To solve this problem, we need to understand the concept of an "ideal company" as defined by the question. An ideal company is a subset of the student group where all female students in the subset are liked by all male students within the same subset, and no additional student can be added to this group without breaking this condition.

Given:
- There are $9$ female students.
- There are $15$ students in total, which means there are $15 - 9 = 6$ male students.

### Understanding the Structure:

The structure of an ideal company is primarily governed by the fact that all males in the subset like all females in the subset. This means we can select any number of males from the $6$ available, and for each selection of males, we can select any number of females from the $9$ such that they all meet the condition.

### Calculating the Number of Ideal Companies:

1. Choose any subset of the 6 male students. Since each male student either is in the subset or isn't, there are $2^6$ ways to select the male students.

2. Select females based on the condition. Importantly, if there are no males, any subset of the females is possible (including the empty set). For each non-empty set of males, there is at least one possible subgroup of females (at least the empty set since there's no requirement for females to be actually present), but adding any more females still satisfies the condition given in the question because no constraint is imposed on it from their side. The safest assumption is that adding females happens freely once all males like any females present.

3. Combining the Selections:
- Since males can choose any situation from $2^6$,
- and each male subset can have any specific or empty subset of females without additional constraints,

The power set condition implicitly mixed while counting females with males gives a general indicator form as $2^6$, but as decided above since it has to include no unilateral additions by females other than blank, $2^0$, or varied inside settings.

Since females have no further subdivisions outside selected males containing them as decided, it's clear: $2^6 + 2^3$.

### Result:

The total number of different ideal companies that can be formed is:
$$\boxed{512}.$$