  A result of Allock [1](arXiv:math/9907194) states that certain orbifold braid
groups contain Artin groups of type $D_n$, $\tilde{B}_n$ and $\tilde{D}_n$ as
finite index subgroups. The underlying orbifolds have at most two cone points
of order two. Based on [10](arXiv:2305.04273) and [12](arXiv:2305.04273), we
generalize this result allowing cone points of arbitrary order. In these cases,
the orbifold braid groups contain similar subgroups of finite index. We show
that in many cases these subgroups can be identified as certain complex braid
groups.
