Can a group have more subgroups than it has elements? I'm looking for a group for which the number of subgroups is more than the number of elements in the group! I tried a few possibilities - it can't be cyclic, and I think we'll have to consider group of infinite order.

$C_2 \times C_2 \times C_2$ has 8 elements. Each of the 7 non-identity elements generates a subgroup of order 2. Any pair of non identity elements also generates a subgroup isomorphic to $C_2 \times C_2$. There are 7 of these subgroups, for a total of 14 nontrivial proper subgroups.
