An Indonesian family bought its eight-year-old daughter to a suicide bomb attack it launched on Monday on the police headquarters in the country's second-largest city, authorities said, a day after members of another family conducted coordinated suicide bombings on three city churches that killed 12 people.

National police chief Tito Karnavian said the girl, who was with two of the attackers on a motorcycle, survived being thrown by the blast at Surabaya's police headquarters.

The attack killed the four perpetrators. Six civilians and four officers were wounded.

The militant Islamic State (IS) group claimed responsibility for the church bombings in a statement carried by its Aamaq news agency. Karnavian, however, said earlier police comments that the family had spent time in Syria were incorrect.

He said the church bombers and the police headquarters attackers were friends, as were another family whose homemade bombs exploded in their apartment Sunday night.

The flurry of bombings raised concerns that previously beaten-down militant networks in the world's most populous Muslim-majority nation have been reinvigorated by the return of some of the estimated 1,100 Indonesians who went to fight with the militant Islamic State (IS) group in Syria.

Experts have warned for several years that when those fighters return, they could pose a significant threat.

The use of children in the attacks has been particularly horrifying to people.

“This is terrifying,” said Taufik Andrie, executive director of an institute that runs programmes to help paroled militants reject extremism and rejoin society. “This is showing how extremist ideology can entrap children. Children have no choice. They can't comprehend the decisions involved.”

All told, 25 people have died since Sunday including a total of 13 militants and their children.

The blast came just hours after police said one family, including girls aged 9 and 12, had carried out the church bombings.

The flurry of attacks have raised concerns that previously beaten down militant networks in the world's most populous Muslim-majority nation have been reinvigorated by the return of some of the estimated 1,100 Indonesians who went to fight with the militant Islamic State (IS) group in Syria.

Experts have warned for several years that when those fighters return, they could pose a significant threat.

Police say the family that carried out Sunday's suicide bombings had indeed returned to Indonesia from Syria, and IS claimed responsibility for those bombings in a statement carried by its Aamaq news agency.

CCTV footage of the attack on Surabaya's police headquarters showed at least one explosion after two motorcycles, each with two people aboard, drove into a security checkpoint. The motorbikes, which moved closely together, pulled up alongside a car and four officers manning opposite sides of the checkpoint.

Two men, apparently civilians, were walking into the checkpoint just metres from the motorbikes at the moment of the explosion, which a split second later was followed by a second possible blast.

Indonesia's president Joko “Jokowi” Widodo condemned the string of attacks as “barbaric” and vowed that authorities would root out and “destroy” militants.

National police chief Tito Karnavian has said the father of the family that carried out the church bombings was head of the Surabaya cell of Jemaah Anshorut Daulah, an Indonesian militant network affiliated with IS that has been implicated in attacks in Indonesia in the past year. All six members of the family were killed.

The IS statement claiming the attacks didn't mention anything about families or children taking part and said there were only three attackers. The group also claimed responsibility for a hostage-taking ordeal last week by imprisoned militants at a detention centre near Jakarta in which six officers were killed.

Separately on Sunday, three members of another family were killed when homemade bombs exploded at an apartment in Sidoarjo, a town bordering Surabaya, police say.

Indonesia's deadliest terrorist attack occurred in 2002, when bombs exploded on the tourist island of Bali, killing 202 people in one night, mostly foreigners.

The church attacks occurred within minutes of each other, according to Mangera.

Karnavian said the father drove a bomb-laden car into the city's Pentecostal church. The mother, with her two daughters, attacked the Christian Church of Diponegoro, he said. Based on their remains, Karnavian said the mother and daughters were all wearing explosives around their waists.

The sons aged 16 and 18 rode a motorcycle onto the grounds of the Santa Maria Church and detonated their explosives there.