Step 1: Understanding the Concept:
Solubility of alkaline earth metal hydroxides ($Group 2$) depends on the lattice energy and hydration energy of the ions.
For these hydroxides, the solubility increases significantly down the group from beryllium to barium.
Step 3: Detailed Explanation:
As we move down Group 2:
1. Both lattice energy and hydration energy decrease as the size of the metal cation ($M^{2+}$) increases.
2. However, the lattice energy decreases much more slowly than the hydration energy in the case of small cations like $Be^{2+}$.
3. Consequently, $Be(OH)_{2}$ has a very high lattice energy compared to its hydration energy, making it practically insoluble in water.
4. $Mg(OH)_{2}$ is sparingly soluble, while $Ba(OH)_{2}$ is the most soluble in this group.
5. Additionally, $Be(OH)_{2}$ is amphoteric, whereas others are basic.
Step 4: Final Answer:
The hydroxide not soluble in water is beryllium hydroxide.