The Government Employees Insurance Co., the largest automobile insurer in the Washington area, raised its premium rates for District of Columbia residents 8.9 percent yesterday, completing a round of increases that affect all policyholders in both the city and suburbs.

Geico's premium rates rose 10.5 percent in Maryland in September and 9.4 percent in Virginia in October.

With 35,315 automobile policyholders in the District, Geico insures one out of every five insured cars in the city. Its written premium volume of $138 million last year equaled the next three largest insurers combined. It has 159,000 policyholders in Maryland and 143,000 in Virginia, but could not say how many policyholders it has in the Washington suburbs.

Most policyholders will not feel the increases until it is time for them to renew, according to O. M. (Tony) Nicely, assistant vice president of the Chevy Chase-based insurance firm. Most Geico policies are written for one-year terms.

A typical District policyholder pays about $400 a year for a range of coverages, according to figures submitted by Geico to the D.C. Insurance Department. The new increase would push that premium to about $436.

Under D.C. insurance law, a company's proposals for increased premiums go into effect automatically after being filed with the Insurance Department. gThey may be challenged by the insurance superintendent and rolled back if he determines them to be inadequate, excessive or unduly discriminatory among customers.

Nicely said Geico's rates are not going up as fast as the increased cost of claim payments to policyholders. The cost of repairing damaged cars rose 15 percent and payments of bodily injury claims rose 19 percent last year, he said.

Since Geico last increased its rates in the District by 9 percent in April 1977 and reduced them by 3.8 percent seven months later, Insurance Superintendent James R. Montgomery III did not challenge the increase that took effect yesterday.

However, city residents were advised by the department in a legal notice published last week that they have 30 days to inspect the information filed by Geico and file comments. The department office is at 614 H St. NW.

In Virginia, where the insurance rate law is similar to the District's, Geico's most recent previous increase was 6.4 percent in December 1978.

In Maryland, where insurance companies must get their applications for higher premiums approved before they can take effect, Geico requested a 15.9 percent increase earlier this year. That was cut to the 10.5 percent increase that took effect in September. The last previous increase in Maryland was in 1976.

Nicely said Geico probably will file another application for higher rates in Maryland soon.

In the District, two more of the five largest insurers have raised their rates this year. The premiums of Nationwide Insurance, which ranks third, went up 10.5 percent in July, while fourth-ranking Dairyland Insurance raised rates 10.3 percent in October.

State Farm Mutual, the city's second-largest insurer, and Allstate, which ranks fifth, last increased their rates in 1976.