3x3:d:k:3584:3584:3841:3841:3841:6146:6146:6146:4376:3584:4100:4100:3333:3333:4376:6146:4376:3078:3584:5895:5895:1544:5401:2825:4376:3078:3078:2570:5895:5895:1544:5643:5401:2825:5401:3084:2570:3085:10254:1544:5643:5643:5401:3084:3084:3085:10254:2319:10254:1808:1808:4881:4626:4626:4627:4627:10254:2319:10254:4881:4881:4626:4626:4627:10254:5908:10254:1557:1557:1557:3606:3607:10254:5908:5908:5908:2307:2307:3606:3606:3607:

1. 16(2)n1 = {79}
-> 13(2)n2 = {58}
-> Outies n3 = r12c6 = +8(2) = [71] or {26}

2. 6(3)c4 = {123}
Outies c1234 = r127c5 = +19(3)
Since r1c5 not from (58), r2c5 from (58), and r7c5 not 5 (40(8) No 5)...
... -> H19(3) from [658], <289>, or <487>

Trying H19(3) = <487>
Whichever of (47) is in r7c5 can only go in c4 in r1c4
But 15(3)r1c3 cannot contain both (47)

Trying H19(3) = [658]
Puts r2c4 = 8, r12c6 = [71], and 22(3)n5 = [{79}6]
Also puts r4c6 = 8, r6c6 = 5, and r6c4 = 4
But this leaves no value for r1c4 since it cannot be 9

-> r127c5 = <289> and r2c4 = 5
-> 22(3)n5 = [{67}9]

3. Trying r127c5 = [289]
Puts r12c6 = [71] and 15(3)r1c3 = [492]
-> r3c456 = [3{46}]
But this leaves no solution for remaining Innies n1 = r3c23 = +11(2)
-> r127c5 = [982]

4. -> 15(3)r1c3 = [249]
-> 14(4)n1 = {1346}
-> r3c23 = {58}

5. Also -> 6(3)r8c5 = [{13}2]
-> 9(2)n8 = {45}
-> In n5 - One of (45) in r4c6 and the other in r6c5
-> (NS) r6c4 = 8
-> r7c6 = 8
-> r789c4 = {679}

6. Also (HS 2 in n7) r9c2 = 2
-> 2 in n4 in 10(2)n4 = {28}
Remaining outies n1 = r4c23 = +10(2)
Outies r6789 = r5c23 = +9(2) = [54] or [36]
-> r4c23 cannot be {46}
12(2)n4 from [57] or [39]
-> r4c23 cannot be {37}
-> r4c23 = {19}
-> 12(2)n4 = [57]
-> r5c3 = 4
-> r6c23 = {36}
-> 7(2)n5 = [52]
-> r4c6 = 4 and 9(2)n8 = [45]
Also -> 6(3)c4 = [231]
Also r123c6 = [716]
-> r3c5 = 3 and r8c56 = [13]
Also r4c7 = 5

7. Also r67c7 = [47] and r6c89 = {19}
-> r45c5 = [76]
Also r7c89 = {35}
-> 14(2)n9 = [68]
Also remaining innies D/ in n7 = +11(3)
-> Outie n7 = r9c4 = 7
-> (HS 7 in 40(8)) r8c2 = 7
-> D/ in n7 = <173>
-> 16(2)n1 = [97]
-> r4c23 = [19]
-> 1 in n1 in c1
-> D/ in n7 = [173]
-> r1c2 = 3
-> r6c23 = [63]
etc.

Prelims

a) R2C23 = {79}
b) R2C45 = {49/58/67}, no 1,2,3
c) 11(2) cage at R3C6 = {29/38/47/56}, no 1
d) R45C1 = {19/28/37/46}, no 5
e) 12(2) cage at R5C2 = {39/48/57}, no 1,2,6
f) 9(2) cage at R6C3 = {18/27/36/45}, no 9
g) R6C56 = {16/25/34}, no 7,8,9
h) R89C9 = {59/68}
i) R9C56 = {18/27/36/45}, no 9
j) 6(3) cage at R3C4 = {123}
k) 22(3) cage at R4C5 = {589/679}
l) 19(3) cage at R6C7 = {289/379/469/478/568}, no 1
m) 6(3) cage at R8C5 = {123}
n) 14(4) cage at R1C1 = {1238/1247/1256/1346/2345}, no 9
o) 40(8) cage at R5C3 = {12346789}, no 5

1a. Naked pair {79} in R2C23, locked for R2 and N1, clean-up: no 4,6 in R2C45
1b. Naked pair {58} in R2C45, locked for R2 and N2, clean-up: no 3,6 in R4C7
1c. Naked triple {123} in 6(3) cage at R3C4, locked for C4, clear-up: no 6,7,8 in R6C3
1d. Naked triple {123} in 6(3) cage at R8C5, locked for R8
1e. 22(3) cage at R4C5 = {589/679}, 9 locked for N5
1f. Killer triple 1,2,3 in R45C4 + R6C56, locked for N5

