Chesterfield FC CEO Graham Bean says a number of serious allegations, including one of racial abuse, have been made after incidents involving Town fans during today's clash with Ebbsfleet.
There were a number of flash-points involving home supporters and away keeper Nathan Ashmore.
He confronted fans at the Kop end towards the end of the game and some supporters ran onto the pitch after Chesterfield's dramatic injury time equaliser.
The club's CEO condemned the actions that he says 'overshadow' the result.
"After what has been a tremendous comeback this afternoon, the result has been overshadowed by a number of serious allegations which have been made, resulting in the detention of at least one supporter," he said.
"There are a number of allegations, one of racial abuse to a player, one of physical contact with an opposition player and that is already the subject of an investigation by the police. The FA have already been informed. This does not make good headlines for the club."
Fans invaded the pitch during a recent game and Mr Bean warned of potential FA sanctions, including points deductions in extreme cases.
He believes after today, Chesterfield's situation has become an extreme case.
"Yet another pitch incursion means that we will inevitably face an FA investigation. It is likely that we will have to appear before the FA.
"I cannot stress enough how serious this is now becoming. As a club we totally condemn the actions of a few and anybody who is found to have been involved in these matters will face very serious consequences from the club.
"I as the chief executive, the board and the owner, we are not prepared to tolerate this type of behaviour and if we have to deal with this in a very severe way, to get the message through, then that is what we're going to have to do.
"As I said a couple of weeks ago, in extreme cases the FA can deduct points. We are now in a situation where this is an extreme case.
"Supporters, from now until the end of the season have to fully understand the implications of the behaviour of a minority connected to this club are doing to the reputation of it."
The FA investigated an incident during a November Ebbsfleet game, when Ashmore jumped into the crowd to confront Boreham Wood fans over abuse he received.