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In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $\mathrm{L}\left(G\right)$ and sublattices of $\mathrm{L}\left(G\right)$. It is proved that the collections of all pronormal subgroups of ${\mathrm{A}}_n$ and S${}_n$ do not form sublattices of respective $\mathrm{L}\left({\mathrm{A}}_n\right)$ and $\mathrm{L}\left({\mathrm{S}}_n\right)$, whereas the collection of all pronormal subgroups $\mathrm{LPrN}\left({\mathrm{Dic}}_n\right)$ of a dicyclic group is a sublattice of $\mathrm{L}\left({\mathrm{Dic}}_n\right)$. Furthermore, it is shown that $\mathrm{L}\left({\mathrm{Dic}}_n\right)$ and $\mathrm{LPrN}({\mathrm{Dic}}_n$) are lower semimodular lattices.