This is not really a proper answer - just speculation about the case $c=3$.

It looks as though the simple groups with $c=3$ are ${\rm PSL}(2,q)$ for $q=5,7$, and $2^k$ with $k$ prime. I have not checked that completely, but there is certainly enough known about maximal subgroups of simple groups to do so, and I am also confident that there are no more almost simple groups (i.e. groups with nonabelian simple socle) with $c=3$. Note also that, in the case of ${\rm PSL}(2,7)$ only, two of the three classes are fused under an automorphism of $G$.

Assuming that is correct, a non-solvable group $G$ with $c=3$ would have to be a nonsplit extension of a nilpotent group $N = \Phi(G)$ by one of the above simple groups.

I see from the answer of M.Farrokhi D. G. that this result has been proved by V. A. Belonogov.

By checking databases of perfect groups, I see that there are examples with $G/\Phi(G) = A_5$ and $|\Phi(G)| = 2, 2^5, 2^9, 3^4, 3^8, 5^3, 5^6$.

In fact, there is a result in group cohomology that says that, for an finite group $G$ with order divisible by a prime $p$, and any $n>0$, there is an irreducible module $M$ over ${\mathbb F}_p$ with $H^n(G,M) \ne 0$. So by applying this repeatedly with $n=2$, you can start with any of the above simple groups $S$ and a prime $p$ dividing $|S|$, and keep constructing downward extensions with finite $p$-groups, to produce examples $G$ with $c=3$, $G/\Phi(G) \cong S$, and $\Phi(G)$ a $p$-group of arbitrarily large order.

$\mathbf{Added\ later}$: following discussion in the comments, it seems relevant to summarize how maximal subgroups of a finite group $G$ are related to those of its nonabelian composition factors. This theory can be found in two papers published in the 1980s, one by Aschbacher and Scott, and one by Fletcher and Kovács, which are independent. The references are at the end. It is also closely related to the O'Nan-Scott Theorem, because maximal subgroups of $G$ correspond to its primitive permutation representations.

Let $M/N$ be a chief factor of $G$. If $M/N$ is elementary abelian, then its complements $C/N$ in $G/N$ (if any) are maximal and correspond to the affine case of O'Nan-Scott.

The remaining maximal subgroups of $G$ are related to nonabelian chief factors $M/N$. We are considering complements that contain $N$ but not $M$, so we may as well assume that $N=1$. Note that $M \cap C_G(M) = 1$. Suppose first that $C_G(M) = 1$. Then $G \le {\rm Aut}(M) \cong {\rm Aut}(S) \wr S_n$, where $M \cong S^n$ with $S$ nonabelian simple.

Let $S_1$ be one of the copies of $S$ in $M$, and let $N_1 = N_G(S_1)$, and $C_1 = C_G(S_1)$. So $N_1/C_1$ is isomorphic to a subgroup of ${\rm Aut}(S)$ containing $S$, i.e. an almost simple group with socle $S$. It turns out that each conjugacy class $C$ of maximal subgroups of $N_1/C_1$ corresponds to a conjugacy class of maximal subgroups of $G$ whose intersection with $N_1$ projects onto the groups in $C$. They correspond to primitive groups of product type in O'Nan-Scott.

In particular, the number of conjugacy classes of maximal subgroups of $G$ is greater than or equal to the number for $N_1/C_1$ and it is greater than this when $n>1$, because there are maximal subgroups containing $M$. In particular, if $G$ has $3$ such classes, then we must have $n=1$, and $G = N_1$, which is how we can reduce this problem to the almost simple case when $c=3$.

Of course there are usually other maximal subgroups associated with $M$ that do not contain $M$. There may be some whose intersection with $M$ maps onto the direct factors $S_i$ of $M$, and these correspond to primitive groups of diagonal type in O'Nan-Scott. Occasionally there are some that complement $M$: the twisted wreath product type. Finally, when $C_G(M) \ne 1$, and there is a chief factor $M'$ of $C$ isomorphic to $M$, then there may be maximal subgroups that intersect $M \times M'$ in a diagonal subgroup: these correspond to the primitive groups with two distinct minimal normal subgroups in O'Nan-Scott.

The code in Magma and GAP for computing maximal subgroups of finite groups uses this theory, which is how I got involved in it. It has the disadvantage that it needs to know the maximal subgroups of all almost simple groups, which means that data libraries have to be compiled containing this information, which can never be complete, but it is difficult to imagine a better way of doing this.

$\mathbf{References}$

M. Aschbacher and L. Scott. Maximal subgroups of finite groups. J. Algebra, 92:44–80, 1985.

F. Gross and L.G. Kovács. On normal subgroups which are direct products. J. Algebra, 90:133–168, 1984.