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I would guess that $g(n)$ approaches 1 as $g$ n$ approaches infinity.

The topic of the growth rate of the number of isomorphism classes of groups of order $n$ has been discussed previously, for example in:

http://mathoverflow.net/questions/21265/

Both from looking at these results and from known results for $n \le 2000$, there is very strong evidence that most finite groups are 2-groups. Unfortunately, the imprecision in the results proved is big enough to make it very unlikely that this will be proved any time soon.

The known lower bound on the number of groups of order $2^n$, due to G. Higman, which is about $2^{2n^3/27}$, is proved by considering groups $G$ in which $G'=Z(G)$ and $G/Z(G)'$ and $Z(G)$ are both elementary abelian 2-groups, and where $|Z(G)|^2$ is approximately equal to $|G/Z(G)|$. Sims proved that the total number of 2-groups of a given order is roughly the same as this, but the error term is in the exponent, so it has not been proved that almost all 2-groups are of this form, and that might not be correct.

But looking at groups $G$ of that form, we can see that they will virtually all have quotient groups that are not isomorphic to subgroup - I am sure that claim could be proved. If you quotient out a central subgroup of order 2, then you will usually end up with another $n$-generator 2-group $H$ of order $|G|/2$ with $Z(H) = H'$, and such an $H$ cannot be subgroup of $G$, because any $n$ elements of $G \setminus G'$ that generate $G/G'$ also generate $G$.
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I would guess that $g(n)$ approaches 1 as $g$ approaches infinity.

The topic of the growth rate of the number of isomorphism classes of groups of order $n$ has been discussed previously, for example in:

http://mathoverflow.net/questions/21265/

Both from looking at these results and from known results for $n \le 2000$, there is very strong evidence that most finite groups are 2-groups. Unfortunately, the imprecision in the results proved is big enough to make it very unlikely that this will be proved any time soon.

The known lower bound on the number of groups of order $2^n$, due to G. Higman, which is about $2^{2n^3/27}$, is proved by considering groups $G$ in which $G'=Z(G)$ and $G/Z(G)'$ and $Z(G)$ are both elementary abelian 2-groups, and where $|Z(G)|^2$ is approximately equal to $|G/Z(G)|$. Sims proved that the total number of 2-groups of a given order is roughly the same as this, but the error term is in the exponent, so it has not been proved that almost all 2-groups are of this form, and that might not be correct.

But looking at groups $G$ of that form, we can see that they will virtually all have quotient groups that are not isomorphic to subgroup - I am sure that claim could be proved. If you quotient out a central subgroup of order 2, then you will usually end up with another $n$-generator 2-group $H$ of order $|G|/2$ with $Z(H) = H'$, and such an $H$ cannot be subgroup of $G$, because any $n$ elements of $G \setminus G'$ that generate $G/G'$ also generate $G$.