MySQL SELECT slow, but only 2 x 300K rows and indexes

Have the MySQL SELECT query below that is awfully slow.

It takes ~1.0 seconds to execute despite having only 300K rows and indexes, so I would love to find a way to get it to execute faster since it's a query that needs to be run again and again.

The query:

SELECT p.id, p.image, c.name, s.name, MIN(p.saleprice)
FROM products p 
JOIN shops s ON p.shopid = s.id 
JOIN products_category pc ON p.id = pc.product_id 
JOIN categories c ON pc.category_id = c.id
WHERE p.brand_id > 0
AND pc.category_id = 46
AND pc.active = 1
AND p.price > 0
AND p.saleprice > 0
AND p.saleprice < p.price
AND (p.last_seen > DATE_SUB(NOW(), INTERVAL 2 DAY))
GROUP BY p.image

The query returns 960 rows.

The table products has 300.000 rows and these columns + one index:

CREATE TABLE `products` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `name` varchar(512) DEFAULT NULL,
  `shopid` int(11) DEFAULT NULL,
  `image` varchar(512) DEFAULT NULL,
  `price` int(11) DEFAULT NULL,
  `saleprice` int(11) DEFAULT NULL,
  `last_seen` datetime DEFAULT NULL,
  `brand_id` int(11) DEFAULT NULL,
  PRIMARY KEY (`id`),
  KEY `brand_id etc` (`brand_id`,`shopid`,`last_seen`,`price`,`saleprice`)
) ENGINE=InnoDB AUTO_INCREMENT=391671 DEFAULT CHARSET=utf8

The table products_categories also has 300.000 rows and these columns + two indexes:

CREATE TABLE `products_category` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `product_id` int(11) DEFAULT NULL,
  `category_id` int(11) DEFAULT NULL,
  `active` int(11) DEFAULT NULL,
  PRIMARY KEY (`id`),
  KEY `category_id etc` (`category_id`,`active`),
  KEY `product_id etc` (`product_id`,`active`)
) ENGINE=InnoDB AUTO_INCREMENT=373364 DEFAULT CHARSET=utf8

The shops table. It has 15 rows and these columns:

CREATE TABLE `shops` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `name` varchar(256) DEFAULT NULL,
  `active` int(11) DEFAULT '1',
  PRIMARY KEY (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=56 DEFAULT CHARSET=utf8

Based on similar questions here, I have tried nesting things with an inner select:

SELECT p.id, p.image, c.name, s.name, MIN(p.saleprice)
FROM 
(SELECT * FROM products WHERE p.brand_id > 0 AND price > 0 AND saleprice > 0 AND saleprice < price AND (p.last_seen > DATE_SUB(NOW(), INTERVAL 3 DAY))) p 
JOIN shops s ON p.shopid = s.id 
JOIN products_category pc ON p.id = pc.product_id 
JOIN categories c ON pc.category_id = c.id 
WHERE pc.category_id = 46
AND pc.active = 1
GROUP BY p.image

It didn't help. The version with the inner select takes ~1,3 seconds to execute.

The problem seems to be the join between products and products_category, i.e. the two big tables with 300K rows each.

The GROUP BY p.image is necessary (some products are just different sizes, so we use p.image to filter these out).

Maybe there's a trick I can do with my indexes? Or can any of you spot something else I should optimize?

EXPLAIN of the query:

id  select_type table   partitions  type    possible_keys                   key             key_len ref             rows    filtered    Extra
1   SIMPLE      c       \N          const   PRIMARY                         PRIMARY         4       const           1       100.00      Using temporary; Using filesort
1   SIMPLE      pc      \N          ref     category_id etc,product_id etc  category_id etc 10      const,const     43104   100.00      Using where
1   SIMPLE      p       \N          eq_ref  PRIMARY,brand_id etc            PRIMARY         4       pc.product_id   1       5.00        Using where
1   SIMPLE      s       \N          eq_ref  PRIMARY                         PRIMARY         4       p.shopid        1       100.00      \N