Every group G is a subgroup of Isometry group of its Cayley graph.
What is essential property of being an Isometry group? Lie group?
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Every group is the full group of isometries of a connected, locally connected, complete metric space:
Being a group of symmetries is the same thing as being a group.
You may also be interested to know that every group is the full automorphism group of a graph, not just a subgroup. References for this and various refinements are given at the wikipedia page for Frucht's theorem.