Group #141890 — subgroups

Subgroups — #141890

Order 512.

348 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₄×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×#346	C₂	1	1	yes	maximal, characteristic
3	#8807	C₂	1	1	yes	maximal, characteristic
4	#7178	C₂	1	4	yes	maximal
5	C₂×#499	C₄	1	1	yes	characteristic
6	C₄×#125	—	2	1	no	—
7	C₂×#346	—	2	1	no	—
8	C₂×#346	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	1	no	—
10	#1991	—	2	1	no	—
11	#1942	C₄	1	2	yes	—
12	#1941	—	2	1	no	—
13	#1846	C₄	1	1	yes	characteristic
14	#1831	—	2	1	no	—
15	C₄×#107	C₂²	1	2	yes	—
16	C₄×#106	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#335	—	2	1	no	—
19	C₂×#334	C₂²	1	2	yes	—
20	C₂×#125	—	2	1	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	#499	—	2	2	no	—
23	#499	C₈	1	4	yes	complemented
24	C₂×#107	C₂×C₄	1	2	yes	—
25	C₂×#107	C₂×C₄	1	2	yes	—
26	C₂×#106	—	2	1	no	—
27	C₂×#106	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	#389	—	2	2	no	—
36	#387	—	4	2	no	—
37	#382	—	2	1	no	—
38	C₄×C₄⋊C₄	—	2	2	no	—
39	C₄×C₄⋊C₄	—	4	2	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	#346	—	2	4	no	—
42	#346	—	2	2	no	—
43	#346	Q₈	1	4	yes	—
44	C₄×C₁₆	—	2	4	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	#336	—	4	2	no	—
48	#335	—	4	2	no	—
49	#334	—	2	4	no	—
50	C₂²×C₈	—	4	2	no	—
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#125	—	2	4	no	—
56	#125	—	4	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄²	—	4	2	no	—
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	D₈	1	2	yes	—
65	C₂×C₄²	SD₁₆	1	2	yes	—
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	—	2	4	no	—
69	C₂×C₁₆	—	2	4	no	—
70	#107	—	2	2	no	—
71	#107	—	2	4	no	—
72	#107	C₂×C₈	1	4	yes	—
73	#106	—	2	4	no	—
74	#106	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	C₂×C₈	1	2	yes	—
81	C₄×C₈	M₁₆	1	2	yes	—
82	C₄×C₈	SD₁₆	1	2	yes	—
83	C₄×C₈	Q₁₆	1	2	yes	—
84	C₄×C₈	C₂×Q₈	1	2	yes	—
85	C₄×C₈	#41	1	2	yes	—
86	#95	—	4	2	no	—
87	#95	—	4	2	no	—
88	#95	—	4	2	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	#102	1	2	yes	—
92	C₂²×C₄	#102	1	2	yes	—
93	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	8	4	no	—
101	C₂×C₈	—	8	2	no	—
102	C₂×C₈	C₂²⋊C₈	1	2	yes	—
103	C₂×C₈	#103	1	2	yes	—
104	C₂×C₈	#103	1	2	yes	—
105	C₂×C₈	#104	1	2	yes	—
106	C₂×C₈	C₄×Q₈	1	2	yes	—
107	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₈	#131	1	2	yes	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	8	4	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	—	4	2	no	—
121	C₄²	—	8	4	no	—
122	C₄²	C₂²⋊C₈	1	2	yes	—
123	C₄²	#104	1	2	yes	—
124	C₄²	#124	1	1	yes	characteristic
125	C₄²	C₄⋊Q₈	1	1	yes	characteristic
126	C₄²	C₂×SD₁₆	1	1	yes	characteristic
127	C₄²	C₂×Q₁₆	1	1	yes	characteristic
128	C₁₆	—	2	8	no	—
129	C₂³	C₂×#102	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	D₄⋊C₈	1	4	yes	—
144	C₂×C₄	#391	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₄	#408	1	1	yes	characteristic
147	C₂×C₄	C₂×#103	1	1	yes	characteristic
148	C₂×C₄	C₂×#104	1	1	yes	characteristic
149	C₂×C₄	C₂×#104	1	1	yes	characteristic
150	C₂×C₄	#486	1	1	yes	characteristic
151	C₂×C₄	#487	1	1	yes	characteristic
152	C₂×C₄	#498	1	1	yes	characteristic
153	C₂×C₄	#500	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	16	2	no	—
157	C₈	Q₈⋊C₈	1	4	yes	—
158	C₈	C₈×Q₈	1	2	yes	—
159	C₈	#446	1	2	yes	—
160	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
161	C₂²	#1492	1	1	yes	characteristic
162	C₂²	#1990	1	1	yes	characteristic
163	C₂²	#2145	1	2	yes	—
164	C₂²	#2148	1	2	yes	—
165	C₄	—	2	4	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	16	2	no	—
173	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
174	C₄	#1493	1	1	yes	characteristic
175	C₄	#1984	1	1	yes	characteristic
176	C₄	#1994	1	1	yes	characteristic
177	C₂	#9870	1	1	yes	characteristic
178	C₂	#11597	1	2	yes	—
179	C₂	#11682	1	2	yes	—
180	C₂	#13288	1	1	yes	characteristic
181	C₂	#13289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#499

