President Carter's effort to put a short leash on his admirals has run into their determination to maintain U.S. naval supremacy against arms-control pressures inside the administration.

Adm. Thomas B. Hayward, newly installed Chief of Naval Operations, is caught in the middle. On Aug. 18, he gave verbal orders to senior officers not to speak out against the president's veto of the defense bill (which authorized a new nuclear aircraft carrier of the Nimitz class). But those orders evoked such anger in the Navy that Hayward's private memo to flag officers a week later was an exercise in ambiguity that avoided clear support of the president.

In truth, the naval brass clearly support an override of the president's veto with nothing less at stake than naval superiority - not mere equivalence with the Soviet fleet. Arms-control advocates dominating the national-security bureaucracy oppose naval supremacy as inimical to U.S.-Soviet strategic balance. To admirals and arms controllers alike, the nuclear carrier is integral to naval supremacy.

The conflict resulted in peculiar circumstances attending the retirement of Hayward's predecessor, Adm. James L. Holloway III, two months ago. Holloway, who advocates a nuclear carrier, was denied the visit with the president traditionally accorded to a military chief completing his term.

Even more revealing was Holloway's experience in making farewell remarks at Annapolis July l. Submitting prepared remarks to theoffice of the secretary of defense June 28, the admiral was informed that five references to "maritime superiority", must be eliminated or softened. An aide to Holloway snapped that Pentagon civilans would have midshipmen exhorting their football team to "tie Army" next year.

Holloway appealed the blue-pencilling, making clear he would talk about "maritime superiority" at Annapolis, with or without approval. Deputy Assistant Secretary of Defense TonLambert rescinded all but one of the changes but insisted on keeping one.

Hayward had been no less an advocate than Holloway of naval supremacy and the nuclear carrier, but encountered new White House determination to leash the Joint Chiefs of Staff. That resolve hardened after the defense-bill veto became a centerpiece of the president's campaign for political rehabilitation.

According, meeting with several assistant chiefs and other flag officers at the Pentagon Aug. 18, Hayward supported the veto and ordered the officers not to help the override. His motives seem clear: not to jeopardize White House backing for a conventionally powered (non-nuclear) carrier next year. Besides, the president's veto then, seemed sure to be sustained.

But Hayward was rocked by criticism from naval officers over his capitulation to Mr.Carter just as support for the override was building in Congress. The result was an Aug.25 "confidential memorandum from Hayward (signed "Tom") for "eyes only" of the Navy's flag officers. It is a masterpiece of obfuscation.

"Our fundamental goal at this point must be to preserve the momentum already obtained for the early authorization and funding of a follow-on carrier to replace (the) Midway," Hayward told his fellow admirals. "My position has been, and remains, that we must start as soon as possible on a replacement for Midway, and that the replacement must be at least as capable as the ship replaced." This Delphic utterance falls obviously short of Defense Secretary Harold Brown's desire that a conventional carrier be endorsed.

Nor does Hayward's memo flatly instruct the admirals to stay clear of the veto override fight, saying instead: "It is important that we refrain from any actions that could jeopardize the existing basic support for a follow-on carrier, or diminish our influence over decisions relating to it." This could be interpreted as follows: If you lobby to override the veto, don't get caught.

Although hard-nose admirals feel Hayward abandoned naval interests, it is doubtful whether he could do more under the intense pressure he faces - as when he attended the Army Forces Policy Council meeting Aug.28. Deputy Defense Secretary Charles Duncan instructed all present to help sustain the president's veto (informing them they would be coordinated by Richard Moe, political aide to Vice President Mondale).

The admirals view such activity as incompatible not only with traditions of American naval strategy, as laid down by Adm. Alfred Mahan, but with Defense Department Directive 5100.1 instructing the Navy "to gain and maintain general navalsupremacy." In a difficult hour, Hayward's memo avoids violating that trust.