In 1984 Ihrig & Golubitsky generalized previous work by Sattinger and Busse et.al. 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.