Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that
there exists a subgroup of $G$ with order $d$ **or** $n/d$?

Of course this does not hold in full generality. -- In particular one quickly finds that e.g. ${\rm PSL}(2,8)$, ${\rm PSL}(2,11)$, ${\rm PSL}(2,13)$, ${\rm SL}(2,11)$, ${\rm SL}(2,13)$, ${\rm PSL}(2,17)$, ${\rm A}_7$, ${\rm PSL}(2,19)$, ${\rm A}_5 \times {\rm A}_5$, etc. are counterexamples.

Though does the assertion hold for

solvable groups?

solvable groups with derived length less than a certain bound $> 2$?

groups whose order has at most 2 distinct prime divisors?

A quick computation with GAP shows that any counterexample must have order $\geq 192$.