We analyse the Higgs branch of 4d \mathcal{N}=2 𝒩 = 2 SQCD gauge theories with non-connected gauge groups \widetilde{\mathrm{SU}}(N) = \mathrm{SU}(N) \rtimes_{I,II} \mathbb{Z}_2 S U ̃ ( N ) = S U ( N ) ⋊ I , I I ℤ 2 whose study was initiated in . We derive the Hasse diagrams corresponding to the Higgs mechanism using adapted characters for representations of non-connected groups. We propose 3d \mathcal{N}=4 𝒩 = 4 magnetic quivers for the Higgs branches in the type I I discrete gauging case, in the form of recently introduced wreathed quivers, and provide extensive checks by means of Coulomb branch Hilbert series computations.