I think in the subgroup lattice of the non-solvable group $A_5 \times \mathbb{Z}/2^N\mathbb{Z}$, the interval between $\{1\} \times \{1\}$ and $\{ 1 \} \times \mathbb{Z}/2^N\mathbb{Z}$ is a chain of length $N$.
Edit: For any $N$, you can also find primes $p$ and $q$ such that $q \equiv \pm 1$ mod $p^N$, so there are cyclic groups of order $p^N$ inside any group of Lie type over $\mathbb{F}_q$ that has a split torus (or just a copy of where you use $\mathbb{G}_m$). +1$ in the split case and $-1$ in the non-split case). These will yield intervals that are chains of length $N$. Similarly, the alternating group $A_{p^N}$ contains a cyclic group of order $p^N$.
