TITLE: The Möbius number of the nonabelian finite simple groups
QUESTION [13 upvotes]: Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The  Möbius function $\mu$ for $L$ is defined recursively by: for  $\forall a,b \in L$ with $a<b$,  $\mu(b,b) = 1$ and $\mu(a,b) = -\sum_{a<c\le b}\mu(c,b)$.
The Möbius number of $L$ is defined by $\mu(\hat{0},\hat{1})$.     
Define the Möbius number $\mu(G)$ of a finite group $G$ to be the Möbius number of its subgroup lattice. For the nonabelian finite simple groups of small order, we observe that $$\mu(G) \in |G| \mathbb{Z}$$
Question: Is it true for any nonabelian finite simple group? 
Remark: It is true if $|G| < 100000$ (see the table below).
It's false for nonsimple groups because for $p$ (odd) prime, $\mu(S_p) = p!/2$ ([S]) and  $\mu(C^2_p) = p$.

Table for the nonabelian finite simple groups of order $< 100000$:
$$ \begin{array}{c|c|c|c|c|c}
G & |G| & \mu(G) & \mu(G)/|G| & |Out(G)| & \newline 
 \hline  
  A_5 & 60     &  -60  & -1 & 2 & \newline 
 \hline  
  PSL(2,7) & 168    &  0 & 0 & 2&   \newline 
 \hline
  A_6 & 360     &   720   & 2 & 4 & \newline 
 \hline
  PSL(2,8) & 504     &   -504   & -1 & 3  \newline 
 \hline
PSL(2,11) & 660     &  660 & 1 & 2 \newline 
 \hline
PSL(2,13) & 1092     &  -1092 & -1 & 2 \newline 
 \hline
  PSL(2,17) & 2448    &   0 & 0 & 2 \newline 
 \hline
  A_7 & 2520        &  2520 & 1 & 2  \newline 
 \hline
  PSL(2,19)& 3420     &   3420 & 1 & 2 \newline 
 \hline
  PSL(2,16)& 4080     &  0 & 0 & 4 \newline 
 \hline
  PSL(3,3)& 5616      &  0 & 0 & 2 \newline
 \hline
   PSU(3,3)& 6048     &  0 & 0 & 2 \newline
 \hline
  PSL(2,23)& 6072     &  0 & 0 & 2 \newline
 \hline
   PSL(2,25)& 7800    &  0 & 0 & 4 \newline
 \hline
   M_{11} & 7920      &   -7920 & -1 & 1 \newline
 \hline
  PSL(2,27)& 9828     &   9828 & 1 & 6 \newline
 \hline
   PSL(2,29) & 12180 &   12180 & 1 & 2 \newline
 \hline
  PSL(2,27)& 14880 &   29760 & 2 & 2 \newline
 \hline
  A_8& 20160 &   20160 & 1 & 2 \newline
 \hline
  PSL(3,4)& 20160 &   -120960 & -6 & 12 \newline
 \hline
  PSL(2,37)& 25308&   -25308& -1 & 2 \newline
 \hline
PSp(4,3)& 25920& -25920& -1   & 2  \newline
 \hline
Sz(8)& 29120& -29120& -1   & 3  \newline
 \hline
PSL(2,32)& 32736& -32736& -1   & 5  \newline
 \hline
PSL(2,41)& 34440& 68880& 2   & 2  \newline
 \hline
PSL(2,43)& 39732& -39732& -1   & 2  \newline
 \hline
PSL(2,47)& 51888& 0& 0   &  2 \newline
 \hline
PSL(2,49)& 58800& 117600& 2   &  4 \newline
 \hline
PSU(3,4)& 62400& 0& 0   & 4  \newline
 \hline
 PSL(2,53)& 74412& -74412& -1  & 2  \newline
 \hline
M_{12}   & 95040 &  95040  & 1  & 2  \newline
\end{array}$$

REPLY [17 votes]: The answer to your question is ``yes".  Something more general was proved by Hawkes, Isaacs and \"Ozaydin in a 1989 paper in the Rocky Mountain Journal of Mathematics.
CORRECTION: As Sebastien Palcoux notes below, this result is due to Kratzer and Th\'evenaz
More precise results for the groups $PSL_2(p)$ were known to Philip Hall, and $PSL_2(q)$ was handled by Martin Downs, as were the Suzuki groups.
Some results on symmetric groups appear in a paper of mine, and other classes of ``small" almost simple groups are addressed in my thesis,
http://www.math.wustl.edu/~shareshi/thmsri.ps
(Sorry, it is a postscript file and some non-mathematical pages are repeated.)