Find the number of subsets of $\{ 1,2,3,4,5,6,7,8 \}$ that are subsets of neither $\{1,2,3,4,5\}$ nor $\{4,5,6,7,8\}$.
1. **Calculate the total number of subsets of the set $\{1,2,3,4,5,6,7,8\}$:**
   \[
   \text{Total subsets} = 2^8 = 256
   \]

2. **Calculate the number of subsets of the set $\{1,2,3,4,5\}$:**
   \[
   \text{Subsets of } \{1,2,3,4,5\} = 2^5 = 32
   \]

3. **Calculate the number of subsets of the set $\{4,5,6,7,8\}$:**
   \[
   \text{Subsets of } \{4,5,6,7,8\} = 2^5 = 32
   \]

4. **Calculate the number of subsets of the intersection $\{1,2,3,4,5\} \cap \{4,5,6,7,8\} = \{4,5\}$:**
   \[
   \text{Subsets of } \{4,5\} = 2^2 = 4
   \]

5. **Apply the principle of inclusion-exclusion to find the number of subsets that are subsets of neither $\{1,2,3,4,5\}$ nor $\{4,5,6,7,8\}$:**
   \[
   \text{Subsets of neither} = 2^8 - 2^5 - 2^5 + 2^2
   \]
   \[
   = 256 - 32 - 32 + 4
   \]
   \[
   = 256 - 64 + 4
   \]
   \[
   = 196
   \]

The final answer is $\boxed{196}$.