C₂

yes

maximal, characteristic

C₄×#346

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#7178

C₂

yes

maximal

C₂×#499

C₄

yes

—

—

C₂²

yes

—

C₂×C₄×C₁₆

—

#1991

—

#1942

C₄

yes

—

#1941

—

#1846

C₄

yes

characteristic

#1831

—

C₄×#107

C₂²

yes

—

C₄×#106

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

C₂²

yes

—

C₂×#125

—

C₂²×C₁₆

—

#499

—

#499

C₈

yes

complemented

C₂×#107

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

D₄

yes

—

—

Q₈

yes

—

C₄×C₁₆

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×C₈

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#102

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊C₈

yes

—

103

C₂×C₈

#103

yes

—

104

C₂×C₈

#103

yes

—

105

C₂×C₈

#104

yes

—

106

C₂×C₈

C₄×Q₈

yes

—

107

C₂×C₈

C₂²⋊Q₈

yes

—

108

C₂×C₈

C₂²×C₈

yes

characteristic

109

C₂×C₈

C₂×M₁₆

yes

characteristic

110

C₂×C₈

#131

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²⋊C₈

yes

—

123

C₄²

#104

yes

—

124

C₄²

#124

yes

characteristic

125

C₄²

C₄⋊Q₈

yes

characteristic

126

C₄²

C₂×SD₁₆

yes

characteristic

127

C₄²

C₂×Q₁₆

yes

characteristic

128

C₁₆

—

129

C₂³

C₂×#102

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

D₄⋊C₈

yes

—

144

C₂×C₄

#391

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

146

C₂×C₄

#408

yes

characteristic

147

C₂×C₄

C₂×#103

yes

characteristic

148

C₂×C₄

C₂×#104

yes

characteristic

149

C₂×C₄

C₂×#104

yes

characteristic

150

C₂×C₄

#486

yes

characteristic

151

C₂×C₄

#487

yes

characteristic

152

C₂×C₄

#498

yes

characteristic

153

C₂×C₄

#500

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

Q₈⋊C₈

yes

—

158

C₈

C₈×Q₈

yes

—

159

C₈

#446

yes

—

160

C₂²

C₂×D₄⋊C₈

yes

characteristic

161

C₂²

#1492

yes

characteristic

162

C₂²

#1990

yes

characteristic

163

C₂²

#2145

yes

—

164

C₂²

#2148

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₂×Q₈⋊C₈

yes

characteristic

174

C₄

#1493

yes

characteristic

175

C₄

#1984

yes

characteristic

176

C₄

#1994

yes

characteristic

177

C₂

#9870

yes

characteristic

178

C₂

#11597

yes

—

179

C₂

#11682

yes

—

180

C₂

#13288

yes

characteristic

181

C₂

#13289

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
23. #499	C₈	semidirect

H ≅ G/N

type

23. #499

C₈

semidirect

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

subgroup	index
C₄×#499	2
C₄×#346	2
#8807	2
#7178	2

subgroup

index

Subgroups — #141890

Conjugacy classes of subgroups

Complements

Maximal subgroups