Group #29537382 — subgroups

Subgroups — Q₈×Q₃₂×C₆₁ #29537382

Order 15616.

460 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₆₁	C₂	1	2	yes	maximal
2	C₆₁×#2252	C₂	1	3	yes	maximal
3	C₆₁×#2227	C₂	1	6	yes	maximal
4	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₆₁	C₂	1	3	yes	maximal
6	Q₈²×C₆₁	—	2	2	no	—
7	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#500	C₂²	1	6	yes	—
9	C₆₁×#477	—	2	6	no	—
10	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₆₁	C₂²	1	6	yes	—
12	C₆₁×#366	C₂²	1	3	yes	—
13	C₆₁×#366	C₂²	1	6	yes	—
14	C₆₁×C₄⋊C₁₆	C₂²	1	3	yes	—
15	C₆₁×#358	C₂²	1	6	yes	—
16	C₄×C₁₆×C₆₁	C₂²	1	3	yes	—
17	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
18	C₄⋊Q₈×C₆₁	—	2	6	no	—
19	C₄⋊Q₈×C₆₁	—	4	6	no	—
20	C₄×Q₈×C₆₁	—	2	6	no	—
21	C₄×Q₈×C₆₁	—	4	2	no	—
22	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
23	Q₃₂×C₆₁	Q₈	1	4	yes	direct product factor
24	C₂×C₁₆×C₆₁	C₂³	1	3	yes	—
25	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
26	#107×C₆₁	—	2	6	no	—
27	#107×C₆₁	C₂³	1	6	yes	—
28	C₄⋊C₈×C₆₁	D₄	1	3	yes	—
29	#103×C₆₁	—	2	6	no	—
30	C₄×C₈×C₆₁	C₂³	1	3	yes	—
31	C₂×Q₈×C₆₁	—	2	2	no	—
32	C₂×Q₈×C₆₁	—	4	6	no	—
33	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
34	Q₁₆×C₆₁	—	2	2	no	—
35	Q₁₆×C₆₁	C₂×Q₈	1	4	yes	—
36	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
37	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	2	6	no	—
39	C₄⋊C₄×C₆₁	—	4	6	no	—
40	C₄⋊C₄×C₆₁	—	4	6	no	—
41	C₄⋊C₄×C₆₁	D₈	1	3	yes	—
42	C₄²×C₆₁	—	4	6	no	—
43	C₄²×C₆₁	C₂×D₄	1	3	yes	—
44	C₁₆×C₆₁	—	2	3	no	—
45	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
46	Q₈×C₆₁	—	2	4	no	—
47	Q₈×C₆₁	—	4	2	no	—
48	Q₈×C₆₁	—	16	6	no	—
49	Q₈×C₆₁	Q₃₂	1	4	yes	direct product factor
50	C₂×C₄×C₆₁	—	4	2	no	—
51	C₂×C₄×C₆₁	—	4	6	no	—
52	C₂×C₄×C₆₁	C₂×D₈	1	3	yes	—
53	C₂×C₄×C₆₁	C₂×D₈	1	3	yes	—
54	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
55	C₈×C₆₁	—	2	3	no	—
56	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
57	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
58	Q₈×Q₃₂	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
60	C₄×C₆₁	—	2	3	no	—
61	C₄×C₆₁	—	4	4	no	—
62	C₄×C₆₁	—	8	6	no	—
63	C₄×C₆₁	C₂×Q₃₂	1	6	yes	—
64	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
65	C₄×C₆₁	#578	1	1	yes	characteristic
66	Q₈×Q₁₆	C₂×C₆₁	1	2	yes	—
67	#2252	C₂×C₆₁	1	3	yes	—
68	#2227	C₂×C₆₁	1	6	yes	—
