Bounds on total number of subspaces of finite field

Question

Are there asymptotic upper and lower bounds on the "total" number of subspaces of $\mathbf{F}^n_q$? $q$ is fixed and $n$ increases

The following two answers are helpful but dont answer the question.

How to count number of bases and subspaces of a given dimension in a vector space over a finite field?

https://mathoverflow.net/questions/89934/sum-of-gaussian-binomial-coefficients

Answer 1

When $q=2$, these numbers are tabulated at A006116 where many references and links are given. Maybe some of those lead to other values of $q$ and more general questions. It says the number is asymptotic to $c2^{n^2/4}$, where $c = {\rm EllipticTheta}[3,0,1/2] / {\rm QPochhammer}[1/2,1/2] = 7.3719688014613\dots$ if $n$ is even and $c = {\rm EllipticTheta}[2,0,1/2] / {\rm QPochhammer}[1/2,1/2] = 7.3719494907662\dots$ if $n$ is odd.

When $q=3$, the numbers are tabulated at A006117, and again there's an asymptotic formula of the form $c3^{n^2/4}$, where $c$ is given in terms of EllipticTheta and QPochhammer. I'm sure there's more where those came from.

Answer 2

One simple way is to lower bound for $k=n/2$. Then for the specific term, denominator is at most $q^{n^2/4}$, and numerator is at least $(0.1q^n)^{n/2}$, which gives lower bound of $q^{\Omega(n^2)}$. Same can be done for upper bound. A better asymptotic bound would be nice.