Can a group have more subgroups than it has elements?

Consider the product of $n \gt 2$ copies of $\mathbb{Z}_2$, a group of order $2^n$. Each nonzero element in this (additive) group has order two, so in addition to the trivial subgroup, there are $2^n - 1$ subgroups of order two.

Of course there are also proper subgroups of order greater than two, so more subgroups than elements.

$C_2 \times C_2 \times C_2$ 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.