Group #140507 — subgroups

Subgroups — #140507

Order 512.

416 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	4	yes	maximal
5	C₂×C₈×Q₈	C₂²	1	2	yes	—
6	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₁₆	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	2	no	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	#1984	C₄	1	2	yes	—
12	#1939	—	2	2	no	—
13	#1842	C₄	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	2	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	Q₈⋊C₁₆	—	2	8	no	—
17	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
18	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	—	2	2	no	—
21	C₈×Q₈	—	2	4	no	—
22	C₈×Q₈	—	2	4	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	—	2	2	no	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#391	C₈	1	2	yes	—
32	#389	—	2	2	no	—
33	#384	C₈	1	1	yes	characteristic
34	C₄×C₄⋊C₄	—	2	2	no	—
35	C₄³	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₁₆	—	2	4	no	—
37	C₄⋊C₁₆	—	4	4	no	—
38	C₄×C₁₆	—	2	8	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	D₄	1	4	yes	—
41	#336	—	2	4	no	—
42	C₂²×Q₈	C₂×C₈	1	2	yes	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	2	no	—
48	C₄⋊Q₈	C₁₆	1	4	yes	complemented
49	C₄×Q₈	—	2	4	no	—
50	C₄×Q₈	—	2	4	no	—
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
55	C₂×C₄²	—	2	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄²	M₁₆	1	2	yes	—
60	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₁₆	—	2	8	no	—
62	C₂×C₁₆	—	2	4	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	—	4	4	no	—
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	4	4	no	—
67	C₄⋊C₈	—	4	4	no	—
68	C₄×C₈	—	2	4	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	—	4	2	no	—
72	C₄×C₈	SD₁₆	1	4	yes	—
73	C₄×C₈	Q₁₆	1	4	yes	—
74	C₄×C₈	C₂×D₄	1	2	yes	—
75	C₄×C₈	#41	1	2	yes	—
76	#95	—	2	4	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	C₂×C₁₆	1	4	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
84	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
85	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
87	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	#103	1	4	yes	—
94	C₂×C₈	#121	1	2	yes	—
95	C₂×C₈	#124	1	1	yes	characteristic
96	C₂×C₈	#127	1	1	yes	characteristic
97	C₂×C₈	C₂×SD₁₆	1	2	yes	—
98	C₂×C₈	C₂×Q₁₆	1	2	yes	—
99	C₄⋊C₄	—	2	8	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄²	—	2	4	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	#103	1	4	yes	—
108	C₄²	#104	1	4	yes	—
109	C₄²	C₂×C₁₆	1	2	yes	—
110	C₄²	M₃₂	1	2	yes	—
111	C₄²	C₄×D₄	1	2	yes	—
112	C₄²	#131	1	2	yes	—
113	C₁₆	—	2	8	no	—
114	C₁₆	—	8	2	no	—
115	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	Q₈	—	2	8	no	—
117	Q₈	—	4	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	Q₈⋊C₈	1	4	yes	—
127	C₂×C₄	Q₈⋊C₈	1	4	yes	—
128	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
129	C₂×C₄	#350	1	4	yes	—
130	C₂×C₄	#390	1	1	yes	characteristic
131	C₂×C₄	#408	1	1	yes	characteristic
132	C₂×C₄	C₂×#103	1	2	yes	—
133	C₂×C₄	C₂×#104	1	2	yes	—
134	C₂×C₄	C₈×D₄	1	2	yes	—
135	C₂×C₄	#436	1	2	yes	—
136	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
137	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
138	C₂×C₄	#504	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	#487	1	2	yes	—
143	C₈	#492	1	1	yes	characteristic
144	C₈	#494	1	1	yes	characteristic
145	C₂²	C₂×Q₈⋊C₈	1	2	yes	—
146	C₂²	#1954	1	1	yes	characteristic
147	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
148	C₂²	#2113	1	1	yes	characteristic
149	C₂²	C₂×#350	1	2	yes	—
150	C₄	—	2	4	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	8	2	no	—
154	C₄	Q₈⋊C₁₆	1	8	yes	complemented
155	C₄	#1493	1	2	yes	—
156	C₄	#1956	1	1	yes	characteristic
157	C₄	#1962	1	1	yes	characteristic
158	C₄	D₄×C₁₆	1	2	yes	—
159	C₄	#2169	1	2	yes	—
160	C₂	#9637	1	1	yes	characteristic
161	C₂	C₂×Q₈⋊C₁₆	1	2	yes	—
162	C₂	#11608	1	2	yes	—
163	C₂	#13189	1	1	yes	characteristic
164	C₂	#13191	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×C₄⋊Q₈

