[uxvvgrap] Infinite non-simple families of sporadic groups

Consider the sporadic groups in the classification of finite simple groups.  Are there sporadic groups which are members of interesting families of finite groups, families that may contain both simple and non-simple groups?  Such families might be analogous to the various infinite families that make up the not-sporadic finite simple groups, except these families are allowed to have non-simple members.  Perhaps a particular sporadic group is the smallest group in an interesting family, and by something resembling dumb luck, it happens to be the only one in the family that is simple.

Examining just the simple members of a family might be missing the forest for the trees: the interesting features of a group could be the features it shares with its family, unrelated to the fact that it is simple.  Or, pedagogically explaining a sporadic group might be better done by first describing its family then describing where the group fits in within its family.

As an analogy, consider the even numbers.  The only prime among the evens is 2.  Studying only the number 2 omits how interesting even numbers in general can be, for example, how only graphs with even numbers of nodes can have perfect matchings.

Inspired by the finite simple groups which are not sporadic all belonging to infinite families, as already mentioned.