Group #9272 — subgroups

Subgroups — #9272

Order 256.

214 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 71 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	#3229	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	1	yes	maximal, characteristic
3	#1966	C₂	1	1	yes	maximal, characteristic
4	#1550	C₂	1	1	yes	maximal, characteristic
5	C₂×#332	C₂	1	1	yes	maximal, characteristic
6	#1301	C₂	1	1	yes	maximal, characteristic
7	#1283	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#498	—	2	1	no	—
11	#487	—	2	1	no	—
12	#483	C₄	1	4	yes	—
13	#475	—	2	2	no	—
14	#446	—	2	2	no	—
15	#446	—	2	1	no	—
16	C₈×Q₈	—	2	1	no	—
17	#431	C₂²	1	1	yes	characteristic
18	#424	—	2	1	no	—
19	#423	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×#101	—	2	1	no	—
23	C₂×#101	—	2	1	no	—
24	#405	—	2	1	no	—
25	C₄×M₁₆	C₂²	1	1	yes	characteristic
26	#333	—	2	1	no	—
27	#332	—	2	1	no	—
28	#332	—	2	1	no	—
29	#332	—	2	1	no	—
30	C₂²×Q₈	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₄⋊Q₈	D₄	1	1	yes	characteristic
42	#125	—	2	1	no	—
43	C₂²⋊Q₈	—	2	1	no	—
44	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
45	C₄×Q₈	—	2	1	no	—
46	C₄×Q₈	D₄	1	2	yes	—
47	#117	—	2	1	no	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	#106	—	2	4	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₄⋊C₈	D₄	1	1	yes	characteristic
62	#103	—	2	4	no	—
63	#101	—	4	2	no	—
64	#101	—	4	2	no	—
65	C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	#97	—	2	2	no	—
67	#97	—	2	1	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	4	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	1	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	#41	1	1	yes	characteristic
83	M₁₆	—	2	1	no	—
84	M₁₆	—	2	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	M₁₆	SD₁₆	1	2	yes	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	8	1	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	#41	1	1	yes	characteristic
115	C₂³	C₄×D₄	1	1	yes	characteristic
116	Q₈	—	4	2	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	8	1	no	—
120	Q₈	—	8	1	no	—
121	Q₈	—	8	1	no	—
122	Q₈	—	8	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄	#120	1	1	yes	characteristic
134	C₂×C₄	#121	1	1	yes	characteristic
135	C₂×C₄	#121	1	1	yes	characteristic
136	C₂×C₄	#123	1	1	yes	characteristic
137	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄	#135	1	1	yes	characteristic
139	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄	#137	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	4	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	8	1	no	—
147	C₈	—	8	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#386	1	1	yes	characteristic
151	C₂²	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²	#440	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	8	2	no	—
159	C₄	#448	1	1	yes	characteristic
160	C₄	#449	1	1	yes	characteristic
161	C₄	#460	1	1	yes	characteristic
162	C₄	#464	1	1	yes	characteristic
163	C₄	#471	1	1	yes	characteristic
164	C₄	#473	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#1880	1	1	yes	characteristic
168	C₂	#1908	1	1	yes	characteristic
169	C₂	#1915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3229

C₂

yes

maximal, characteristic

#2962

C₂

yes

maximal, characteristic

#1966

C₂

yes

maximal, characteristic

#1550

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal, characteristic

#1283

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

C₄

yes

—

#475

—

#446

—

#446

—

C₈×Q₈

—

#431

C₂²

yes

characteristic

#424

—

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

—

—

C₄×M₁₆

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

—

—

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂²⋊C₄

yes

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×D₄

yes

characteristic

112

C₄²

C₂×D₄

yes

characteristic

113

C₄²

C₂×D₄

yes

characteristic

114

C₄²

#41

yes

characteristic

115

C₂³

C₄×D₄

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

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

C₂×C₂²⋊C₄

yes

characteristic

132

C₂×C₄

C₄×D₄

yes

characteristic

133

C₂×C₄

#120

yes

characteristic

134

C₂×C₄

#121

yes

characteristic

135

C₂×C₄

#121

yes

characteristic

136

C₂×C₄

#123

yes

characteristic

137

C₂×C₄

C₂×SD₁₆

yes

characteristic

138

C₂×C₄

#135

yes

characteristic

139

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

140

C₂×C₄

#137

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

#386

yes

characteristic

151

C₂²

C₄×SD₁₆

yes

characteristic

152

C₂²

#440

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#448

yes

characteristic

160

C₄

#449

yes

characteristic

161

C₄

#460

yes

characteristic

162

C₄

#464

yes

characteristic

163

C₄

#471

yes

characteristic

164

C₄

#473

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#1880

yes

characteristic

168

C₂

#1908

yes

characteristic

169

C₂

#1915

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
#3229	2
#2962	2
#1966	2
#1550	2
C₂×#332	2
#1301	2
#1283	2

subgroup

index

Subgroups — #9272

Conjugacy classes of subgroups

Maximal subgroups