There are infinite graphs which contain all finite graphs as induced subgraphs, e.g. the Rado graph or the coprimeness graph on the naturals.


  Are there infinite groups which
  contain all finite groups as
  subgroups?


Yes, plenty. The group only has to contain all finite permutation groups. Perhaps the most
straightforward example would be the permutations of a countable set. That is bijections
which fix all but a finite set.