Group #125318 — subgroups

Subgroups — #125318

Order 512.

267 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 80 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	#33295	C₂	1	1	yes	maximal, characteristic
2	#12658	C₂	1	1	yes	maximal, characteristic
3	#12594	C₂	1	1	yes	maximal, characteristic
4	#7130	C₂	1	4	yes	maximal
5	#2859	C₄	1	1	yes	characteristic
6	#2859	C₄	1	1	yes	characteristic
7	#2150	—	2	1	no	—
8	C₂×#347	—	2	1	no	—
9	C₄×M₃₂	—	2	1	no	—
10	#1992	C₄	1	2	yes	—
11	#1992	C₂²	1	2	yes	—
12	#1991	—	2	1	no	—
13	#1991	—	2	1	no	—
14	#1836	—	2	1	no	—
15	#1836	—	2	1	no	—
16	C₄×#97	C₂²	1	1	yes	characteristic
17	#1375	—	2	1	no	—
18	#1375	C₂²	1	2	yes	—
19	#1375	C₂²	1	2	yes	—
20	C₂×#125	—	2	1	no	—
21	C₂×#125	C₂×C₄	1	2	yes	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×#97	C₂×C₄	1	1	yes	characteristic
25	C₂×#97	D₄	1	1	yes	characteristic
26	C₂×#97	D₄	1	1	yes	characteristic
27	C₂×#97	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	#389	—	4	2	no	—
31	#389	—	4	2	no	—
32	#384	—	2	1	no	—
33	#384	C₂×C₄	1	2	yes	—
34	#382	—	2	1	no	—
35	#382	—	2	1	no	—
36	C₄³	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₁₆	—	2	4	no	—
38	#347	—	4	2	no	—
39	#346	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	#340	—	2	2	no	—
42	#340	—	2	2	no	—
43	#340	—	2	2	no	—
44	#340	—	2	2	no	—
45	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	#125	—	2	4	no	—
50	#125	—	4	2	no	—
51	#125	—	4	2	no	—
52	#125	—	4	1	no	—
53	#125	—	4	1	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄²	—	2	2	no	—
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
64	M₃₂	—	4	2	no	—
65	M₃₂	—	4	2	no	—
66	M₃₂	—	4	2	no	—
67	M₃₂	—	4	2	no	—
68	C₂×C₁₆	—	2	4	no	—
69	C₂×C₁₆	—	2	4	no	—
70	#97	—	2	2	no	—
71	#97	—	2	2	no	—
72	#97	—	4	1	no	—
73	#97	SD₁₆	1	2	yes	—
74	#97	Q₁₆	1	2	yes	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₄×C₈	—	2	2	no	—
78	#95	—	4	2	no	—
79	#95	—	4	2	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	4	2	no	—
83	C₂²×C₄	—	4	2	no	—
84	C₂²×C₄	#100	1	2	yes	—
85	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	#103	1	2	yes	—
96	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₈	#137	1	2	yes	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	8	4	no	—
104	C₄⋊C₄	—	8	4	no	—
105	C₄⋊C₄	—	8	4	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	8	2	no	—
111	C₄²	#103	1	2	yes	—
112	C₄²	#104	1	2	yes	—
113	C₄²	#104	1	1	yes	characteristic
114	C₄²	#104	1	1	yes	characteristic
115	C₄²	#131	1	2	yes	—
116	C₁₆	—	4	4	no	—
117	C₁₆	—	4	4	no	—
118	C₂³	C₂×#100	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	8	4	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	#329	1	2	yes	—
129	C₂×C₄	#329	1	2	yes	—
130	C₂×C₄	#408	1	1	yes	characteristic
131	C₂×C₄	#410	1	1	yes	characteristic
132	C₂×C₄	C₂×#103	1	1	yes	characteristic
133	C₂×C₄	#419	1	1	yes	characteristic
134	C₂×C₄	C₂×#104	1	1	yes	characteristic
135	C₂×C₄	C₂×#104	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#1469	1	1	yes	characteristic
143	C₂²	C₂×#329	1	1	yes	characteristic
144	C₂²	#1548	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	16	2	no	—
150	C₄	—	16	2	no	—
151	C₄	#1485	1	1	yes	characteristic
152	C₄	#1493	1	1	yes	characteristic
153	C₄	#1503	1	1	yes	characteristic
154	C₄	#1506	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#7682	1	1	yes	characteristic
158	C₂	#11911	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33295

C₂

yes

maximal, characteristic

#12658

C₂

yes

maximal, characteristic

#12594

C₂

yes

maximal, characteristic

#7130

C₂

yes

maximal

#2859

C₄

yes

characteristic

#2859

C₄

yes

characteristic

#2150

—

C₂×#347

—

C₄×M₃₂

—

#1992

C₄

yes

—

#1992

C₂²

yes

—

—

—

—

—

C₄×#97

C₂²

yes

characteristic

#1375

—

#1375

C₂²

yes

—

#1375

C₂²

yes

—

C₂×#125

—

C₂×#125

C₂×C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

C₂³

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

—

C₂×C₄

yes

—

#382

—

#382

—

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

#347

—

#346

—

C₄×C₁₆

—

—

—

—

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

SD₁₆

yes

—

#97

Q₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#100

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#103

yes

—

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₂×C₈

#137

yes

—

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

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

#103

yes

—

112

C₄²

#104

yes

—

113

C₄²

#104

yes

characteristic

114

C₄²

#104

yes

characteristic

115

C₄²

#131

yes

—

116

C₁₆

—

117

C₁₆

—

118

C₂³

C₂×#100

yes

characteristic

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

#329

yes

—

129

C₂×C₄

#329

yes

—

130

C₂×C₄

#408

yes

characteristic

131

C₂×C₄

#410

yes

characteristic

132

C₂×C₄

C₂×#103

yes

characteristic

133

C₂×C₄

#419

yes

characteristic

134

C₂×C₄

C₂×#104

yes

characteristic

135

C₂×C₄

C₂×#104

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

#1469

yes

characteristic

143

C₂²

C₂×#329

yes

characteristic

144

C₂²

#1548

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#1485

yes

characteristic

152

C₄

#1493

yes

characteristic

153

C₄

#1503

yes

characteristic

154

C₄

#1506

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#7682

yes

characteristic

158

C₂

#11911

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#329
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#329

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×#97

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33295	2
#12658	2
#12594	2
#7130	2

subgroup

index

Subgroups — #125318

Conjugacy classes of subgroups

Maximal subgroups