Group #135013 — subgroups

Subgroups — #135013

Order 512.

262 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12891	C₂	1	1	yes	maximal, characteristic
3	#12008	C₂	1	1	yes	maximal, characteristic
4	#8659	C₂	1	2	yes	maximal
5	#7276	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×C₁₆	—	2	1	no	—
8	Q₈×C₁₆	—	2	1	no	—
9	Q₈×C₁₆	C₄	1	2	yes	—
10	Q₈×C₁₆	C₄	1	2	yes	—
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	—	2	1	no	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	2	yes	—
16	#1564	C₂²	1	2	yes	—
17	#1487	—	2	2	no	—
18	#1381	—	2	1	no	—
19	#1381	—	2	1	no	—
20	#1372	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1301	—	2	2	no	—
23	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	—	2	1	no	—
25	C₂²×C₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	C₈×Q₈	—	2	1	no	—
28	C₈×Q₈	—	2	1	no	—
29	C₈×Q₈	C₂×C₄	1	2	yes	—
30	C₈×Q₈	Q₈	1	2	yes	—
31	#431	—	2	2	no	—
32	C₂×#108	—	2	2	no	—
33	C₂×#108	—	2	2	no	—
34	#423	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	2	no	—
41	C₄⋊C₁₆	—	2	1	no	—
42	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₄×C₁₆	—	2	2	no	—
44	C₄×C₁₆	—	2	2	no	—
45	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₁₆	D₄	1	1	yes	characteristic
48	C₄×C₁₆	D₄	1	1	yes	characteristic
49	#335	—	2	2	no	—
50	#334	—	2	2	no	—
51	Q₈⋊C₈	—	4	4	no	—
52	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
53	C₂×M₁₆	—	4	2	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	#41	1	1	yes	characteristic
57	C₄×Q₈	—	2	1	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	C₄⋊C₄	1	2	yes	—
60	C₄×Q₈	D₈	1	2	yes	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
63	C₂×C₄²	—	2	1	no	—
64	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	4	2	no	—
69	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
73	#108	—	4	2	no	—
74	#108	—	4	2	no	—
75	#108	—	8	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	2	1	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
83	C₂²⋊C₈	—	4	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	4	2	no	—
87	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂×D₄	1	1	yes	characteristic
89	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	#107	1	2	yes	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	#121	1	1	yes	characteristic
97	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₄	#135	1	1	yes	characteristic
99	M₁₆	—	8	2	no	—
100	M₁₆	—	8	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	8	2	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈	#117	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₈	#135	1	2	yes	—
111	C₂×C₈	#137	1	2	yes	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	#107	1	2	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₄²	#121	1	1	yes	characteristic
120	C₄²	C₂×D₈	1	1	yes	characteristic
121	C₄²	#135	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	2	no	—
127	C₂³	#467	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	2	no	—
130	Q₈	#368	1	4	yes	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#380	1	1	yes	characteristic
138	C₂×C₄	C₂×#107	1	1	yes	characteristic
139	C₂×C₄	#427	1	1	yes	characteristic
140	C₂×C₄	#464	1	2	yes	—
141	C₂×C₄	#466	1	1	yes	characteristic
142	C₂×C₄	#466	1	1	yes	characteristic
143	C₂×C₄	#467	1	1	yes	characteristic
144	C₂×C₄	#471	1	2	yes	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	2	no	—
148	C₈	—	16	2	no	—
149	C₈	—	16	2	no	—
150	C₈	#416	1	1	yes	characteristic
151	C₈	#419	1	1	yes	characteristic
152	C₈	#479	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#1684	1	2	yes	—
156	C₂²	#1808	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	2	no	—
162	C₄	#1682	1	2	yes	—
163	C₄	#1805	1	1	yes	characteristic
164	C₄	#1815	1	1	yes	characteristic
165	C₄	C₂×#368	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#8649	1	1	yes	characteristic
168	C₂	#12837	1	1	yes	characteristic
169	C₂	#12841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#12891

C₂

yes

maximal, characteristic

#12008

C₂

yes

maximal, characteristic

#8659

C₂

yes

maximal

#7276

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

Q₈×C₁₆

—

Q₈×C₁₆

—

Q₈×C₁₆

C₄

yes

—

Q₈×C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1564

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₂×C₄

yes

—

C₈×Q₈

Q₈

yes

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

#335

—

#334

—

Q₈⋊C₈

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₄⋊C₄

yes

—

C₄×Q₈

D₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₄⋊C₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#107

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

107

C₂×C₈

#117

yes

characteristic

108

C₂×C₈

C₂²⋊Q₈

yes

characteristic

109

C₂×C₈

C₂²⋊Q₈

yes

characteristic

110

C₂×C₈

#135

yes

—

111

C₂×C₈

#137

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

#107

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×C₄⋊C₄

yes

characteristic

119

C₄²

#121

yes

characteristic

120

C₄²

C₂×D₈

yes

characteristic

121

C₄²

#135

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

127

C₂³

#467

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

#368

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#380

yes

characteristic

138

C₂×C₄

C₂×#107

yes

characteristic

139

C₂×C₄

#427

yes

characteristic

140

C₂×C₄

#464

yes

—

141

C₂×C₄

#466

yes

characteristic

142

C₂×C₄

#466

yes

characteristic

143

C₂×C₄

#467

yes

characteristic

144

C₂×C₄

#471

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

#416

yes

characteristic

151

C₈

#419

yes

characteristic

152

C₈

#479

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

#1684

yes

—

156

C₂²

#1808

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#1682

yes

—

163

C₄

#1805

yes

characteristic

164

C₄

#1815

yes

characteristic

165

C₄

C₂×#368

yes

—

166

C₂

—

167

C₂

#8649

yes

characteristic

168

C₂

#12837

yes

characteristic

169

C₂

#12841

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂³	#467
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#466

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
C₂×Q₈×C₁₆	2
#12891	2
#12008	2
#8659	2
#7276	2

subgroup

index

C₂×Q₈×C₁₆

Subgroups — #135013

Conjugacy classes of subgroups

Maximal subgroups