Subgroup as quotient of normal subgroups

Question

$N \lhd G$ then we get the group $G/N$.

1. Can any subgroup of $G$ be written in the form $G/N$ ?
2. Can we always identify $G/N$ with a subgroup of $G$ and if so is it unique? (like for internal direct product we can do.)

Thanks in Advance.

Answer 1

Answer to 1. No: take $G$ a non-abelian simple group. Answer to 2. No: take $G$ the quaternion group of order $8$ and $N=Z(G)$.

Answer 2

For positive results in the direction of your question, check : https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem

Your question 2 basically is satisfied if $G \cong N (G/N)$. The Schur–Zassenhaus theorem (link above) says that if $G$ is a finite group and $N$ is a normal subgroup such that $\gcd(|N|,|G|/|N|)=1$ then $G$ is a semi-direct product of $N$ and $G/N$. Inparticular there is a subgroup isomorphic to $G/N$. For infinite groups there is already an answer which gives counter example.

check out: https://en.wikipedia.org/wiki/Semidirect_product#Q8 for counter example to your question 2. Same example as given in another answer.