Group #142109 — subgroups

Subgroups — #142109

Order 512.

318 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 122 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	#7269	C₂	1	4	yes	maximal
5	C₂×#499	—	2	1	no	—
6	C₂×#499	—	2	1	no	—
7	#2859	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	C₄×M₃₂	—	2	1	no	—
13	#1846	—	2	1	no	—
14	#1846	—	2	1	no	—
15	#1845	C₂²	1	2	yes	—
16	#1844	C₄	1	2	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1376	—	2	1	no	—
19	C₂×#125	—	2	2	no	—
20	C₂×M₃₂	—	2	2	no	—
21	C₂×M₃₂	—	2	2	no	—
22	#499	—	4	1	no	—
23	#499	—	4	1	no	—
24	#499	—	4	2	no	—
25	#499	—	4	2	no	—
26	#499	—	4	1	no	—
27	#499	—	4	1	no	—
28	C₂×#107	—	2	2	no	—
29	C₂×#107	—	2	2	no	—
30	C₂×#106	—	2	2	no	—
31	C₂×#106	—	2	2	no	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#389	—	4	4	no	—
39	#384	C₂×C₄	1	4	yes	—
40	#382	—	2	2	no	—
41	C₄³	C₂×C₄	1	1	yes	characteristic
42	#364	—	2	2	no	—
43	#364	—	2	4	no	—
44	#364	—	4	2	no	—
45	#364	Q₈	1	4	yes	—
46	C₄⋊C₁₆	—	4	2	no	—
47	#346	—	4	1	no	—
48	C₄×C₁₆	—	4	1	no	—
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	#125	—	4	2	no	—
54	#125	—	4	2	no	—
55	#125	—	4	4	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	—	4	4	no	—
58	C₂×C₄⋊C₄	—	4	4	no	—
59	C₂×C₄²	—	2	2	no	—
60	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
63	M₃₂	—	2	4	no	—
64	M₃₂	—	4	2	no	—
65	M₃₂	—	4	2	no	—
66	M₃₂	—	4	2	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	—	4	2	no	—
69	#107	—	4	2	no	—
70	#107	—	4	2	no	—
71	#107	—	4	2	no	—
72	#107	—	4	2	no	—
73	#106	—	4	4	no	—
74	#106	—	4	2	no	—
75	#106	—	4	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	D₈	1	1	yes	characteristic
79	C₄×C₈	D₈	1	1	yes	characteristic
80	C₄×C₈	D₈	1	2	yes	—
81	C₄×C₈	SD₁₆	1	2	yes	—
82	C₄×C₈	SD₁₆	1	1	yes	characteristic
83	C₄×C₈	SD₁₆	1	1	yes	characteristic
84	C₄×C₈	SD₁₆	1	2	yes	—
85	C₄×C₈	Q₁₆	1	2	yes	—
86	C₄×C₈	C₂×Q₈	1	2	yes	—
87	C₄×C₈	#41	1	2	yes	—
88	#95	—	4	4	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	4	4	no	—
92	C₂²×C₄	#100	1	2	yes	—
93	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	#101	1	2	yes	—
97	C₂×C₈	#102	1	2	yes	—
98	C₂×C₈	#102	1	1	yes	characteristic
99	C₂×C₈	#102	1	1	yes	characteristic
100	C₂×C₈	#103	1	2	yes	—
101	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₈	#124	1	1	yes	characteristic
103	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₈	C₂×D₈	1	1	yes	characteristic
105	C₂×C₈	C₂×D₈	1	1	yes	characteristic
106	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₈	#137	1	2	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	8	8	no	—
114	C₄⋊C₄	—	8	4	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	4	4	no	—
119	C₄²	#102	1	1	yes	characteristic
120	C₄²	#102	1	1	yes	characteristic
121	C₄²	#102	1	2	yes	—
122	C₄²	#103	1	2	yes	—
123	C₄²	C₄×Q₈	1	2	yes	—
124	C₁₆	—	4	4	no	—
125	C₁₆	—	8	2	no	—
126	C₂³	C₂×#100	1	1	yes	characteristic
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	8	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	#331	1	2	yes	—
135	C₂×C₄	#331	1	2	yes	—
136	C₂×C₄	#391	1	1	yes	characteristic
137	C₂×C₄	C₂×#101	1	1	yes	characteristic
138	C₂×C₄	C₂×#102	1	1	yes	characteristic
139	C₂×C₄	C₂×#102	1	1	yes	characteristic
140	C₂×C₄	C₂×#103	1	1	yes	characteristic
141	C₂×C₄	#419	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	#332	1	2	yes	—
144	C₈	#332	1	2	yes	—
145	C₈	#474	1	2	yes	—
146	C₈	#475	1	2	yes	—
147	C₈	#486	1	1	yes	characteristic
148	C₈	#487	1	1	yes	characteristic
149	C₈	#498	1	1	yes	characteristic
150	C₈	#500	1	1	yes	characteristic
151	C₂²	—	2	2	no	—
152	C₂²	C₂×#331	1	1	yes	characteristic
153	C₂²	#1548	1	1	yes	characteristic
154	C₂²	#1997	1	1	yes	characteristic
155	C₄	—	2	4	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	16	4	no	—
160	C₄	C₂×#332	1	1	yes	characteristic
161	C₄	#1546	1	1	yes	characteristic
162	C₄	#1988	1	1	yes	characteristic
163	C₄	#1990	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#9898	1	1	yes	characteristic
167	C₂	#11920	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

