Group #142123 — subgroups

Subgroups — #142123

Order 512.

251 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 90 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	#33260	C₂	1	1	yes	maximal, characteristic
2	#12881	C₂	1	1	yes	maximal, characteristic
3	#12656	C₂	1	1	yes	maximal, characteristic
4	#7271	C₂	1	2	yes	maximal
5	#7268	C₂	1	2	yes	maximal
6	#2647	C₄	1	2	yes	—
7	C₂×#364	C₂²	1	2	yes	—
8	C₂×#364	C₂²	1	2	yes	—
9	C₄×M₃₂	—	2	1	no	—
10	#1944	—	2	1	no	—
11	#1943	—	2	1	no	—
12	#1941	—	2	1	no	—
13	#1941	—	2	1	no	—
14	#1846	C₂²	1	1	yes	characteristic
15	#1844	C₂²	1	1	yes	characteristic
16	#1836	C₄	1	2	yes	—
17	#1726	C₂²	1	1	yes	characteristic
18	#1377	—	2	1	no	—
19	#1376	—	2	1	no	—
20	#1375	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×#107	—	2	1	no	—
24	C₂×#106	—	2	1	no	—
25	C₂×#106	—	2	1	no	—
26	C₂×#106	—	2	1	no	—
27	C₂×#97	C₂×C₄	1	1	yes	characteristic
28	C₂×#97	D₄	1	1	yes	characteristic
29	C₂×#97	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	#400	—	4	2	no	—
34	#398	—	4	2	no	—
35	#389	—	2	2	no	—
36	#389	—	2	2	no	—
37	#384	C₂×C₄	1	2	yes	—
38	#382	C₂×C₄	1	2	yes	—
39	#376	C₂×C₄	1	1	yes	characteristic
40	#364	—	2	2	no	—
41	#364	—	2	4	no	—
42	#364	Q₈	1	4	yes	—
43	#346	—	4	1	no	—
44	C₄×C₁₆	—	4	1	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	4	2	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
59	M₃₂	—	2	4	no	—
60	M₃₂	—	4	2	no	—
61	M₃₂	—	4	2	no	—
62	M₃₂	—	4	2	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	#107	—	4	2	no	—
66	#106	—	4	2	no	—
67	#106	—	4	2	no	—
68	#106	—	4	1	no	—
69	#106	—	4	1	no	—
70	#97	—	2	2	no	—
71	#97	—	2	2	no	—
72	#97	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	C₂×Q₈	1	2	yes	—
77	C₄×C₈	#41	1	2	yes	—
78	#95	—	2	2	no	—
79	#95	—	4	2	no	—
80	#95	—	4	2	no	—
81	#95	—	4	2	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	—	4	2	no	—
86	C₂²×C₄	#100	1	2	yes	—
87	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#101	1	2	yes	—
94	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₈	#124	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₈	#135	1	2	yes	—
98	C₂×C₈	#135	1	2	yes	—
99	C₂×C₈	#135	1	2	yes	—
100	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₈	#137	1	1	yes	characteristic
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	16	2	no	—
107	C₄⋊C₄	—	16	2	no	—
108	C₄⋊C₄	—	16	2	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	C₄×Q₈	1	2	yes	—
115	C₁₆	—	4	4	no	—
116	C₁₆	—	8	2	no	—
117	C₂³	C₂×#100	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	#391	1	1	yes	characteristic
131	C₂×C₄	C₂×#101	1	1	yes	characteristic
132	C₂×C₄	#416	1	1	yes	characteristic
133	C₂×C₄	#416	1	1	yes	characteristic
134	C₂×C₄	#416	1	1	yes	characteristic
135	C₂×C₄	#417	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	4	1	no	—
138	C₈	#478	1	2	yes	—
139	C₈	#479	1	2	yes	—
140	C₈	#488	1	2	yes	—
141	C₈	#499	1	2	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	#1543	1	1	yes	characteristic
144	C₂²	#1551	1	1	yes	characteristic
145	C₂²	#1997	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	1	no	—
149	C₄	—	16	2	no	—
150	C₄	—	16	2	no	—
151	C₄	#1544	1	1	yes	characteristic
152	C₄	#1552	1	1	yes	characteristic
153	C₄	#1989	1	1	yes	characteristic
154	C₄	#1991	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#9901	1	1	yes	characteristic
158	C₂	#11991	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33260

C₂

yes

maximal, characteristic

#12881

C₂

yes

maximal, characteristic

#12656

C₂

yes

maximal, characteristic

#7271

C₂

yes

maximal

#7268

C₂

yes

maximal

#2647

C₄

yes

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

C₄×M₃₂

—

—

—

—

—

C₂²

yes

characteristic

#1844

C₂²

yes

characteristic

#1836

C₄

yes

—

#1726

C₂²

yes

—

—

—

C₂×M₃₂

—

C₂×M₃₂

—

—

—

—

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

#382

C₂×C₄

yes

—

#376

C₂×C₄

yes

—

—

Q₈

yes

—

#346

—

C₄×C₁₆

—

#340

—

#340

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

—

—

—

—

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

#101

yes

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

#135

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#135

yes

—

100

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

101

C₂×C₈

#137

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₄×Q₈

yes

—

115

C₁₆

—

116

C₁₆

—

117

C₂³

C₂×#100

yes

characteristic

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

#391

yes

characteristic

131

C₂×C₄

C₂×#101

yes

characteristic

132

C₂×C₄

#416

yes

characteristic

133

C₂×C₄

#416

yes

characteristic

134

C₂×C₄

#416

yes

characteristic

135

C₂×C₄

#417

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

#478

yes

—

139

C₈

#479

yes

—

140

C₈

#488

yes

—

141

C₈

#499

yes

—

142

C₂²

—

143

C₂²

#1543

yes

characteristic

144

C₂²

#1551

yes

characteristic

145

C₂²

#1997

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#1544

yes

characteristic

152

C₄

#1552

yes

characteristic

153

C₄

#1989

yes

characteristic

154

C₄

#1991

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#9901

yes

characteristic

158

C₂

#11991

yes

—

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1551

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ 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
#33260	2
#12881	2
#12656	2
#7271	2
#7268	2

subgroup

index

Subgroups — #142123

Conjugacy classes of subgroups

Maximal subgroups