Simple groups satisfying some conditions

Question

We are looking for an example of simple groups $G$ of order $n$ such that the following condition (*) does not hold:

(*) for every factorization $n=ab$ there exists a non-trivial subgroup of $G$ whose index is a divisor of $a$ or $b$.

Note that:

1. If $G$ has a subgroup of prime index, then (*) holds, and so the simple groups of orders 60, 168, 660 etc., are removed from the list (of finite simple groups).

2. Also, (*) holds for the simple group of order 360 (with no subgroup of prime index).

Therefore, order of the first example, if exists, must be a number equal or greater than 2448.

Note. I've made a correction for this question now, since the mentioned item 3 ( The simple groups of orders 504 and 1092 also admit (*)) was wrong. Indeed, the first three examples are $PSL(2,8)$ of order 504 with $(a,b)=(12,42), (21,24)$, $PSL(2,13)$ of order 1092 with $(a,b)=(21,52)$, and (as it is mentioned by Derek Holt) $PSL(2,17)$ of order 2448 with $(a,b)=(48,51)$. Therefore, this question has been answered.

Any idea for the problem?

Thanks in advance.