There are apparently $3072$ ways to draw this flower. But why?

First you have to draw the petals. There are $4!=24$ ways to choose the order of the petals and $2^4=16$ ways to choose the direction you go around each petal. Then you go down the stem to the leaves. There are $2! \cdot 2^2=8$ ways to draw the leaves. Finally you draw the lower stem. $24 \cdot 16 \cdot 8=3072$

At the beginning you could go 8 different ways, then you could go 6 different ways, then you could go 4 and 2 different ways but in the down of the picture you could go at first 4 different ways and 2 at the end. $8\cdot6\cdot4\cdot2\cdot4\cdot2 = 3072$

Looking at the picture, there are 4 phases.

1. Draw the petals
2. Draw the upper stem
3. Draw the leaves
4. Draw the lower stem

Lets label these $A,B,C,D$. Clearly, the total number of ways to draw the flower is simply;

$$Total = A \times B \times C \times D$$

We can see that the upper and lower stem are un-ambiguous; i.e. there is only one way to draw them. Thus $B=D=1$. So our equation becomes

$$Total = A \times C$$

Now lets look at the leaves first. The factors in it are:

a. Which leaf do you draw first?
b. What direction do you use for the first leaf?
c. What direction do you use for the second leaf?

There are 2 possibilities for each, so $C=2\times 2 \times 2=8$.

Now lets look at the petals.

a. Which petal do you draw first? (4 choices)
b. Which petal do you draw second? (3 choices)
c. Which petal do you draw third? (2 choices)
d. Which direction do you draw the X petal? (2 choices for each petal)

So $A=4 \times 3 \times 2 \times 1 \times 2^4=24 \times 16=384$.

And

$$Total=A \times C = 384 \times 8 = 3072$$