$L_{10}$-free $\{p,q\}$-groups

Abstract

If $L$ is a lattice, a group is called $L$-free if its subgroup lattice has no sublattice isomorphic to $L$. It is easy to see that $L_{10}$, the subgroup lattice of the dihedral group of order 8, is the largest lattice $L$ such that every finite $L$-free $p$-group is modular. In this paper we continue the study of $L_{10}$-free groups. We determine all finite $L_{10}$-free $\{p,q\}$-groups for primes $p$ and $q$, except those of order $2^{\alpha}3^{\beta}$ with normal Sylow $3$-subgroup