WASHINGTON — They have had their fleeting moments on cable television. Their closed-door run-ins with Speaker John A. Boehner spill occasionally into the pages of Capitol Hill newspapers. But outside their districts, and sometimes even within them, few have heard of the conservative cadre of House Republicans who have led the charge to shut down the government.

In contrast to 1995, when Speaker Newt Gingrich led his band of “revolutionary” Republicans into the last battle that shuttered the federal government, this time a small but powerful group of outspoken conservative hard-liners is leading its leaders — and increasingly angering a widening group of fellow Republicans.

“We’ve passed the witching hour of midnight, and the sky didn’t fall, nothing caved in,” said Representative Steve King, Republican of Iowa, who still believes Republicans can achieve “the end of Obamacare.” “Now the pressure will build on both sides, and the American people will weigh in.”

Mr. King is part of a hard-core group of about two dozen or so of the most conservative House members who stand in the way of a middle path for Mr. Boehner that could keep most of his party unified while pressuring the Senate to compromise. Their numbers may be small, but they are large enough to threaten the speaker’s job if he were to turn to Democrats to pass a spending bill that reopened the government without walloping the health law. Their strategy is to yield no ground until they are able to pass legislation reining in the health care law; if the federal government stays closed, so be it.