Group #141695 — subgroups

Subgroups — #141695

Order 512.

350 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 126 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#13287	C₂	1	4	yes	maximal
3	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#8814	C₂	1	1	yes	maximal, characteristic
5	#3170	—	2	1	no	—
6	#3170	—	2	1	no	—
7	#3160	C₂²	1	2	yes	—
8	C₄×M₃₂	C₂²	1	2	yes	—
9	#1845	—	2	1	no	—
10	#1844	C₂²	1	2	yes	—
11	#1834	—	2	1	no	—
12	#1713	C₄	1	1	yes	characteristic, complemented
13	#1713	C₄	1	1	yes	characteristic, complemented
14	#1710	C₄	1	2	yes	complemented
15	#1707	—	2	1	no	—
16	#1707	—	2	1	no	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1386	—	2	8	no	—
19	#1311	—	2	1	no	—
20	C₈×C₁₆	—	2	1	no	—
21	#533	—	2	2	no	—
22	C₂×M₃₂	D₄	1	4	yes	—
23	#500	—	2	1	no	—
24	#500	—	2	1	no	—
25	#500	C₂×C₄	1	2	yes	—
26	#500	C₂×C₄	1	2	yes	—
27	#499	—	2	4	no	—
28	#498	—	2	2	no	—
29	#494	—	2	1	no	—
30	#494	—	2	1	no	—
31	#487	—	2	2	no	—
32	#487	—	2	2	no	—
33	#467	—	4	2	no	—
34	#467	—	4	2	no	—
35	C₄×Q₁₆	—	4	2	no	—
36	C₄×Q₁₆	—	4	2	no	—
37	#427	—	2	4	no	—
38	C₂×#107	—	4	2	no	—
39	C₂×#106	—	2	4	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#384	—	2	2	no	—
44	#346	—	2	2	no	—
45	C₄×C₁₆	—	2	2	no	—
46	#340	—	4	2	no	—
47	C₈²	C₂×C₄	1	1	yes	characteristic
48	C₈²	C₂×C₄	1	1	yes	characteristic
49	C₈²	D₄	1	1	yes	characteristic
50	C₈²	D₄	1	1	yes	characteristic
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂²×C₈	D₈	1	2	yes	—
54	C₂²×C₈	D₈	1	2	yes	—
55	C₂²×C₈	C₂×D₄	1	2	yes	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	—	2	2	no	—
58	#125	—	4	2	no	—
59	C₂²⋊Q₈	—	4	2	no	—
60	C₂²⋊Q₈	—	4	2	no	—
61	C₄×Q₈	—	4	2	no	—
62	C₄×Q₈	—	4	2	no	—
63	#117	—	4	2	no	—
64	C₂×C₄⋊C₄	—	4	2	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	M₃₂	—	2	4	no	—
69	M₃₂	Q₁₆	1	8	yes	complemented
70	C₂×C₁₆	—	2	4	no	—
71	#107	—	2	4	no	—
72	#107	—	2	4	no	—
73	#107	—	4	2	no	—
74	#107	—	4	2	no	—
75	#107	—	8	4	no	—
76	#106	—	4	4	no	—
77	#106	—	4	8	no	—
78	#103	—	4	4	no	—
79	#103	—	4	4	no	—
80	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₄×C₈	SD₁₆	1	2	yes	—
85	C₄×C₈	SD₁₆	1	2	yes	—
86	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	#41	1	2	yes	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂²×C₄	—	8	2	no	—
92	C₂²×C₄	#127	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	Q₁₆	—	8	4	no	—
96	Q₁₆	—	8	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	#102	1	2	yes	—
100	C₂×C₈	#102	1	2	yes	—
101	C₂×C₈	#102	1	2	yes	—
102	C₂×C₈	#102	1	2	yes	—
103	C₂×C₈	C₄×D₄	1	2	yes	—
104	C₂×C₈	#121	1	2	yes	—
105	C₂×C₈	C₂×Q₁₆	1	4	yes	—
106	C₂×C₈	#135	1	4	yes	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	8	4	no	—
112	C₄⋊C₄	—	8	4	no	—
113	C₄⋊C₄	—	8	2	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₂²⋊C₄	—	8	2	no	—
117	C₄²	—	8	2	no	—
118	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₄²	#124	1	1	yes	characteristic
120	C₄²	C₂×SD₁₆	1	1	yes	characteristic
121	C₄²	C₂×SD₁₆	1	1	yes	characteristic
122	C₁₆	—	2	8	no	—
123	C₂³	#493	1	1	yes	characteristic
124	Q₈	—	8	4	no	—
125	Q₈	—	8	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	2	no	—
131	C₂×C₄	—	16	2	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	#390	1	1	yes	characteristic
134	C₂×C₄	C₂×#102	1	1	yes	characteristic
135	C₂×C₄	C₂×#102	1	1	yes	characteristic
136	C₂×C₄	#486	1	1	yes	characteristic
137	C₂×C₄	#486	1	1	yes	characteristic
138	C₂×C₄	#492	1	1	yes	characteristic
139	C₂×C₄	#494	1	2	yes	—
140	C₂×C₄	#495	1	2	yes	—
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	#360	1	4	yes	—
145	C₈	C₄×Q₁₆	1	4	yes	—
146	C₈	#467	1	4	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#1709	1	1	yes	characteristic
150	C₂²	#1713	1	1	yes	characteristic
151	C₂²	#1957	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	16	2	no	—
155	C₄	—	16	2	no	—
156	C₄	—	32	2	no	—
157	C₄	#1707	1	2	yes	—
158	C₄	#1969	1	2	yes	—
159	C₄	C₂×#360	1	2	yes	—
160	C₄	#2145	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	#9825	1	1	yes	characteristic
164	C₂	#13210	1	1	yes	characteristic
165	C₂	#13211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19604