69	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic
70	C₄×Q₃₂	C₂×C₆₁	1	3	yes	—
71	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
72	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
73	C₂×C₆₁	#3415	1	1	yes	characteristic
74	Q₈²	—	2	2	no	—
75	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
76	#500	C₂²×C₆₁	1	6	yes	—
77	#477	—	2	6	no	—
78	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
79	C₄×Q₁₆	C₂²×C₆₁	1	6	yes	—
80	#366	C₂²×C₆₁	1	3	yes	—
81	#366	C₂²×C₆₁	1	6	yes	—
82	C₄⋊C₁₆	C₂²×C₆₁	1	3	yes	—
83	#358	C₂²×C₆₁	1	6	yes	—
84	C₄×C₁₆	C₂²×C₆₁	1	3	yes	—
85	C₆₁	Q₈×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
87	C₄⋊Q₈	—	2	6	no	—
88	C₄⋊Q₈	—	4	6	no	—
89	C₄×Q₈	—	2	6	no	—
90	C₄×Q₈	—	4	2	no	—
91	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
92	Q₃₂	Q₈×C₆₁	1	4	yes	direct product factor
93	C₂×C₁₆	C₂³×C₆₁	1	3	yes	—
94	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
95	#107	—	2	6	no	—
96	#107	C₂³×C₆₁	1	6	yes	—
97	C₄⋊C₈	D₄×C₆₁	1	3	yes	—
98	#103	—	2	6	no	—
99	C₄×C₈	C₂³×C₆₁	1	3	yes	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	4	6	no	—
102	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
103	Q₁₆	—	2	2	no	—
104	Q₁₆	C₂×Q₈×C₆₁	1	4	yes	—
105	C₂×C₈	C₂×D₄×C₆₁	1	3	yes	—
106	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
107	C₄⋊C₄	—	2	6	no	—
108	C₄⋊C₄	—	4	6	no	—
109	C₄⋊C₄	—	4	6	no	—
110	C₄⋊C₄	D₈×C₆₁	1	3	yes	—
111	C₄²	—	4	6	no	—
112	C₄²	C₂×D₄×C₆₁	1	3	yes	—
113	C₁₆	—	2	3	no	—
114	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
115	Q₈	—	2	4	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	16	6	no	—
118	Q₈	Q₃₂×C₆₁	1	4	yes	direct product factor
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	6	no	—
121	C₂×C₄	C₂×D₈×C₆₁	1	3	yes	—
122	C₂×C₄	C₂×D₈×C₆₁	1	3	yes	—
123	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
124	C₈	—	2	3	no	—
125	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
126	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
127	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	6	no	—
131	C₄	C₂×Q₃₂×C₆₁	1	6	yes	—
132	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
133	C₄	C₆₁×#578	1	1	yes	characteristic
134	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
135	C₂	C₂²×Q₃₂×C₆₁	1	1	yes	characteristic
136	C₂	C₆₁×#3415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Q₁₆×C₆₁

