Group #140352 — subgroups

Subgroups — #140352

Order 512.

239 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 106 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	#13163	C₂	1	2	yes	maximal
2	#11996	C₂	1	1	yes	maximal, characteristic
3	#11402	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7169	C₂	1	2	yes	maximal
6	C₂×#364	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	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	#1376	—	2	2	no	—
15	#1367	—	2	1	no	—
16	#1367	—	2	1	no	—
17	#1312	—	2	1	no	—
18	#1312	—	2	1	no	—
19	#1279	C₄	1	2	yes	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	D₄	1	2	yes	—
23	C₂×#107	—	2	1	no	—
24	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	#364	—	4	4	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	#346	—	2	1	no	—
34	#346	Q₈	1	2	yes	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	#340	—	2	1	no	—
38	#336	—	2	2	no	—
39	#335	C₈	1	2	yes	—
40	#335	C₈	1	2	yes	—
41	C₈²	D₄	1	1	yes	characteristic
42	C₈²	D₄	1	1	yes	characteristic
43	C₈²	Q₈	1	1	yes	characteristic
44	C₈²	Q₈	1	1	yes	characteristic
45	C₂²×C₈	—	2	2	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	M₁₆	1	1	yes	characteristic
48	C₂²×C₈	M₁₆	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄²	—	2	2	no	—
53	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
54	M₃₂	—	2	2	no	—
55	M₃₂	—	2	2	no	—
56	M₃₂	—	4	1	no	—
57	M₃₂	—	4	1	no	—
58	M₃₂	—	4	2	no	—
59	M₃₂	—	4	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	4	2	no	—
63	#107	—	2	4	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	C₂×C₈	1	4	yes	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	—	4	2	no	—
69	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×C₈	1	1	yes	characteristic
74	C₄×C₈	C₂×C₈	1	1	yes	characteristic
75	C₄×C₈	C₂×D₄	1	1	yes	characteristic
76	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₄×C₈	#41	1	1	yes	characteristic
79	#95	—	2	2	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₄	#131	1	1	yes	characteristic
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₈	#121	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₈	#125	1	1	yes	characteristic
101	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₈	#137	1	2	yes	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄×Q₈	1	1	yes	characteristic
110	C₄²	C₂²×C₈	1	1	yes	characteristic
111	C₄²	#131	1	1	yes	characteristic
112	C₁₆	—	4	2	no	—
113	C₁₆	—	4	2	no	—
114	C₁₆	—	8	2	no	—
115	C₂³	#436	1	1	yes	characteristic
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	#389	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₄	#424	1	1	yes	characteristic
125	C₂×C₄	C₈×D₄	1	1	yes	characteristic
126	C₂×C₄	#441	1	1	yes	characteristic
127	C₂×C₄	#442	1	1	yes	characteristic
128	C₂×C₄	#444	1	1	yes	characteristic
129	C₂×C₄	#444	1	1	yes	characteristic
130	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
131	C₂×C₄	#446	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	4	4	no	—
140	C₈	#364	1	2	yes	complemented
141	C₈	#468	1	2	yes	—
142	C₈	#501	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#1593	1	1	yes	characteristic
146	C₂²	#1594	1	1	yes	characteristic
147	C₂²	#1939	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	8	4	no	—
152	C₄	#1579	1	1	yes	characteristic
153	C₄	#1580	1	1	yes	characteristic
154	C₄	#1943	1	1	yes	characteristic
155	C₄	#1946	1	1	yes	characteristic
156	C₄	C₂×#364	1	1	yes	characteristic
157	C₄	#2153	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#9595	1	1	yes	characteristic
161	C₂	#13163	1	1	yes	characteristic
162	C₂	#13164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13163

C₂

yes

maximal

#11996

C₂

yes

maximal, characteristic

#11402

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal, characteristic

#7169

C₂

yes

maximal

C₂×#364

—

#2150

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×#107

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#335

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

—

—

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

—

C₂×#107

—

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

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

#336

—

#335

C₈

yes

—

#335

C₈

yes

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

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

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

100

C₂×C₈

#125

yes

characteristic

101

C₂×C₈

C₄⋊Q₈

yes

characteristic

102

C₂×C₈

AGL(1,ℤ/8)

yes

—

103

C₂×C₈

#137

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×C₄⋊C₄

yes

characteristic

109

C₄²

C₄×Q₈

yes

characteristic

110

C₄²

C₂²×C₈

yes

characteristic

111

C₄²

#131

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₂³

#436

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#389

yes

characteristic

123

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

124

C₂×C₄

#424

yes

characteristic

125

C₂×C₄

C₈×D₄

yes

characteristic

126

C₂×C₄

#441

yes

characteristic

127

C₂×C₄

#442

yes

characteristic

128

C₂×C₄

#444

yes

characteristic

129

C₂×C₄

#444

yes

characteristic

130

C₂×C₄

C₈×Q₈

yes

characteristic

131

C₂×C₄

#446

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

#364

yes

complemented

141

C₈

#468

yes

—

142

C₈

#501

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

#1593

yes

characteristic

146

C₂²

#1594

yes

characteristic

147

C₂²

#1939

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#1579

yes

characteristic

153

C₄

#1580

yes

characteristic

154

C₄

#1943

yes

characteristic

155

C₄

#1946

yes

characteristic

156

C₄

C₂×#364

yes

characteristic

157

C₄

#2153

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

#9595

yes

characteristic

161

C₂

#13163

yes

characteristic

162

C₂

#13164

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#121

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
140. C₈	#364	semidirect

H ≅ G/N

type

140. C₈

#364

semidirect

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

subgroup	index
#13163	2
#11996	2
#11402	2
C₈×#107	2
#7169	2

subgroup

index

Subgroups — #140352

Conjugacy classes of subgroups

Complements

Maximal subgroups