Ratio of number of subgroups to the order of a finite group Let $\mathcal{G}$ be the set of finite groups and for $G \in \mathcal{G}$, let $S(G)$ be the set of subgroups of $G$. I am interested in the ratio $R(G)=|S(G)|/|G|$. It is easy to show that by picking $G$ appropriately, $R(G)$ can be made arbitrarily large or arbitrarily close to zero. I am interested in some deeper properties of the set $R=(R(G) : G \in \mathcal{G})$, such as: 
(1) For which $x \in \mathbb{R}$ do there exist sequences of finite groups $G_1, G_2, ...$ such that the sequence $R(G_1), R(G_2), ... $ converges to $x$?
(2) Does $R$ contain a (non-empty) interval $(a,b) \in \mathbb{Q}$?
(3) Which integers belong to $R$?
(4) How do these properties change when $\mathcal{G}$ is replaced by the set of finite abelian groups? Finite simple groups? 
 A: With regards to Q1 (and part of Q4), the numbers of the form $R(G)$ are dense in $\mathbb{R}_{\ge 0}$ even when $G$ is restricted to be abelian.
Some first results. $R(G \times H) = R(G) R(H)$ if $\gcd(|G|, |H|) = 1$. We also have $R(C_p) = \frac{2}{p}$ ($p$ is always a prime) and
$$R(C_p^4) = \frac{1}{p^4} \sum_{k=0}^4 {4 \choose k}_p = 1 + \frac{3}{p} + \frac{4}{p^2} + \frac{3}{p^3} + \frac{5}{p^4}$$
where ${n \choose k}_p$ is a Gaussian binomial coefficient. 
Lemma: Let $a_1, a_2, ... $ be a sequence of positive reals such that $\lim_{n \to \infty} a_n = 0$ but such that $\sum a_n$ diverges. Then the set of sums of finite subsequences of the $a_i$ is dense in $\mathbb{R}_{\ge 0}$.
Proof. Let $r \in \mathbb{R}_{\ge 0}$ and fix $\epsilon > 0$. Choose $N$ such that $a_n < \epsilon$ for all $n \ge N$. Then the partial sums starting from $a_N$ diverge but begin less than $\epsilon$ and increase by at most $\epsilon$ at each step, so the conclusion follows. $\Box$
Applying the lemma to the sequence $\log R(C_p^4)$ (which satisfies the hypotheses of the lemma using the fact that $\sum \frac{1}{p}$ diverges), we conclude that the numbers of the form $R(G)$ where $G$ is a product of groups of the form $C_p^4$ for distinct primes $p$ are dense in $\mathbb{R}_{\ge 1}$, and if we allow in addition the groups of the form $C_p$, then the conclusion follows.
Q2 appears to be potentially very difficult and I have not thought about Q3.  
