Let $G=P \ltimes QR=QM$ be a non-nilpotent solvable group, where $\mathbb{Z}_{2^n} \cong P=C_G(P)=N_G(P) \in {\rm Syl}_2(G)(n \ge 2)$, $\mathbb{Z}_{q} \cong Q \in {\rm Syl}_q(G)$, $\mathbb{Z}_{r} \times \mathbb{Z}_{r} \cong R \in {\rm Syl}_r(G)$ and $M=P \ltimes R$ is a non-abelian maximal subgroup of $G$.
If $Q \ntrianglelefteq G$ and $R \trianglelefteq G$, then i want to find three non-normal non-conjugate subgroups of $G$ which are not cyclic.
My try: Clearly, $M$ is a non-normal subgroup of $G$ which is not cyclic. Also we conclude from Frattini argument that $N_G(Q)=P \ltimes Q$ is a non-normal non-cyclic subgroup of $G$.
Since $G/QR \cong P$, $G^{\prime} \le QR$. By e Theorem we see that $Q=[Q,P] \le G^{\prime}$ and $R=[P,P] \le G^{\prime}$. Thus $G^{\prime} =Q \ltimes R$. I need to find another non-normal non-cyclic subgroup.