Yes, if the organization has the following structure:
Board With A Volunteer Chairperson
President/CEO With Full Authority for Operations
Executive Director for Division A
Executive Director for Division B
However this structure can be confusing to persons in the nonprofit arena. The executive director should have final authority for all operational matters related to the organization, except those designated for the board in the bylaws such as pensions plan changes. Under this structure the president/CEO -- a full-time employee, not the executive directors, has operational responsibility. This leads to external confusion as to the authority of the executive directors, which might lead to legal issues.
The big question is who carries the CEO title? Some nonprofits, in their early stages, have a volunteer, part-time President/CEO with operational authority, as well as board chair responsibilities. This signifies the volunteer, representing the will of the board, can have final authority in all daily issues. It is not a good structure because the CEO title might require the volunteer to have personal liabilities that other board members don't have.
In addition, it can lead to board micromanagement, a culture that is difficult to change as the organization grows. Such an arrangement can even directly hamper growth because experienced chief executives often won't apply for senior positions that have such arrangements, with decisions involving an additional board layer.
I know of one nonprofit that organizationally had its division heads directly reporting to to volunteer board committee chair-people, such as having the CFO reporting to the head of the finance committee. For major operational decisions, the division heads first consulted with the board committee chair then relayed the information to the executive director.
The following structure, based on my experiences, is best:
Board With A Volunteer Chairperson
President/CEO With Full Authority for Operations
Vice President Division A
Vice President Division B
Under this arrangement, the board, as clearly as possible, defines the CEO's operational responsibilities and allows the president/CEO to manage the organization,along with robust board evaluations each year.
Board Members: Try the above arrangement, and you will like it. A great deal of the success of the model develops on bases of organizational trust and the willingness of the senior manager, president/CEO, to accept the full managerial responsibility involved. He/S is not viewed as a "servant" of the board!!