2a. 45 rule on N3 2 outies R12C6 = 8 = [26/62/71]
2b. 45 rule on R6789 2 outies R5C23 = 9 = [36/54/72/81], no 9, no 4 in R5C2, no 3,7,8 in R5C3, clean-up: no 3,8 in R6C3
2c. 45 rule on N1 2 outies R4C23 = 1 innie R1C3 + 8, IOU no 8 in R4C2
2d. 45 rule on C1234 3 outies R127C5 = 19 = {289/478/568} (cannot be {379/469} because R2C5 only contains 5,8), no 1,3, 8 locked for C5, clean-up: no 1 in R9C6
2e. 15(3) cage at R1C3 = {249/456} (cannot be {159/168/258/348/357} because 1,3,5,8 only in R1C3, cannot be {267} which clashes with R1C6), no 1,3,7,8, 4 locked for R3
2f. 5 of {456} must be in R1C3 -> no 6 in R1C3
2g. R12C6 = [62/71] (cannot be [26] which clashes with 15(3) cage)
2h. 3 in N2 only in R3C456, locked for R3
2i. 3 in N1 only in 14(4) cage at R1C1 = {1238/1346} (cannot be {2345} which clashes with R1C3), no 5, 1 locked for N1

3a. 7 in C4 only in R6789C4, CPE no 7 in R7C5
3b. R127C5 (step 2d) = {289/568}, no 4
3c. 6 of {568} must be in R1C5 -> no 6 in R7C5
3d. 22(3) cage at R4C5 = {679} (cannot be {589} which clashes with R127C5, ALS block), locked for N5, clean-up: no 1 in R6C56
3e. 1 in N5 only in R45C4, locked for C4
3f. 8 in N5 only in R4C4 + R6C6, locked for D/
3g. Killer triple 6,7,9 in R127C5 and R45C5, locked for C5, clean-up: no 2,3 in R9C6
3h. Hidden killer triple 6,7,9 in R1C45 and R13C6 for N2, R1C45 contains one of 6,9 (step 2e), R1C6 = {67} -> R3C6 = {679}, R4C7 = {245}
3i. Naked triple {679} in R135C6, locked for C6, clean-up: no 2,3 in R9C5
3j. 6 in N8 only in R789C4, locked for C4

4a. 15(3) cage at R1C3 (step 2e) = {249/456}, R127C5 (step 3b) = {289/568}
4b. R127C5 = {289} (cannot be {568} = [658] because R6C4 = 4 clashes with 15(3) cage = [546]) -> R17C5 = {29}, locked for C5, R2C5 = 8 -> R2C4 = 5, clean-up: no 4 in R6C3, no 5 in R6C6
[Probably clearer as a forcing chain.
Consider placements for R6C6 = {48}
R6C6 = 4 => R1C4 = 9 => 15(3) cage = [492] => R127C5 = [289]
or R6C6 = 8 => R7C5 = {29} => R127C5 = {289}
-> R127C5 = {289} -> R17C5 = {29}, locked for C5, R2C5 = 8 -> R2C4 = 5, clean-up: no 4 in R6C3, no 5 in R6C6]
4c. 15(3) cage = {249} (only remaining combination), locked for R1, 9 locked for N2, clean-up: no 2 in R4C7
4d. Naked pair {67} in R45C5, locked for N5 -> R5C6 = 9, clean-up: no 1 in R4C1
4e. 5 in N1 only in R3C12, locked for R3 and 23(4) cage at R3C2
4f. Killer pair 2,4 in 14(4) cage at R1C1 (step 2i) and R1C3, locked for N1

5a. 3 in R3 only in R3C45, 5 in R3 only in R3C23
5b. 45 rule on R123 5 innies R3C23456 = 24 = {13578/23568}, no 4
[The last key step; fairly straightforward from here]
5c. R1C4 = 4 (hidden single in N2) -> R1C35 = [29], R6C4 = 8, R7C5 = 2, clean-up: no 7 in R5C2 (step 2b), no 5 in R6C1, no 1,5 in R6C3, no 7 in R7C4
5d. 9(2) cage at R6C3 = [36], clean-up: no 7 in R45C1, no 6 in R5C3 (step 2b), no 9 in R6C1, no 4 in R6C56
5e. R6C56 = [52], R4C6 = 4, placed for D/, R4C7 = 5 -> R3C6 = 6, clean-up: no 6 in R5C1
5f. 2 on D/ only in R2C8 + R3C7, locked for N3
5g. 12(3) cage at R2C9 = {138/147}, no 9, 1 locked for N3
5h. R3C7 = 9 (hidden single in N3), placed for D/, R2C6 = 1 -> R1C9 + R2C8 = 7 = [52], clean-up: no 9 in R89C9
[No more clean-ups]
5i. R8C567 = [132], R9C5 = 4 -> R9C6 = 5
5j. R7C6 = 8 -> R67C7 = 11 = {47}, locked for C7
5k. R1C6 = 7, R1C78 + R2C7 = {368}, locked for N3 -> R2C9 = 4, R3C89 = {17}, 1 locked for R3
5l. R3C23 = {58}, 8 locked for N1 and 23(4) cage at R3C2 -> R4C23 = 10 = {19}, locked for R4 and N4
5m. R7C3 + R9C1 = [13] (hidden pair on D/)
5n. R3C1 = 4, R6C1 = 7 -> R5C2 = 5, R5C3 = 4, R6C2 = 6, R8C2 = 7, placed for D/, R5C5 = 6, placed for D\, R2C2 = 9, placed for D\, R9C9 = 8, placed for D\ -> R3C3 = 5, placed for D\

and the rest is naked singles, without using the diagonals.

