring theory a maximal ideal is a proper ideal and I belived a maximal soubgroup only can be a proper subgroup.
Which is the definition of maximal subgroup?