Group #8568 — subgroups

Subgroups — #8568

Order 256.

246 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 97 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	#1842	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	—	2	1	no	—
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	#391	C₄	1	1	yes	characteristic
13	#391	C₄	1	1	yes	characteristic
14	#389	C₄	1	1	yes	characteristic
15	#389	C₄	1	1	yes	characteristic
16	#387	—	2	1	no	—
17	#387	—	2	1	no	—
18	C₄×C₄⋊C₄	—	2	1	no	—
19	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#376	—	2	1	no	—
21	C₄³	C₂²	1	1	yes	characteristic
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	—	4	2	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄×C₈	—	2	2	no	—
52	C₄×C₈	—	2	2	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	#95	—	2	1	no	—
59	#95	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
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	—	2	1	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	4	1	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	SD₁₆	1	2	yes	—
100	C₄²	Q₁₆	1	2	yes	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×Q₈	1	1	yes	characteristic
103	C₄²	#41	1	1	yes	characteristic
104	C₄²	#41	1	1	yes	characteristic
105	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	2	1	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	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	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	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#103	1	2	yes	—
132	C₂×C₄	#103	1	2	yes	—
133	C₂×C₄	#104	1	2	yes	—
134	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#117	1	1	yes	characteristic
136	C₂×C₄	#121	1	1	yes	characteristic
137	C₂×C₄	#121	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄	#131	1	2	yes	—
141	C₂×C₄	#131	1	2	yes	—
142	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄	#137	1	1	yes	characteristic
146	C₈	—	4	4	no	—
147	C₈	—	8	2	no	—
148	C₂²	#380	1	1	yes	characteristic
149	C₂²	#408	1	1	yes	characteristic
150	C₂²	#408	1	1	yes	characteristic
151	C₂²	C₂×#103	1	1	yes	characteristic
152	C₂²	#417	1	1	yes	characteristic
153	C₂²	C₂×#104	1	1	yes	characteristic
154	C₂²	#421	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	4	1	no	—
163	C₄	—	8	1	no	—
164	C₄	—	8	1	no	—
165	C₄	#424	1	2	yes	—
166	C₄	#433	1	2	yes	—
167	C₄	#460	1	1	yes	characteristic
168	C₄	#462	1	1	yes	characteristic
169	C₄	#475	1	1	yes	characteristic
170	C₄	#477	1	1	yes	characteristic
171	C₂	#1481	1	1	yes	characteristic
172	C₂	#1483	1	1	yes	characteristic
173	C₂	#1493	1	1	yes	characteristic
174	C₂	#1495	1	1	yes	characteristic
175	C₂	#1798	1	1	yes	characteristic
176	C₂	#1803	1	1	yes	characteristic
177	C₂	#1806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2295

C₂

yes

maximal, characteristic

#1842

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

—

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

characteristic

#336

—

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

—

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

—

C₄²

SD₁₆

yes

—

100

C₄²

Q₁₆

yes

—

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×Q₈

yes

characteristic

103

C₄²

#41

yes

characteristic

104

C₄²

#41

yes

characteristic

105

C₂³

C₂×C₂²⋊C₄

yes

characteristic

106

Q₈

—

107

Q₈

—

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

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#103

yes

—

132

C₂×C₄

#103

yes

—

133

C₂×C₄

#104

yes

—

134

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

135

C₂×C₄

#117

yes

characteristic

136

C₂×C₄

#121

yes

characteristic

137

C₂×C₄

#121

yes

characteristic

138

C₂×C₄

C₂²⋊Q₈

yes

characteristic

139

C₂×C₄

C₂²⋊Q₈

yes

characteristic

140

C₂×C₄

#131

yes

—

141

C₂×C₄

#131

yes

—

142

C₂×C₄

C₂×SD₁₆

yes

characteristic

143

C₂×C₄

C₂×Q₁₆

yes

characteristic

144

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

145

C₂×C₄

#137

yes

characteristic

146

C₈

—

147

C₈

—

148

C₂²

#380

yes

characteristic

149

C₂²

#408

yes

characteristic

150

C₂²

#408

yes

characteristic

151

C₂²

C₂×#103

yes

characteristic

152

C₂²

#417

yes

characteristic

153

C₂²

C₂×#104

yes

characteristic

154

C₂²

#421

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#424

yes

—

166

C₄

#433

yes

—

167

C₄

#460

yes

characteristic

168

C₄

#462

yes

characteristic

169

C₄

#475

yes

characteristic

170

C₄

#477

yes

characteristic

171

C₂

#1481

yes

characteristic

172

C₂

#1483

yes

characteristic

173

C₂

#1493

yes

characteristic

174

C₂

#1495

yes

characteristic

175

C₂

#1798

yes

characteristic

176

C₂

#1803

yes

characteristic

177

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
#1842	2
C₄×C₄⋊C₈	2
C₂×Q₈⋊C₈	2
#1279	2

subgroup

index

#2295

#1842

C₄×C₄⋊C₈

C₂×Q₈⋊C₈

#1279

Subgroups — #8568

Conjugacy classes of subgroups

Maximal subgroups