Chaining a smaller group inside the pairing friendly group

Let's say there is a bilinear pairing $$G \times G \rightarrow G_t$$ (e.g., for bn128), and let prime $$q$$ be the order of $$G$$. Is it possible to find a prime order group over integers such that its order $$p | q-1$$ (that is $$q$$ is used as the modulus for the "smaller" group)? I checked the order of bn128 and bs381 groups, it seems that the $$q-1$$ has no large prime factors.