To solve this problem, we need to analyze the segments and midpoints given in the problem. Let's define the length of each segment and find the relationship between them step by step.

Step 1: Assign Lengths to Segments

1. Let $AB = x$.
2. Since $C$ is the midpoint of $AB$, $AC = \frac{x}{2}$ and $CB = \frac{x}{2}$.
3. Since $D$ is the midpoint of $AC$, $AD = \frac{AC}{2} = \frac{x}{4}$ and $DC = \frac{AC}{2} = \frac{x}{4}$.
4. Since $E$ is the midpoint of $AD$, $AE = \frac{AD}{2} = \frac{x}{8}$ and $ED = \frac{AD}{2} = \frac{x}{8}$.
5. Since $F$ is the midpoint of $ED$, $EF = \frac{ED}{2} = \frac{x}{16}$ and $FD = \frac{ED}{2} = \frac{x}{16}$.
6. Since $G$ is the midpoint of $EF$, $EG = \frac{EF}{2} = \frac{x}{32}$ and $GF = \frac{EF}{2} = \frac{x}{32}$.
7. Since $H$ is the midpoint of $DB$, $DH = \frac{DB}{2}$.

Step 2: Given Information

We know that $DC = 16$.

Since $DC = \frac{x}{4}$, we have: $\frac{x}{4} = 16$ $x = 64$

Step 3: Find $GH$

We know that: $EF = \frac{x}{16} = \frac{64}{16} = 4$ $G$ is the midpoint of $EF$, so: $EG = GF = \frac{EF}{2} = \frac{4}{2} = 2$

$H$ is the midpoint of $DB$. To find $DB$, consider: $DB = DB = AB - AD - DB = 64 - 16 = 48$ Thus, $H$ is the midpoint of $DB$, so: $DH = HB = \frac{48}{2} = 24$

Now, we find $GH$, the difference between $G$ and $H$: Since $GH = EG - DH$, $GH = 2 - 24 = 22$

Therefore, $GH$ is $2$.