Group #129189 — subgroups

Subgroups — #129189

Order 512.

240 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×#368	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#8293	C₂	1	1	yes	maximal, characteristic
5	C₈×#108	C₂	1	1	yes	maximal, characteristic
6	#7172	C₂	1	1	yes	maximal, characteristic
7	#7172	C₂	1	1	yes	maximal, characteristic
8	C₂×#368	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×#108	C₂²	1	1	yes	characteristic
12	C₄×#108	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1451	—	2	1	no	—
16	C₈×M₁₆	—	2	1	no	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1381	C₄	1	2	yes	—
19	#1372	C₄	1	2	yes	—
20	#1371	C₄	1	2	yes	—
21	#1312	C₄	1	2	yes	—
22	#1312	C₄	1	2	yes	—
23	#1278	—	2	1	no	—
24	#1278	—	2	1	no	—
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	#431	—	2	1	no	—
29	#431	—	2	1	no	—
30	C₂×#108	C₂×C₄	1	2	yes	—
31	C₂×#108	C₂×C₄	1	2	yes	—
32	C₄×M₁₆	—	2	1	no	—
33	C₄×M₁₆	—	2	1	no	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#368	—	2	4	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	#341	—	2	1	no	—
45	#341	—	2	1	no	—
46	#335	C₂×C₄	1	2	yes	—
47	#335	C₂×C₄	1	2	yes	—
48	#334	—	2	1	no	—
49	#334	—	2	1	no	—
50	#322	—	4	1	no	—
51	#322	—	4	1	no	—
52	#322	—	4	1	no	—
53	C₈²	—	4	1	no	—
54	C₈²	C₂×C₄	1	2	yes	—
55	C₈²	Q₈	1	2	yes	—
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	—	2	2	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	Q₁₆	1	2	yes	—
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	#41	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	C₄²	1	1	yes	characteristic
68	C₂×C₁₆	C₄²	1	1	yes	characteristic
69	C₂×C₁₆	C₄²	1	1	yes	characteristic
70	C₂×C₁₆	C₄²	1	1	yes	characteristic
71	#108	—	2	2	no	—
72	#108	—	2	2	no	—
73	#108	—	2	2	no	—
74	#108	M₁₆	1	4	yes	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₂²⋊C₈	—	4	1	no	—
78	C₂²⋊C₈	—	4	1	no	—
79	#97	—	4	1	no	—
80	#97	—	4	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	C₄⋊C₄	1	2	yes	—
87	C₄×C₈	C₄⋊C₄	1	2	yes	—
88	C₄×C₈	D₈	1	1	yes	characteristic
89	C₄×C₈	D₈	1	1	yes	characteristic
90	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
96	C₂²×C₄	#135	1	1	yes	characteristic
97	M₁₆	—	2	4	no	—
98	M₁₆	—	4	2	no	—
99	M₁₆	—	4	2	no	—
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	#107	1	2	yes	—
110	C₂×C₈	#107	1	2	yes	—
111	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₈	#117	1	1	yes	characteristic
113	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂×M₁₆	1	2	yes	—
116	C₂×C₈	#131	1	2	yes	—
117	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄²	C₄×D₄	1	1	yes	characteristic
119	C₄²	C₂×D₈	1	1	yes	characteristic
120	C₄²	#135	1	1	yes	characteristic
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	C₄×Q₁₆	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₂×#107	1	1	yes	characteristic
131	C₂×C₄	#427	1	1	yes	characteristic
132	C₂×C₄	#435	1	2	yes	—
133	C₂×C₄	C₄×D₈	1	1	yes	characteristic
134	C₂×C₄	C₄×D₈	1	1	yes	characteristic
135	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₄	#444	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	4	4	no	—
142	C₈	—	8	2	no	—
143	C₈	—	8	2	no	—
144	C₈	—	8	2	no	—
145	C₈	#368	1	4	yes	complemented
146	C₈	C₄×M₁₆	1	1	yes	characteristic
147	C₈	#405	1	1	yes	characteristic
148	C₈	#446	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#1584	1	2	yes	—
152	C₂²	C₄×#107	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#1581	1	2	yes	—
156	C₄	#1771	1	1	yes	characteristic
157	C₄	#1773	1	1	yes	characteristic
158	C₄	C₂×#368	1	2	yes	—
159	C₄	#2179	1	2	yes	—
160	C₂	—	2	2	no	—
161	C₂	#8291	1	1	yes	characteristic
162	C₂	#12671	1	1	yes	characteristic
163	C₂	C₄×#368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#368

C₂

yes

maximal, characteristic

#12008

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

#8293

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7172

C₂

yes

maximal, characteristic

#7172

C₂

yes

maximal, characteristic

C₂×#368

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1451

—

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

characteristic

#1381

C₄

yes

—

#1372

C₄

yes

—

#1371

C₄

yes

—

#1312

C₄

yes

—

#1312

C₄

yes

—

#1278

—

#1278

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₂×#108

C₂×C₄

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

—

C₂×C₄

yes

—

#335

C₂×C₄

yes

—

—

—

—

—

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

—

—

—

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#107

yes

—

110

C₂×C₈

#107

yes

—

111

C₂×C₈

C₂×C₄²

yes

characteristic

112

C₂×C₈

#117

yes

characteristic

113

C₂×C₈

C₄×Q₈

yes

characteristic

114

C₂×C₈

C₄×Q₈

yes

characteristic

115

C₂×C₈

C₂×M₁₆

yes

—

116

C₂×C₈

#131

yes

—

117

C₄²

C₂×C₄⋊C₄

yes

characteristic

118

C₄²

C₄×D₄

yes

characteristic

119

C₄²

C₂×D₈

yes

characteristic

120

C₄²

#135

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

C₄×Q₁₆

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

130

C₂×C₄

C₂×#107

yes

characteristic

131

C₂×C₄

#427

yes

characteristic

132

C₂×C₄

#435

yes

—

133

C₂×C₄

C₄×D₈

yes

characteristic

134

C₂×C₄

C₄×D₈

yes

characteristic

135

C₂×C₄

C₄×Q₁₆

yes

characteristic

136

C₂×C₄

#444

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

#368

yes

complemented

146

C₈

C₄×M₁₆

yes

characteristic

147

C₈

#405

yes

characteristic

148

C₈

#446

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

#1584

yes

—

152

C₂²

C₄×#107

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

#1581

yes

—

156

C₄

#1771

yes

characteristic

157

C₄

#1773

yes

characteristic

158

C₄

C₂×#368

yes

—

159

C₄

#2179

yes

—

160

C₂

—

161

C₂

#8291

yes

characteristic

162

C₂

#12671

yes

characteristic

163

C₂

C₄×#368

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	C₄×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₈

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₄×#107

prime radical

C₂³

C₄×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
145. C₈	#368	semidirect

H ≅ G/N

type

145. C₈

#368

semidirect

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

subgroup	index
C₄×#368	2
#12008	2
C₂×#1312	2
#8293	2
C₈×#108	2
#7172	2

subgroup

index

Subgroups — #129189

Conjugacy classes of subgroups

Complements

Maximal subgroups