What are the finite groups with 8 or 16 conjugacy classes?

Question

What are the list of finite groups with 8 or 16 conjugacy classes?

I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, $|D_{13}|=26$. Or some people may denote $D_{10}$ as $D_{20}$.) Of course we have trivial examples $Z_8$ and $Z_{16}$ have 8 or 16 conjugacy classes for each.

Are there other examples of non-Abelian groups with 8 or 16 conjugacy classes? I am mostly interested in the non-Abelian groups. Thank you. :o)

Add: Partial answers are fine. (Such as answering what Jack Schmidt points out there are 18 isomorphism classes of 8 conjugacy classes.)

Answer 1

Here are the 101* finite nonabelian groups with 16 conjugacy classes whose order is less than 2000. (*I don't remember if there are any exceptional orders left out of the census.) The names are those given by GAP's structure description, and have not been cleaned up for this answer since there are so many groups.

• SmallGroup(40,1) = C5 : C8
• SmallGroup(40,5) = C4 x D10
• SmallGroup(40,7) = C2 x (C5 : C4)
• SmallGroup(40,13) = C2 x C2 x D10
• SmallGroup(48,31) = C4 x A4
• SmallGroup(48,49) = C2 x C2 x A4
• SmallGroup(52,1) = C13 : C4
• SmallGroup(52,4) = D52
• SmallGroup(58,1) = D58
• SmallGroup(64,41) = (C16 : C2) : C2
• SmallGroup(64,42) = (C16 : C2) : C2
• SmallGroup(64,43) = C2 . ((C8 x C2) : C2) = C8 . (C4 x C2)
• SmallGroup(64,46) = C16 : C4
• SmallGroup(64,134) = ((C4 x C4) : C2) : C2
• SmallGroup(64,135) = ((C4 x C4) : C2) : C2
• SmallGroup(64,136) = ((C4 x C4) : C2) : C2
• SmallGroup(64,137) = ((C4 x C4) : C2) : C2
• SmallGroup(64,138) = (((C4 x C2) : C2) : C2) : C2
• SmallGroup(64,139) = (((C4 x C2) : C2) : C2) : C2
• SmallGroup(64,149) = (C2 x (C8 : C2)) : C2
• SmallGroup(64,150) = (C2 x (C8 : C2)) : C2
• SmallGroup(64,151) = (C2 x Q16) : C2
• SmallGroup(64,152) = (C2 x QD16) : C2
• SmallGroup(64,153) = (C2 x D16) : C2
• SmallGroup(64,154) = (C2 x Q16) : C2
• SmallGroup(64,170) = (Q8 : C4) : C2
• SmallGroup(64,171) = ((C8 x C2) : C2) : C2
• SmallGroup(64,172) = (C2 x C2) . (C2 x D8) = (C4 x C2) . (C2 x C2 x C2)
• SmallGroup(64,177) = (C2 x D16) : C2
• SmallGroup(64,178) = (C2 x Q16) : C2
• SmallGroup(64,182) = C8 : Q8
• SmallGroup(64,190) = (C2 x D16) : C2
• SmallGroup(64,191) = (C2 x Q16) : C2
• SmallGroup(96,66) = SL(2,3) : C4
• SmallGroup(96,67) = SL(2,3) : C4
• SmallGroup(96,68) = C2 x ((C4 x C4) : C3)
• SmallGroup(96,188) = C2 x (C2 . S4 = SL(2,3) . C2)
• SmallGroup(96,189) = C2 x GL(2,3)
• SmallGroup(96,192) = (C2 . S4 = SL(2,3) . C2) : C2
• SmallGroup(96,229) = C2 x ((C2 x C2 x C2 x C2) : C3)
• SmallGroup(100,13) = D10 x D10
• SmallGroup(112,41) = C2 x ((C2 x C2 x C2) : C7)
• SmallGroup(120,39) = A4 x D10
• SmallGroup(136,3) = C17 : C8
• SmallGroup(136,13) = C2 x (C17 : C4)
• SmallGroup(144,184) = A4 x A4
• SmallGroup(156,1) = (C13 : C4) : C3
• SmallGroup(156,8) = C2 x ((C13 : C3) : C2)
• SmallGroup(160,235) = C2 x ((C2 x C2 x C2 x C2) : C5)
• SmallGroup(192,201) = (((C2 x D8) : C2) : C3) : C2
• SmallGroup(192,202) = ((((C4 x C2) : C2) : C2) : C2) : C3
• SmallGroup(196,8) = (C7 x C7) : C4
• SmallGroup(216,88) = ((C3 x C3) : C3) : Q8
• SmallGroup(216,96) = ((C18 x C2) : C3) : C2
• SmallGroup(216,99) = ((C6 x C6) : C3) : C2
• SmallGroup(240,191) = ((C2 x C2 x C2 x C2) : C5) : C3
• SmallGroup(312,51) = ((C26 x C2) : C3) : C2
• SmallGroup(336,210) = C2 x (((C2 x C2 x C2) : C7) : C3)
• SmallGroup(366,1) = (C61 : C3) : C2
• SmallGroup(384,5677) = ((((C4 x C4) : C3) : C2) : C2) : C2
• SmallGroup(384,5678) = ((((C2 x C2 x C2 x C2) : C3) : C2) : C2) : C2
• SmallGroup(384,5863) = ((C2 x ((C2 x C2 x C2 x C2) : C2)) : C2) : C3
• SmallGroup(384,5864) = (((C2 x C2 x Q8) : C2) : C2) : C3
• SmallGroup(384,5865) = ((C2 x C2 x C2) . (C2 x C2 x C2 x C2)) : C3
• SmallGroup(384,5866) = ((C2 x Q8) : Q8) : C3
• SmallGroup(384,18133) = ((((C4 x C4) : C2) : C2) : C3) : C2
• SmallGroup(400,134) = ((C5 x C5) : C4) : C4
• SmallGroup(400,207) = (((C5 x C5) : C4) : C2) : C2
• SmallGroup(400,212) = C2 x ((C5 x C5) : Q8)
• SmallGroup(406,1) = (C29 : C7) : C2
• SmallGroup(448,178) = (C4 x C4 x C4) : C7
• SmallGroup(448,1393) = (C2 x C2 x C2 x C2 x C2 x C2) : C7
• SmallGroup(448,1394) = (C2 x C2 x C2 x C2 x C2 x C2) : C7
• SmallGroup(576,8654) = ((A4 x A4) : C2) : C2
• SmallGroup(576,8661) = (C2 x C2 x C2 x C2 x C2 x C2) : C9
• SmallGroup(588,34) = ((C7 x C7) : C4) : C3
• SmallGroup(600,55) = ((C5 x C5) : C3) : C8
• SmallGroup(600,151) = (((C5 x C5) : C4) : C3) : C2
• SmallGroup(600,152) = C2 x (((C5 x C5) : C3) : C4)
• SmallGroup(610,1) = (C61 : C5) : C2
• SmallGroup(640,21454) = C2 . (((C2 x C2 x C2 x C2) : C5) : C4) = (((C2 x Q8) : C2) : C5) . C4
• SmallGroup(640,21455) = (((C2 x Q8) : C2) : C5) : C4
• SmallGroup(672,1044) = SL(2,7) : C2
• SmallGroup(672,1045) = C2 . (PSL(3,2) : C2) = SL(2,7) . C2
• SmallGroup(672,1257) = (C2 x C2 x ((C2 x C2 x C2) : C7)) : C3
• SmallGroup(864,2666) = ((C2 x ((C3 x C3) : C4)) : C4) : C3
• SmallGroup(864,4666) = (C2 x C2 x ((C3 x C3) : Q8)) : C3
• SmallGroup(1200,947) = (((C5 x C5) : Q8) : C3) : C2
• SmallGroup(1200,950) = C2 x (((C5 x C5) : Q8) : C3)
• SmallGroup(1320,134) = C2 x PSL(2,11)
• SmallGroup(1344,815) = ((C4 x C4 x C4) : C7) : C3
• SmallGroup(1344,11690) = ((C2 x C2 x C2 x C2 x C2 x C2) : C7) : C3
• SmallGroup(1344,11691) = ((C2 x C2 x C2 x C2 x C2 x C2) : C7) : C3
• SmallGroup(1440,4595) = A6 : C4
• SmallGroup(1440,5844) = C2 x (A6 . C2)
• SmallGroup(1452,20) = ((C11 x C11) : C3) : C4
• SmallGroup(1728,47862) = ((C2 x C2 x C2 x C2 x C2 x C2) : C9) : C3
• SmallGroup(1920,240998) = (C2 x C2 x C2 x C2 x C2) : A5
• SmallGroup(1920,240999) = C2 . ((C2 x C2 x C2 x C2) : A5)
• SmallGroup(1920,241004) = ((C2 x Q8) : C2) : A5
• SmallGroup(1944,803) = ((C3 x C3) . ((C3 x C3) : C3) = (C3 x C3 x C3) . (C3 x C3)) : C8

