T	Name(s)	\|G\|	Parity	\|C_S₈(G)\|	Subfields	Other Representations	Resolvents
1	C(8)=8	8 = 2³	-1	8	2T1, 4T1		2: 2T1 4: 4T1
2	4[x]2	8 = 2³	1	8	2T1, 2T1, 2T1, 4T1, 4T1, 4T2		2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2
3	E(8)=2[x]2[x]2	8 = 2³	1	8	2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2		2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
4	D_8(8)=[4]2	8 = 2³	1	8	2T1, 2T1, 2T1, 4T2, 4T3, 4T3	4T3, 4T3	2: 2T1, 2T1, 2T1 4: 4T2
5	Q_8(8)	8 = 2³	1	8	2T1, 2T1, 2T1, 4T2		2: 2T1, 2T1, 2T1 4: 4T2
6	D(8)	16 = 2⁴	-1	2	2T1, 4T3	8T6, 16T13	2: 2T1, 2T1, 2T1 4: 4T2 8: 4T3
7	1/2[2^3]4	16 = 2⁴	-1	4	2T1, 4T1	16T6	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 8T2
8	2D_8(8)=[D(4)]2	16 = 2⁴	-1	2	2T1, 4T3	16T12	2: 2T1, 2T1, 2T1 4: 4T2 8: 4T3
9	E(8):2=D(4)[x]2	16 = 2⁴	1	4	2T1, 2T1, 2T1, 4T2, 4T3, 4T3	8T9, 8T9, 8T9, 16T9	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 8T3
10	[2^2]4	16 = 2⁴	1	4	2T1, 4T1, 4T3, 4T3	8T10, 16T10	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2
11	1/2[2^3]E(4)=Q_8:2	16 = 2⁴	1	4	2T1, 2T1, 2T1, 4T2	8T11, 8T11, 16T11	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 8T3
12	2A_4(8)=[2]A(4)=SL(2,3)	24 = 2³ · 3	1	2	4T4		3: 3T1 12: 4T4
13	E(8):3=A(4)[x]2	24 = 2³ · 3	1	2	2T1, 4T4	6T6, 12T6, 12T7	2: 2T1 3: 3T1 6: 6T1 12: 4T4
14	S(4)[1/2]2=1/2(S_4[x]2)	24 = 2³ · 3	1	2	2T1, 4T5	4T5, 6T7, 6T8, 12T8, 12T9	2: 2T1 6: 3T2
15	[1/4.cD(4)^2]2	32 = 2⁵	-1	2	2T1, 4T3	8T15, 16T35, 16T38, 16T38, 16T45	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 8T3 16: 8T9
16	1/2[2^4]4	32 = 2⁵	-1	2	2T1, 4T1	8T16, 16T36, 16T41, 16T41	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10
17	[4^2]2	32 = 2⁵	-1	4	2T1, 4T3	8T17, 16T28, 16T42	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10
18	E(8):E_4=[2^2]D(4)	32 = 2⁵	1	4	2T1, 4T3, 4T3, 4T3	8T18, 8T18, 8T18, 8T18, 8T18, 8T18, 8T18, 16T39, 16T39, 16T39, 16T39, 16T39, 16T39, 16T46	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3 16: 8T9, 8T9, 8T9
19	E(8):4=[1/4.eD(4)^2]2	32 = 2⁵	1	2	2T1, 4T3	8T19, 8T20, 8T21, 16T33, 16T33, 16T52, 16T53	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10
20	[2^3]4	32 = 2⁵	1	2	2T1, 4T1	8T19, 8T19, 8T21, 16T33, 16T33, 16T52, 16T53	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10
21	1/2[2^4]E(4)=[1/4.dD(4)^2]2	32 = 2⁵	-1	2	2T1, 2T1, 2T1, 4T2	8T19, 8T19, 8T20, 16T33, 16T33, 16T52, 16T53	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10
22	E(8):D_4=[2^3]2^2	32 = 2⁵	1	2	2T1, 2T1, 2T1, 4T2	8T22, 8T22, 8T22, 8T22, 8T22, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3 16: 16T3
23	2S_4(8)=GL(2,3)	48 = 2⁴ · 3	-1	2	4T5	8T23, 16T66	2: 2T1 6: 3T2 24: 4T5
24	E(8):D_6=S(4)[x]2	48 = 2⁴ · 3	1	2	2T1, 4T5	6T11, 6T11, 8T24, 12T21, 12T22, 12T23, 12T23, 12T24, 12T24, 16T61	2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2 12: 6T3 24: 4T5
25	E(8):7=F_56(8)	56 = 2³ · 7	1	1		14T6	7: 7T1
26	1/2[2^4]eD(4)	64 = 2⁶	-1	2	2T1, 4T3	8T26, 8T26, 8T26, 16T135, 16T135, 16T141, 16T141, 16T142, 16T142, 16T152, 16T152	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3 16: 8T9, 8T9, 8T9 32: 8T18
27	[2^4]4	64 = 2⁶	-1	2	2T1, 4T1	8T27, 8T28, 8T28, 16T130, 16T157, 16T157, 16T158, 16T158, 16T159, 16T159, 16T166, 16T170, 16T171, 16T172	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10 32: 8T19
28	1/2[2^4]dD(4)	64 = 2⁶	-1	2	2T1, 4T3	8T27, 8T27, 8T28, 16T130, 16T157, 16T157, 16T158, 16T158, 16T159, 16T159, 16T166, 16T170, 16T171, 16T172	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10 32: 8T19
