The most baffling and threatening domestic question facing Egypt's President Hosni Mubarak's is: How broad and how deep is the Islamic militant opposition to basic Sadat-Mubarak policies?
Mr. Mubarak sees the menace from that quarter as dangerous enough for him to have arrested several hundred more suspected Islamic militants over the past few days. The Egyptian Interior Ministry reported the arrests of 230 Muslim fundamentalists on Oct. 18 alone. The number of people under detention is now reported to be about double the 1,600 rounded up in the late President Sadat's early September crackdown. The majority of those original 1,600 were Muslim fundamentalists.
Mr. Sadat paid for that crackdown with his life - apparently at the hands of a small hard-line militant Islamic group known as Takfir Wal Hijra (Repentance and Holy Flight).
Takfir Wal Hijra is hardly broad-based enough to pose an effective political alternative to the regime. But is there a fundamentalist movement broader in scope that is potentially capable of bringing the regime down - or at least of severely destabilizing it?
The answer is yes - the Muslim Brotherhood, founded in 1928 by the late Hassan Banna. (That ''yes'' must be qualified. Given the nature of contemporary Egyptian society, any coup attempt would need support from within the Army to be successful.)
Saad Eddin Ibrahim, professor of sociology at the American University in Cairo, in a paper given in June at the Institute of Arab Studies in Belmont, Mass., said:
''The Muslim Brotherhood's past history suggests its tremendous potential to become a grass-roots mass movement. Its Islamic ideology has cultural legitimacy. Its political stands on most current issues are in tune with both Egyptian patriotism (independence and nonalignment) and Arab national struggle (anti-Zionism and anti-imperialism). . . .At present the Muslim Brotherhood's image is generally positive. This is evidenced by the landslide victories of the Muslim Brotherhood's supporters in student elections of nearly all Egyptian universities. Historically the latter have been a sensitive barometer of Egyptian public opinion.''
Professor Ibrahim said that ''no one can speak with authority on the Muslim Brotherhood's actual membership size. . . .From the outside, (it looks like) a movement of thousands of youngsters in their 20s led and inspired by figures 40 years their senior.''
The Cairo correspondent of the French newspaper Le Monde, recently expelled from Egypt, has put the across-the-board strength of Islamic militants and their sympathizers in Egypt at an estimated half million.
What made Mr. Sadat vulnerable to these militants was the meagerness of the tangible results - at least in the eyes of the urban middle and lower middle class - of the four main lines of policy that had been distinctively his since the mid-1970s.
The four lines, all a departure from those of President Nasser, were: an open door in the economic field to encouraging the private sector, domestic and foreign; democratization; alliance with the West, particularly the US; and reconciliation with Israel.
Together, they constituted the basis for Mr. Sadat's vision of Egypt's future: modernized, pro-West, strong, and prosperous. Mr. Mubarak's professed intent at the outset of his presidency is to continue to build on the same foundation.
Unfortunately, the Islamic fundamentalists - with the Muslim Brotherhood the most openly vocal of them - have come to see things quite differently. To them, the Sadat-Mubarak policies are leading to social injustice, corruption, the destruction of traditional Muslim values, and the humiliation of Islam at Western and Israeli bidding.
(For the record, it should be noted that in the days of Egypt's close association with the Soviet Union prior to 1972, Islamic militants denounced that alignment with ''atheistic Marxism'' just as they are criticizing Mr. Sadat's subsequent swing to the West and reconciliation with Israel.)
Against this background, it is easier to understand why Mr. Sadat was giving such high priority to securing Israel's withdrawal from the last segment of Egyptian territory in Sinai by April 1982. Completion of that withdrawal would be proof - as he saw it and as Mr. Mubarak now sees it - that the Camp David approach removed an Israeli humiliation from Egyptian soil.
Mr. Sadat's September crackdown was intended partly to silence militant fundamentalists so that their increasingly strident anti-Israel propaganda would give the Israelis no pretext for delaying the April withdrawal.
Mr. Sadat may also have been concerned on two other counts. The first, that the Muslim Brotherhood was having success in forming tactical alliances and coalitions with other groups, some of them secular. And the second, that if it was not checked, the group might feel confident enough to resort to the violence of the '40s and '50s.
Since Mr. Sadat's assassination, some Egyptians have suggested that he should have moved sooner against the brotherhood and other Islamic militants. If there was any reason for his delaying his crackdown until early September, it probably lay in his equivocal relations with the Muslim Brotherhood during most of the 1970s.
When he first took over from President Nasser, Mr. Sadat used the brotherhood - and this strongly anti-Nasser group acquiesced - to weaken the residual strength of last-ditch Nasserites in the armed forces and elsewhere.
This coexistence under the umbrella of democratization worked initially. But by this summer, Muslim Brotherhood attacks on Mr. Sadat in the movement's publications had become so strident, particularly on peace with Israel, that he threw down the gauntlet and challenged them to a showdown. In the process he has been killed. But his successor, President Mubarak, shows signs of willingness to be tougher still.