Group #142134 — subgroups

Subgroups — #142134

Order 512.

308 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 118 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	#33258	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	#8274	C₂	1	1	yes	maximal, characteristic
4	#7274	C₂	1	4	yes	maximal
5	#2859	—	2	1	no	—
6	C₂×C₄⋊C₁₆	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	#1942	—	2	2	no	—
9	#1942	C₄	1	4	yes	—
10	#1845	C₂²	1	2	yes	—
11	#1834	—	2	1	no	—
12	#1767	—	2	1	no	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₂×#335	C₂²	1	2	yes	—
15	#1277	—	2	1	no	—
16	C₂×#125	—	4	1	no	—
17	C₂²×C₁₆	—	2	2	no	—
18	C₂×#107	—	2	2	no	—
19	C₂×#107	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×#97	—	4	1	no	—
22	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	D₄	1	2	yes	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#389	—	2	4	no	—
27	#389	—	4	2	no	—
28	#384	—	2	2	no	—
29	#382	—	4	1	no	—
30	C₄³	D₄	1	1	yes	characteristic
31	C₄⋊C₁₆	—	2	4	no	—
32	C₄⋊C₁₆	—	4	4	no	—
33	#340	—	2	4	no	—
34	#336	—	4	1	no	—
35	#335	—	2	2	no	—
36	#335	Q₈	1	4	yes	—
37	C₂²×C₈	—	4	2	no	—
38	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	#125	—	8	2	no	—
42	#125	—	8	2	no	—
43	C₂×C₄⋊C₄	—	2	4	no	—
44	C₂×C₄⋊C₄	—	4	4	no	—
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	2	no	—
47	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄²	D₈	1	1	yes	characteristic
49	C₂×C₄²	D₈	1	1	yes	characteristic
50	C₂×C₄²	SD₁₆	1	2	yes	—
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₁₆	—	4	2	no	—
53	C₂×C₁₆	—	4	4	no	—
54	#107	—	4	2	no	—
55	#107	—	4	2	no	—
56	#107	—	4	4	no	—
57	C₄⋊C₈	—	2	4	no	—
58	C₄⋊C₈	—	4	2	no	—
59	#97	—	8	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	C₂×Q₈	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	—	4	4	no	—
67	C₂²×C₄	—	4	4	no	—
68	C₂²×C₄	#102	1	2	yes	—
69	C₂²×C₄	#102	1	2	yes	—
70	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	—	8	2	no	—
80	C₂×C₈	C₄×Q₈	1	2	yes	—
81	C₂×C₈	#124	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₈	#137	1	2	yes	—
85	C₄⋊C₄	—	4	4	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₄⋊C₄	—	8	4	no	—
88	C₄⋊C₄	—	8	8	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₄⋊C₄	—	8	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	4	2	no	—
95	C₄²	D₁₆	1	2	yes	—
96	C₄²	SD₃₂	1	2	yes	—
97	C₄²	C₂²⋊Q₈	1	2	yes	—
98	C₄²	#137	1	2	yes	—
99	C₁₆	—	8	2	no	—
100	C₂³	C₂×#102	1	1	yes	characteristic
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	8	4	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	#357	1	4	yes	—
111	C₂×C₄	#391	1	1	yes	characteristic
112	C₂×C₄	#413	1	2	yes	—
113	C₂×C₄	#417	1	2	yes	—
114	C₂×C₄	#419	1	1	yes	characteristic
115	C₂×C₄	#474	1	2	yes	—
116	C₂×C₄	#475	1	2	yes	—
117	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
118	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
119	C₂×C₄	#510	1	2	yes	—
120	C₈	—	2	2	no	—
121	C₈	—	4	2	no	—
122	C₈	—	8	4	no	—
123	C₈	#490	1	1	yes	characteristic
124	C₈	#491	1	1	yes	characteristic
125	C₈	#501	1	2	yes	—
126	C₂²	#1547	1	2	yes	—
127	C₂²	#1988	1	1	yes	characteristic
128	C₂²	C₂×#357	1	1	yes	characteristic
129	C₂²	#2141	1	1	yes	characteristic
130	C₂²	#2148	1	2	yes	—
131	C₄	—	2	4	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	16	4	no	—
135	C₄	#1559	1	2	yes	—
136	C₄	#1992	1	1	yes	characteristic
137	C₄	#1998	1	1	yes	characteristic
138	C₄	#2224	1	2	yes	—
139	C₄	#2225	1	2	yes	—
140	C₂	#9904	1	1	yes	characteristic
141	C₂	#11981	1	2	yes	—
142	C₂	#11987	1	2	yes	—
143	C₂	#13275	1	1	yes	characteristic
144	C₂	#13280	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33258

C₂

yes

maximal, characteristic

#12898

C₂

yes

maximal, characteristic

#8274

C₂

yes

maximal, characteristic

#7274

C₂

yes

maximal

#2859

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1942

—

#1942

C₄

yes

—

#1845

C₂²

yes

—

#1834

—

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

#1277

—

C₂×#125

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#102

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₁₆

yes

—

C₄²

SD₃₂

yes

—

C₄²

C₂²⋊Q₈

yes

—

C₄²

#137

yes

—

C₁₆

—

100

C₂³

C₂×#102

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#357

yes

—

111

C₂×C₄

#391

yes

characteristic

112

C₂×C₄

#413

yes

—

113

C₂×C₄

#417

yes

—

114

C₂×C₄

#419

yes

characteristic

115

C₂×C₄

#474

yes

—

116

C₂×C₄

#475

yes

—

117

C₂×C₄

C₂×D₁₆

yes

characteristic

118

C₂×C₄

C₂×SD₃₂

yes

characteristic

119

C₂×C₄

#510

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

#490

yes

characteristic

124

C₈

#491

yes

characteristic

125

C₈

#501

yes

—

126

C₂²

#1547

yes

—

127

C₂²

#1988

yes

characteristic

128

C₂²

C₂×#357

yes

characteristic

129

C₂²

#2141

yes

characteristic

130

C₂²

#2148

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#1559

yes

—

136

C₄

#1992

yes

characteristic

137

C₄

#1998

yes

characteristic

138

C₄

#2224

yes

—

139

C₄

#2225

yes

—

140

C₂

#9904

yes

characteristic

141

C₂

#11981

yes

—

142

C₂

#11987

yes

—

143

C₂

#13275

yes

characteristic

144

C₂

#13280

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ 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
#33258	2
#12898	2
#8274	2
#7274	2

subgroup

index

Subgroups — #142134

Conjugacy classes of subgroups

Maximal subgroups