Group #8570 — subgroups

Subgroups — #8570

Order 256.

236 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 89 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	#2295	C₂	1	1	yes	maximal, characteristic
2	#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	2	yes	maximal
5	#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	#391	C₄	1	1	yes	characteristic
11	#391	C₄	1	1	yes	characteristic
12	#389	C₄	1	1	yes	characteristic
13	#389	C₄	1	1	yes	characteristic
14	#387	—	2	1	no	—
15	#387	—	2	1	no	—
16	C₄×C₄⋊C₄	—	2	1	no	—
17	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	#376	—	2	1	no	—
19	C₄³	C₂²	1	1	yes	characteristic
20	#336	—	2	1	no	—
21	#336	—	2	1	no	—
22	#336	—	2	1	no	—
23	Q₈⋊C₈	—	2	4	no	—
24	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	—	2	2	no	—
26	C₂²×C₈	—	2	2	no	—
27	C₄×Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	D₄	1	2	yes	—
30	C₄×Q₈	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²	—	2	1	no	—
38	C₂×C₄²	—	2	1	no	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₄²	D₄	1	1	yes	characteristic
43	C₂×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	C₄⋊C₈	—	2	4	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	2	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	—	2	2	no	—
51	#95	—	2	1	no	—
52	#95	—	2	1	no	—
53	#95	—	2	1	no	—
54	#95	—	2	1	no	—
55	#95	—	2	1	no	—
56	#95	—	2	1	no	—
57	#95	—	2	1	no	—
58	C₂×Q₈	—	2	1	no	—
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	—	4	2	no	—
86	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	4	1	no	—
95	C₄²	—	4	2	no	—
96	C₄²	C₂×D₄	1	1	yes	characteristic
97	C₄²	C₂×Q₈	1	1	yes	characteristic
98	C₄²	#41	1	1	yes	characteristic
99	C₄²	#41	1	1	yes	characteristic
100	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#104	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#117	1	1	yes	characteristic
131	C₂×C₄	#121	1	1	yes	characteristic
132	C₂×C₄	#121	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄	#131	1	2	yes	—
136	C₂×C₄	#131	1	2	yes	—
137	C₂×C₄	#135	1	2	yes	—
138	C₂×C₄	#137	1	2	yes	—
139	C₈	—	4	4	no	—
140	C₈	—	8	2	no	—
141	C₂²	#380	1	1	yes	characteristic
142	C₂²	#408	1	1	yes	characteristic
143	C₂²	#408	1	1	yes	characteristic
144	C₂²	#416	1	1	yes	characteristic
145	C₂²	#419	1	1	yes	characteristic
146	C₂²	C₂×#104	1	1	yes	characteristic
147	C₂²	#421	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	8	1	no	—
158	C₄	#424	1	2	yes	—
159	C₄	#433	1	2	yes	—
160	C₄	#464	1	2	yes	—
161	C₄	#479	1	2	yes	—
162	C₂	#1482	1	1	yes	characteristic
163	C₂	#1485	1	1	yes	characteristic
164	C₂	#1494	1	1	yes	characteristic
165	C₂	#1497	1	1	yes	characteristic
166	C₂	#1798	1	1	yes	characteristic
167	C₂	#1805	1	1	yes	characteristic
168	C₂	#1806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2295

C₂

yes

maximal, characteristic

#1831

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

#1279

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

#391

C₄

yes

characteristic

#391

C₄

yes

characteristic

#389

C₄

yes

characteristic

#389

C₄

yes

characteristic

#387

—

#387

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#376

—

C₄³

C₂²

yes

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

C₄×Q₈

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

100

C₂³

C₂×C₂²⋊C₄

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

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

—

127

C₂×C₄

—

128

C₂×C₄

#104

yes

—

129

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

130

C₂×C₄

#117

yes

characteristic

131

C₂×C₄

#121

yes

characteristic

132

C₂×C₄

#121

yes

characteristic

133

C₂×C₄

C₂²⋊Q₈

yes

characteristic

134

C₂×C₄

C₂²⋊Q₈

yes

characteristic

135

C₂×C₄

#131

yes

—

136

C₂×C₄

#131

yes

—

137

C₂×C₄

#135

yes

—

138

C₂×C₄

#137

yes

—

139

C₈

—

140

C₈

—

141

C₂²

#380

yes

characteristic

142

C₂²

#408

yes

characteristic

143

C₂²

#408

yes

characteristic

144

C₂²

#416

yes

characteristic

145

C₂²

#419

yes

characteristic

146

C₂²

C₂×#104

yes

characteristic

147

C₂²

#421

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#424

yes

—

159

C₄

#433

yes

—

160

C₄

#464

yes

—

161

C₄

#479

yes

—

162

C₂

#1482

yes

characteristic

163

C₂

#1485

yes

characteristic

164

C₂

#1494

yes

characteristic

165

C₂

#1497

yes

characteristic

166

C₂

#1798

yes

characteristic

167

C₂

#1805

yes

characteristic

168

C₂

#1806

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#2295	2
#1831	2
C₄×C₄⋊C₈	2
C₂×Q₈⋊C₈	2
#1279	2

subgroup

index

#2295

#1831

C₄×C₄⋊C₈

C₂×Q₈⋊C₈

#1279

Subgroups — #8570

Conjugacy classes of subgroups

Maximal subgroups