C₄×#364

C₂

yes

maximal, characteristic

#7269

C₂

yes

—

—

C₄

yes

—

C₂×#364

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

#2150

—

C₄×M₃₂

—

#1846

—

#1846

—

#1845

C₂²

yes

—

#1844

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1376

—

C₂×#125

—

C₂×M₃₂

—

C₂×M₃₂

—

—

—

—

—

—

—

—

—

—

—

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

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#389

—

#384

C₂×C₄

yes

—

#382

—

C₄³

C₂×C₄

yes

—

—

—

Q₈

yes

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

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

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#100

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#101

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

yes

C₂×C₈

yes

100

C₂×C₈

#103

yes

—

101

C₂×C₈

C₂²⋊Q₈

yes

—

102

C₂×C₈

#124

yes

characteristic

103

C₂×C₈

C₄⋊Q₈

yes

characteristic

104

C₂×C₈

C₂×D₈

yes

characteristic

105

C₂×C₈

C₂×D₈

yes

characteristic

106

C₂×C₈

C₂×SD₁₆

yes

characteristic

107

C₂×C₈

C₂×SD₁₆

yes

characteristic

108

C₂×C₈

C₂×SD₁₆

yes

characteristic

109

C₂×C₈

C₂×Q₁₆

yes

characteristic

110

C₂×C₈

#137

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#102

yes

characteristic

120

C₄²

#102

yes

characteristic

121

C₄²

#102

yes

—

122

C₄²

#103

yes

—

123

C₄²

C₄×Q₈

yes

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

C₂×#100

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#331

yes

—

135

C₂×C₄

#331

yes

—

136

C₂×C₄

#391

yes

characteristic

137

C₂×C₄

C₂×#101

yes

characteristic

138

C₂×C₄

C₂×#102

yes

characteristic

139

C₂×C₄

C₂×#102

yes

characteristic

140

C₂×C₄

C₂×#103

yes

characteristic

141

C₂×C₄

#419

yes

characteristic

142

C₈

—

143

C₈

#332

yes

—

144

C₈

#332

yes

—

145

C₈

#474

yes

—

146

C₈

#475

yes

—

147

C₈

#486

yes

characteristic

148

C₈

#487

yes

characteristic

149

C₈

#498

yes

characteristic

150

C₈

#500

yes

characteristic

151

C₂²

—

152

C₂²

C₂×#331

yes

characteristic

153

C₂²

#1548

yes

characteristic

154

C₂²

#1997

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₂×#332

yes

characteristic

161

C₄

#1546

yes

characteristic

162

C₄

#1988

yes

characteristic

163

C₄

#1990

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#9898

yes

characteristic

167

C₂

#11920

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#331

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33311	2
#12900	2
C₄×#364	2
#7269	2

subgroup

index

Subgroups — #142109

Conjugacy classes of subgroups

Maximal subgroups