Preliminaries by SudokuSolver
Cage 16(2) n1 - cells ={79}
Cage 14(2) n9 - cells only uses 5689
Cage 7(2) n5 - cells do not use 789
Cage 12(2) n4 - cells do not use 126
Cage 13(2) n2 - cells do not use 123
Cage 9(2) n48 - cells do not use 9
Cage 9(2) n8 - cells do not use 9
Cage 10(2) n4 - cells do not use 5
Cage 11(2) n26 - cells do not use 1
Cage 6(3) n89 - cells ={123}
Cage 6(3) n25 - cells ={123}
Cage 22(3) n5 - cells do not use 1234
Cage 19(3) n689 - cells do not use 1
Cage 14(4) n1 - cells do not use 9
Cage 40(8) n4578 - cells ={12346789}

This is a very highly optimised solution. No clean-up done unless written.
1. 16(2)n1 = {79}: both locked for n1 and r2
1a. -> 13(2)n2 = {58}: both locked for n2 and r2

2. 6(3)r3c4 = {123}: all locked for c4

3. "45" on c1234: 3 outies r127c5 = 19
3a. must have 5 or 8 for r2c5
3b. = {289/478/568}(no 1,3)
3c. must have 8: locked for c5
3d. 6 in {568} must be in r1c5 -> no 6 in r7c5

4. "45" on n3: 2 outies r12c6 = 8 = [71]/{26}

5. 15(3)r1c3 must have two of {24679} for r1c45
5a. but {267} blocked by r1c6 = (267)
5b. = {249/456}(no 1,3,7,8)
5c. must have 4: locked for r1

6. 7 in c4 only in r6789c4 -> no 7 in r7c5 (Common Peer Elimination CPE)

7. h19(3)r127c5 = {289/568}(no 4) = 5 or 9

8. 22(3)n5: {59}[8] blocked by h19(3)
8a. = {679} only: all locked for n5
8b. no 1 in 7(2)n5

9. 15(3)r1c3 = {249/456}
9a. {456} = [546] only
9b. -> no 6 in r1c34
9c. note: 9 in r1c4 -> 2 in r1c5

key step
10. r16c4 = {489}
10a. either r1c4 = 9 -> r1c5 = 2
10b. and/or r6c4 = 8 -> no 8 in r7c5
10c. -> [658] blocked from h19(3)r127c5
10c. = {289} only
10d. -> r2c45 = [58]
10e. r17c5 = {29}: both locked for c5
10f. -> r45c5 = {67}: both locked for c5
10g. and r5c6 = 9

11. 15(3)r1c3 = {249} only: 2 and 9 locked for r1
11a. no 6 in r2c6 (h8(2))

12. hidden pair {67} in r13c6 for n2: both locked for c6
12a. 11(2)r3c6 = [65/74]

13. 3 in n2 only in r3: locked for r3

14. 3 in n1 only in 14(4)
14a. but {2345} blocked by r1c3 = (24)
14b. = {1238/1346}(no 5) = 2/4
14c. must have 1: locked for n1

15. killer pair 2,4 in 14(4) & r1c3: both locked for n1

16. 5 in n1 only in 23(4): locked for 23(4) and r3
(note: can now do Andrew's really nice step 5b instead of below)

Crack
17. "45" on r6789: 2 outies r5c23 = 9 = [81/72/36/54](r5c2 = (3578), r5c3 = (1246))
17a. 7(2)n5 = {34}/[52]
17b. if 5 in n4 in r6c1 -> 7(2)n5 = {34} [Andrew noticed both r6c13 do this so part of the next substep is redundant]
17c. or 5 in the h9(2)r5c23 or 9(2)r6c3 -> must have 4
17d. 4 locked for all those 5s -> no 4 in r6c4
17e. r6c4 = 8, placed for d/
17f. 9(2)r6c3 = [27/36/54]

18. r6c3 = (235) and 7(2)n5 = {52]/{34}
18a. -> killer single 3: locked for r6

19. 3 which must be in 40(8)r5c3 is only in n7: 3 locked for n7 and d/

20. 17(4)r1c9 = {1259/1457/2456}
20a. must have 5 -> r1c9 = 5: placed for d/

21. r6c5 = 5 (hsingle n5), r6c6 = 2: placed for d\

Much easier now. Don't forget the diagonals.