Group #129161 — subgroups

Subgroups — #129161

Order 512.

294 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 170 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	C₁₆×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₁₆	—	2	2	no	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#107	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#335	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1367	C₄	1	4	yes	complemented
16	#1312	C₄	1	4	yes	complemented
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	2	no	—
19	C₂²×C₁₆	—	2	1	no	—
20	C₂²×C₁₆	D₄	1	2	yes	—
21	C₂×#107	C₈	1	2	yes	—
22	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	—	2	2	no	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
30	C₄×C₁₆	—	2	1	no	—
31	C₄×C₁₆	—	4	4	no	—
32	C₄×C₁₆	C₂×C₄	1	2	yes	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	Q₈	1	2	yes	—
35	#340	C₈	1	2	yes	—
36	#336	—	2	2	no	—
37	#335	C₈	1	4	yes	—
38	C₈²	C₂×C₄	1	2	yes	—
39	C₈²	C₂×C₄	1	2	yes	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₈	#41	1	1	yes	characteristic
46	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
47	C₂×C₄²	—	2	2	no	—
48	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	4	1	no	—
54	C₂×C₁₆	—	4	1	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	4	2	no	—
57	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
58	#107	C₁₆	1	8	yes	complemented
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	C₂×C₈	1	4	yes	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	4	2	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	C₄²	1	2	yes	—
65	C₄×C₈	C₄²	1	2	yes	—
66	C₄×C₈	C₂×C₈	1	2	yes	—
67	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₄×C₈	#41	1	1	yes	characteristic
71	#95	—	2	2	no	—
72	C₂²×C₄	—	2	4	no	—
73	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
75	C₂²×C₄	#131	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	C₄×C₈	1	2	yes	—
85	C₂×C₈	C₄×C₈	1	2	yes	—
86	C₂×C₈	C₂×C₁₆	1	4	yes	—
87	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₈	#117	1	1	yes	characteristic
89	C₂×C₈	C₄×D₄	1	1	yes	characteristic
90	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₈	#137	1	2	yes	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
95	C₄²	—	4	2	no	—
96	C₄²	—	4	2	no	—
97	C₄²	C₂×C₄²	1	1	yes	characteristic
98	C₄²	C₄×Q₈	1	1	yes	characteristic
99	C₄²	C₂²×C₈	1	1	yes	characteristic
100	C₄²	#131	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₁₆	—	4	2	no	—
103	C₁₆	—	4	4	no	—
104	C₂³	C₈×D₄	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	8	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂×C₄	#405	1	1	yes	characteristic
112	C₂×C₄	C₈×D₄	1	1	yes	characteristic
113	C₂×C₄	#441	1	1	yes	characteristic
114	C₂×C₄	#442	1	1	yes	characteristic
115	C₂×C₄	#444	1	1	yes	characteristic
116	C₂×C₄	#444	1	1	yes	characteristic
117	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
118	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₁₆	1	2	yes	—
120	C₂×C₄	#504	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	4	no	—
126	C₈	C₄×C₁₆	1	4	yes	complemented
127	C₈	#428	1	2	yes	—
128	C₈	#440	1	2	yes	—
129	C₂²	#1579	1	1	yes	characteristic
130	C₂²	#1580	1	1	yes	characteristic
131	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
132	C₂²	D₄×C₁₆	1	2	yes	—
133	C₂²	#2171	1	2	yes	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	8	no	—
137	C₄	#1579	1	1	yes	characteristic
138	C₄	#1580	1	1	yes	characteristic
139	C₄	#1773	1	1	yes	characteristic
140	C₄	#1776	1	1	yes	characteristic
141	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
142	C₄	#2108	1	1	yes	characteristic
143	C₄	Q₈×C₁₆	1	2	yes	—
144	C₂	#8284	1	1	yes	characteristic
145	C₂	#12059	1	2	yes	—
146	C₂	#12062	1	2	yes	—
147	C₂	C₁₆×C₄⋊C₄	1	1	yes	characteristic
148	C₂	#12665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×C₄⋊C₄

C₂

yes

maximal

C₂×#1367

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal, characteristic

#7167

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#107

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#335

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1374

—

#1367

C₄

yes

complemented

#1312

C₄

yes

complemented

#1279

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

C₂×#107

C₈

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

—

#336

—

#335

C₈

yes

—

C₈²

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

#107

C₁₆

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

100

C₄²

#131

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₂³

C₈×D₄

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

110

C₂×C₄

C₂×C₄×C₈

yes

characteristic

111

C₂×C₄

#405

yes

characteristic

112

C₂×C₄

C₈×D₄

yes

characteristic

113

C₂×C₄

#441

yes

characteristic

114

C₂×C₄

#442

yes

characteristic

115

C₂×C₄

#444

yes

characteristic

116

C₂×C₄

#444

yes

characteristic

117

C₂×C₄

C₈×Q₈

yes

characteristic

118

C₂×C₄

C₈×Q₈

yes

characteristic

119

C₂×C₄

C₂²×C₁₆

yes

—

120

C₂×C₄

#504

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

C₄×C₁₆

yes

complemented

127

C₈

#428

yes

—

128

C₈

#440

yes

—

129

C₂²

#1579

yes

characteristic

130

C₂²

#1580

yes

characteristic

131

C₂²

C₈×C₄⋊C₄

yes

characteristic

132

C₂²

D₄×C₁₆

yes

—

133

C₂²

#2171

yes

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#1579

yes

characteristic

138

C₄

#1580

yes

characteristic

139

C₄

#1773

yes

characteristic

140

C₄

#1776

yes

characteristic

141

C₄

C₂×C₄×C₁₆

yes

characteristic

142

C₄

#2108

yes

characteristic

143

C₄

Q₈×C₁₆

yes

—

144

C₂

#8284

yes

characteristic

145

C₂

#12059

yes

—

146

C₂

#12062

yes

—

147

C₂

C₁₆×C₄⋊C₄

yes

characteristic

148

C₂

#12665

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₄

C₂×C₄×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. #1367	C₄	semidirect
16. #1312	C₄	semidirect
58. #107	C₁₆	semidirect
126. C₈	C₄×C₁₆	semidirect

H ≅ G/N

type

15. #1367

C₄

semidirect

16. #1312

C₄

semidirect

58. #107

C₁₆

semidirect

126. C₈

C₄×C₁₆

semidirect

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

subgroup	index
C₁₆×C₄⋊C₄	2
C₂×#1367	2
C₂×#1312	2
C₈×#107	2
#7167	2

subgroup

index

C₁₆×C₄⋊C₄

Subgroups — #129161

Conjugacy classes of subgroups

Complements

Maximal subgroups