There may be multiple subgroups that are pairwise permutable complements

Statement

Let $n$ be any natural number. Then, there exists a finite group $G$ with subgroups $H_1,H_2,\dots ,H_n$ such that $H_i$ and $H_j$ are Permutable complements (?) in $G$ for any $i\ne j$.

Proof

Let $p$ be a prime such that $p+1\ge n$. Let $G$ be the group $\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}$: in other words, $G$ is an elementary Abelian group of order $p^2$. $G$ has $(p+1)$ cyclic subgroups of order $p$, and any two of these are permutable complements.