I don't know whether it is a good title for this question or not. Please if possible suggest a different title.

Let $G$ be a finite group and $L$ be a minimal subgroup of $G$ of order $p$ for some prime $p$. If there is only one nontrivial proper subgroup of $G$ containing properly $L$, then one of the following cases occurs:

a) $G\cong D_{8}$

b) $G\cong Z_{p}\times Z_{q^2}$, for some primes(not necessarily distinct) $p, q$

c) $G\cong Z_{p}\ltimes(Z_{p}\times Z_{p})$, for some prime $p\neq2$, where $Z_{p}$ acts nontrivially on $Z_{p}\times Z_{p}$

d) $G\cong Z_{q}\ltimes(Z_{p}\times Z_{p})$, for some primes $p, q$ and $q\mid(p^{2}-1)$, where $Z_{q}$ acts nontrivially on $Z_{p}\times Z_{p}$

e) $G\cong L_{2}(q)$, for some prime $q$ and $q\not\equiv \pm 1, \pmod 8$.

Some examples of the last case, (e) is $L=Z_{5}$ in $G=A_{5}$, $L=Z_{11}$ in $G=L_{2}(11)$, $L=Z_{7}$ in $G=L_{2}(13)$ and etc.

Is there any other possible structure for $G$?