Answer 2

Here are the 18 isomorphism classes of finite groups with 8 conjugacy classes:

• SmallGroup( 20, 1) = $\operatorname{AGL}(1,5)$
• SmallGroup( 20, 4) = $D_{10}$
• SmallGroup( 24, 13) = $C_2 \times A_4$
• SmallGroup( 26, 1) = $D_{13}$
• SmallGroup( 48, 3) = $C_3 \ltimes (C_4 \times C_4)$
• SmallGroup( 48, 28) = $\operatorname{SL}(2,3) \mathsf{Y} C_4$
• SmallGroup( 48, 29) = $\operatorname{GL}(2,3)$
• SmallGroup( 48, 50) = $C_3 \ltimes (C_2^4)$
• SmallGroup( 56, 11) = $\operatorname{AGL}(1,8)$
• SmallGroup( 68, 3) = $C_4 \ltimes C_{17}$
• SmallGroup( 78, 1) = $C_6 \ltimes C_{13}$
• SmallGroup( 80, 49) = $C_5 \ltimes C_2^4$
• SmallGroup(168, 43) = $\operatorname{A\Gamma L}(1,8)$
• SmallGroup(200, 44) = $Q_8 \ltimes (C_5 \times C_5)$
• SmallGroup(300, 23) = $C_4 \ltimes C_3 \ltimes C_5^2$
• SmallGroup(600,150) = $\operatorname{SL}(2,3) \ltimes C_5^2$
• SmallGroup(660, 13) = $\operatorname{PSL}(2,11)$
• SmallGroup(720,765) = $M_{10}$, the Mathieu group on 10 points

This list (with a computer-free proof of correctness) can be found on page 310 of (Vera-López–Vera-López, 1985).

• Vera López, Antonio; Vera López, Juan. “Classification of finite groups according to the number of conjugacy classes.” Israel J. Math. 51 (1985), no. 4, 305–338. MR804489 DOI:10.1007/BF02764723