29	E(8):D_8=[2^3]D(4)	64 = 2⁶	1	2	2T1, 4T3	8T29, 8T29, 8T29, 8T29, 8T29, 8T31, 8T31, 16T127, 16T128, 16T128, 16T128, 16T129, 16T129, 16T129, 16T147, 16T149, 16T149, 16T149, 16T149, 16T149, 16T149, 16T150, 16T150, 16T150	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3 16: 8T9, 8T9, 8T9 32: 8T18
30	1/2[2^4]cD(4)	64 = 2⁶	-1	2	2T1, 4T3	8T30, 8T30, 8T30, 16T143, 16T143, 16T167, 16T167, 16T168, 16T168, 16T169, 16T169	2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10 32: 8T19
31	[2^4]E(4)	64 = 2⁶	-1	2	2T1, 2T1, 2T1, 4T2	8T29, 8T29, 8T29, 8T29, 8T29, 8T29, 8T31, 16T127, 16T128, 16T128, 16T128, 16T129, 16T129, 16T129, 16T147, 16T149, 16T149, 16T149, 16T149, 16T149, 16T149, 16T150, 16T150, 16T150	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3 16: 8T9, 8T9, 8T9 32: 8T18
32	[2^3]A(4)	96 = 2⁵ · 3	1	2	4T4	8T32, 8T32	3: 3T1 12: 4T4, 4T4, 4T4, 4T4, 4T4 48: 12T32
33	E(8):A_4=[1/3.A(4)^2]2=E(4):6	96 = 2⁵ · 3	1	1	2T1	8T33, 12T58, 12T58, 12T59, 12T59, 16T183	2: 2T1 3: 3T1 6: 6T1 12: 4T4 24: 6T6
34	1/2[E(4)^2:S_3]2=E(4)^2:D_6	96 = 2⁵ · 3	1	1	2T1	12T66, 12T66, 12T66, 12T67, 12T68, 12T68, 12T68, 12T69, 16T194	2: 2T1 6: 3T2 24: 4T5, 4T5, 4T5
35	[2^4]D(4)	128 = 2⁷	-1	2	2T1, 4T3	8T35, 8T35, 8T35, 8T35, 8T35, 8T35, 8T35, 16T376, 16T376, 16T376, 16T376, 16T388, 16T388, 16T388, 16T388, 16T390, 16T390, 16T390, 16T390, 16T391, 16T391, 16T391, 16T391, 16T393, 16T393, 16T393, 16T393, 16T395, 16T395, 16T395, 16T395, 16T396, 16T396, 16T396, 16T396, 16T401, 16T401, 16T401, 16T401	2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3 16: 8T9, 8T9, 8T9 32: 8T18 64: 8T29
36	E(8):F_21	168 = 2³ · 3 · 7	1	1		14T11	3: 3T1 21: 7T3
37	L(8)=PSL(2,7)	168 = 2³ · 3 · 7	1	1		7T5, 7T5, 14T10, 14T10, 21T14
38	[2^4]A(4)	192 = 2⁶ · 3	-1	2	4T4	8T38, 16T425, 16T427	2: 2T1 3: 3T1 6: 6T1 12: 4T4 24: 6T6 96: 8T33
39	[2^3]S(4)	192 = 2⁶ · 3	1	2	4T5	8T39, 8T39, 8T39, 8T39, 8T39, 16T442, 16T442, 16T442	2: 2T1 6: 3T2 24: 4T5, 4T5, 4T5 96: 8T34
40	1/2[2^4]S(4)	192 = 2⁶ · 3	-1	2	4T5	8T40, 16T444, 16T445	2: 2T1 6: 3T2 24: 4T5, 4T5, 4T5 96: 8T34
41	E(8):S_4=[E(4)^2:S_3]2=E(4)^2:D_12	192 = 2⁶ · 3	1	1	2T1	8T41, 12T108, 12T108, 12T109, 12T109, 12T110, 12T110, 12T111, 12T111, 16T435, 16T435, 16T436	2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2 12: 6T3 24: 4T5 48: 6T11
42	[A(4)^2]2	288 = 2⁵ · 3²	1	1	2T1	12T126, 12T128, 12T129, 16T708, 18T112, 18T113	2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5
43	L(8):2=PGL(2,7)	336 = 2⁴ · 3 · 7	-1	1		14T16, 16T713, 21T20	2: 2T1
44	[2^4]S(4)	384 = 2⁷ · 3	-1	2	4T5	8T44, 8T44, 8T44, 16T736, 16T736, 16T743, 16T743, 16T748, 16T748, 16T752, 16T752	2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2 12: 6T3 24: 4T5 48: 6T11 192: 8T41
45	[1/2.S(4)^2]2	576 = 2⁶ · 3²	1	1	2T1	12T161, 12T163, 12T165, 12T165, 16T1032, 16T1034, 18T179, 18T180, 18T185, 18T185	2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2, 3T2 12: 6T3, 6T3 36: 6T9
46	1/2[S(4)^2]2	576 = 2⁶ · 3²	-1	1	2T1	12T160, 12T162, 16T1030, 16T1031, 18T182, 18T184	2: 2T1 4: 4T1 36: 6T10
47	[S(4)^2]2	1152 = 2⁷ · 3²	-1	1	2T1	12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T273, 18T274, 18T275	2: 2T1, 2T1, 2T1 4: 4T2 8: 4T3 72: 6T13
48	E(8):L_7=AL(8)	1344 = 2⁶ · 3 · 7	1	1		8T48, 14T34, 14T34	168: 7T5
49	A8	20160 = 2⁶ · 3² · 5 · 7	1	1		15T72, 15T72
50	S8	40320 = 2⁷ · 3² · 5 · 7	-1	1		16T1838	2: 2T1

Transitive Subgroups of S8

Transitive Subgroups of S₈