C₂

yes

maximal, characteristic

#12898

C₂

yes

maximal, characteristic

C₄²×C₁₆

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₁₆

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

—

C₄×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

—

#1984

C₄

yes

—

#1939

—

#1842

C₄

yes

characteristic

C₈×C₄⋊C₄

—

C₄²×C₈

C₂²

yes

characteristic

Q₈⋊C₁₆

—

C₂×C₄⋊Q₈

C₈

yes

characteristic

C₂×C₄×Q₈

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#391

C₈

yes

—

#389

—

#384

C₈

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

#336

—

C₂²×Q₈

C₂×C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₁₆

yes

complemented

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#103

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

#103

yes

—

108

C₄²

#104

yes

—

109

C₄²

C₂×C₁₆

yes

—

110

C₄²

M₃₂

yes

—

111

C₄²

C₄×D₄

yes

—

112

C₄²

#131

yes

—

113

C₁₆

—

114

C₁₆

—

115

C₂³

C₂×C₂²⋊C₈

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

Q₈⋊C₈

yes

—

127

C₂×C₄

Q₈⋊C₈

yes

—

128

C₂×C₄

C₂²⋊C₁₆

yes

—

129

C₂×C₄

#350

yes

—

130

C₂×C₄

#390

yes

characteristic

131

C₂×C₄

#408

yes

characteristic

132

C₂×C₄

C₂×#103

yes

—

133

C₂×C₄

C₂×#104

yes

—

134

C₂×C₄

C₈×D₄

yes

—

135

C₂×C₄

#436

yes

—

136

C₂×C₄

C₂²×C₁₆

yes

characteristic

137

C₂×C₄

C₂×M₃₂

yes

characteristic

138

C₂×C₄

#504

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#487

yes

—

143

C₈

#492

yes

characteristic

144

C₈

#494

yes

characteristic

145

C₂²

C₂×Q₈⋊C₈

yes

—

146

C₂²

#1954

yes

characteristic

147

C₂²

C₂×C₂²⋊C₁₆

yes

characteristic

148

C₂²

#2113

yes

characteristic

149

C₂²

C₂×#350

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

Q₈⋊C₁₆

yes

complemented

155

C₄

#1493

yes

—

156

C₄

#1956

yes

characteristic

157

C₄

#1962

yes

characteristic

158

C₄

D₄×C₁₆

yes

—

159

C₄

#2169

yes

—

160

C₂

#9637

yes

characteristic

161

C₂

C₂×Q₈⋊C₁₆

yes

—

162

C₂

#11608

yes

—

163

C₂

#13189

yes

characteristic

164

C₂

#13191

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
48. C₄⋊Q₈	C₁₆	semidirect
154. C₄	Q₈⋊C₁₆	semidirect

H ≅ G/N

type

48. C₄⋊Q₈

C₁₆

semidirect

154. C₄

Q₈⋊C₁₆

semidirect

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

subgroup	index
C₈×C₄⋊Q₈	2
#12898	2
C₄²×C₁₆	2
C₂×Q₈⋊C₁₆	2

subgroup

index

C₈×C₄⋊Q₈

#12898

C₄²×C₁₆

C₂×Q₈⋊C₁₆

Subgroups — #140507

Conjugacy classes of subgroups

Complements

Maximal subgroups