C₂

yes

maximal, characteristic

#13287

C₂

yes

maximal

C₈×M₃₂

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

#3170

—

#3170

—

#3160

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1845

—

#1844

C₂²

yes

—

#1834

—

#1713

C₄

yes

characteristic, complemented

#1713

C₄

yes

characteristic, complemented

#1710

C₄

yes

complemented

#1707

—

#1707

—

C₂×C₈²

C₂²

yes

characteristic

#1386

—

#1311

—

C₈×C₁₆

—

#533

—

C₂×M₃₂

D₄

yes

—

#500

—

#500

—

#500

C₂×C₄

yes

—

#500

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#427

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#346

—

C₄×C₁₆

—

#340

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

complemented

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

100

C₂×C₈

#102

yes

—

101

C₂×C₈

#102

yes

—

102

C₂×C₈

#102

yes

—

103

C₂×C₈

C₄×D₄

yes

—

104

C₂×C₈

#121

yes

—

105

C₂×C₈

C₂×Q₁₆

yes

—

106

C₂×C₈

#135

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂×C₂²⋊C₄

yes

characteristic

119

C₄²

#124

yes

characteristic

120

C₄²

C₂×SD₁₆

yes

characteristic

121

C₄²

C₂×SD₁₆

yes

characteristic

122

C₁₆

—

123

C₂³

#493

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#390

yes

characteristic

134

C₂×C₄

C₂×#102

yes

characteristic

135

C₂×C₄

C₂×#102

yes

characteristic

136

C₂×C₄

#486

yes

characteristic

137

C₂×C₄

#486

yes

characteristic

138

C₂×C₄

#492

yes

characteristic

139

C₂×C₄

#494

yes

—

140

C₂×C₄

#495

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#360

yes

—

145

C₈

C₄×Q₁₆

yes

—

146

C₈

#467

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

#1709

yes

characteristic

150

C₂²

#1713

yes

characteristic

151

C₂²

#1957

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#1707

yes

—

158

C₄

#1969

yes

—

159

C₄

C₂×#360

yes

—

160

C₄

#2145

yes

—

161

C₂

—

162

C₂

—

163

C₂

#9825

yes

characteristic

164

C₂

#13210

yes

characteristic

165

C₂

#13211

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1957

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#493

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #1713	C₄	semidirect
13. #1713	C₄	semidirect
14. #1710	C₄	semidirect
69. M₃₂	Q₁₆	semidirect

H ≅ G/N

type

12. #1713

C₄

semidirect

13. #1713

C₄

semidirect

14. #1710

C₄

semidirect

69. M₃₂

Q₁₆

semidirect

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

subgroup	index
#19604	2
#13287	2
C₈×M₃₂	2
#8814	2

subgroup

index

#19604

#13287

C₈×M₃₂

#8814

Subgroups — #141695

Conjugacy classes of subgroups

Complements

Maximal subgroups