C₂

yes

maximal

C₆₁×#2252

C₂

yes

maximal

C₆₁×#2227

C₂

yes

maximal

Q₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₆₁

C₂

yes

maximal

Q₈²×C₆₁

—

C₂×Q₃₂×C₆₁

C₂²

yes

characteristic

C₆₁×#500

C₂²

yes

—

C₆₁×#477

—

C₈×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₁₆×C₆₁

C₂²

yes

—

C₆₁×#366

C₂²

yes

—

C₆₁×#366

C₂²

yes

—

C₆₁×C₄⋊C₁₆

C₂²

yes

—

C₆₁×#358

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂³

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

D₄

yes

characteristic

Q₃₂×C₆₁

Q₈

yes

direct product factor

C₂×C₁₆×C₆₁

C₂³

yes

—

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

#107×C₆₁

C₂³

yes

—

C₄⋊C₈×C₆₁

D₄

yes

—

#103×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₈

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₈

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×Q₈

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

Q₃₂

yes

direct product factor

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂²×D₄

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

C₂²×Q₈

yes

characteristic

C₈×C₆₁

2₋¹⁺⁴

yes

characteristic

Q₈×Q₃₂

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

C₂²×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₃₂

yes

—

C₄×C₆₁

D₄×Q₈

yes

characteristic

C₄×C₆₁

#578

yes

characteristic

Q₈×Q₁₆

C₂×C₆₁

yes

—

#2252

C₂×C₆₁

yes

—

#2227

C₂×C₆₁

yes

—

Q₈×C₁₆

C₂×C₆₁

yes

characteristic

C₄×Q₃₂

C₂×C₆₁

yes

—

C₂×C₆₁

Q₈×D₈

yes

characteristic

C₂×C₆₁

C₂²×Q₃₂

yes

characteristic

C₂×C₆₁

#3415

yes

characteristic

Q₈²

—

C₂×Q₃₂

C₂²×C₆₁

yes

characteristic

#500

C₂²×C₆₁

yes

—

#477

—

C₈×Q₈

C₂²×C₆₁

yes

characteristic

C₄×Q₁₆

C₂²×C₆₁

yes

—

#366

C₂²×C₆₁

yes

—

#366

C₂²×C₆₁

yes

—

C₄⋊C₁₆

C₂²×C₆₁

yes

—

#358

C₂²×C₆₁

yes

—

C₄×C₁₆

C₂²×C₆₁

yes

—

C₆₁

Q₈×Q₃₂

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

C₂³×C₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄×C₆₁

yes

characteristic

Q₃₂

Q₈×C₆₁

yes

direct product factor

C₂×C₁₆

C₂³×C₆₁

yes

—

C₂×C₁₆

C₂³×C₆₁

yes

characteristic

#107

—

#107

C₂³×C₆₁

yes

—

C₄⋊C₈

D₄×C₆₁

yes

—

#103

—

C₄×C₈

C₂³×C₆₁

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

D₈×C₆₁

yes

characteristic

103

Q₁₆

—

104

Q₁₆

C₂×Q₈×C₆₁

yes

—

105

C₂×C₈

C₂×D₄×C₆₁

yes

—

106

C₂×C₈

C₂⁴×C₆₁

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

D₈×C₆₁

yes

—

111

C₄²

—

112

C₄²

C₂×D₄×C₆₁

yes

—

113

C₁₆

—

114

C₁₆

C₂×Q₈×C₆₁

yes

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

Q₃₂×C₆₁

yes

direct product factor

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×D₈×C₆₁

yes

—

122

C₂×C₄

C₂×D₈×C₆₁

yes

—

123

C₂×C₄

C₂²×D₄×C₆₁

yes

characteristic

124

C₈

—

125

C₈

C₂²×Q₈×C₆₁

yes

characteristic

126

C₈

2₋¹⁺⁴×C₆₁

yes

characteristic

127

C₂²

C₂²×D₈×C₆₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×Q₃₂×C₆₁

yes

—

132

C₄

D₄×Q₈×C₆₁

yes

characteristic

133

C₄

C₆₁×#578

yes

characteristic

134

C₂

Q₈×D₈×C₆₁

yes

characteristic

135

C₂

C₂²×Q₃₂×C₆₁

yes

characteristic

136

C₂

C₆₁×#3415

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₆₁

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₆₁

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

N	H ≅ G/N	type
23. Q₃₂×C₆₁	Q₈	direct (N × H)
49. Q₈×C₆₁	Q₃₂	direct (N × H)
58. Q₈×Q₃₂	C₆₁	direct (N × H)
85. C₆₁	Q₈×Q₃₂	direct (N × H)
92. Q₃₂	Q₈×C₆₁	direct (N × H)
118. Q₈	Q₃₂×C₆₁	direct (N × H)

H ≅ G/N

type

23. Q₃₂×C₆₁

Q₈

direct (N × H)

49. Q₈×C₆₁

Q₃₂

direct (N × H)

58. Q₈×Q₃₂

C₆₁

direct (N × H)

85. C₆₁

Q₈×Q₃₂

direct (N × H)

92. Q₃₂

Q₈×C₆₁

direct (N × H)

118. Q₈

Q₃₂×C₆₁

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Q₁₆×C₆₁	2
C₆₁×#2252	2
C₆₁×#2227	2
Q₈×C₁₆×C₆₁	2
C₄×Q₃₂×C₆₁	2
Q₈×Q₃₂	61

subgroup

index

Q₈×Q₁₆×C₆₁

C₆₁×#2252

C₆₁×#2227

Q₈×C₁₆×C₆₁

C₄×Q₃₂×C₆₁

Q₈×Q₃₂

Subgroups — Q₈×Q₃₂×C₆₁ #29537382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Q32×C61 #29537382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₃₂×C₆₁ #29537382