In 1984 Ihrig & Golubitsky generalized previous work by Sattinger and Busse to classify generic bifurcations in the absolutely irreducible representations of O(3). This classification is based on the Equivariant Branching Lemma. Later on Becker & Krämer used this classification to do a similar work for the representations of SO(4). While in the O(3)-case this gives a reasonable complete picture of the possible bifurcation scenario, this is not the case in the SO(4)-case and similarly in the SO(8)-case. In both cases Lauterbach & Matthews [5], respectively Lauterbach constructed infinite families of groups, where the set of group orders in each family forms an arithmetic progression, where in each case the Equivariant Branching Lemma does not apply and the generic bifurcation behavior is a priori unknown. In [5, this behavior was investigated. However there are many more cases, where the Equivariant Branching Lemma does not apply. In this talk we investigate cases of infinite families of groups which have a rich structure (among other propertes their orders do not form an arithmetic progression) and where the bifurcation analysis faces new challenges.