Group #142076 — subgroups

Subgroups — #142076

Order 512.

330 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 122 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	#33313	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	#8266	C₂	1	1	yes	maximal, characteristic
4	#7279	C₂	1	4	yes	maximal
5	C₂×#500	—	2	1	no	—
6	C₂×#498	C₄	1	2	yes	—
7	#2862	—	2	1	no	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	—	2	1	no	—
10	#2150	C₂²	1	2	yes	—
11	#1846	—	2	1	no	—
12	#1846	—	2	1	no	—
13	#1846	C₄	1	2	yes	—
14	#1846	C₂²	1	2	yes	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	2	yes	—
17	#1450	—	2	1	no	—
18	#1276	—	2	1	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×M₃₂	—	2	2	no	—
21	#500	—	4	2	no	—
22	#500	—	4	2	no	—
23	#498	—	4	4	no	—
24	#431	—	2	2	no	—
25	C₂×#107	—	2	2	no	—
26	C₂×#107	—	2	2	no	—
27	C₂×#106	—	2	2	no	—
28	C₂×#106	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	#391	—	4	4	no	—
36	#382	—	2	2	no	—
37	#382	—	2	2	no	—
38	C₄³	D₄	1	1	yes	characteristic
39	#364	—	4	2	no	—
40	#364	—	4	2	no	—
41	C₄⋊C₁₆	—	2	4	no	—
42	#335	—	2	2	no	—
43	#334	Q₈	1	4	yes	complemented
44	#322	—	4	1	no	—
45	C₈²	—	4	1	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₄⋊Q₈	—	4	4	no	—
52	C₄⋊Q₈	—	4	4	no	—
53	C₂×C₄⋊C₄	—	2	4	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄²	—	2	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	M₃₂	—	4	2	no	—
60	M₃₂	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	#107	—	4	4	no	—
63	#107	—	4	2	no	—
64	#107	—	4	2	no	—
65	#106	—	4	4	no	—
66	#106	—	4	4	no	—
67	C₄⋊C₈	—	2	4	no	—
68	C₂²⋊C₈	—	4	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	4	2	no	—
72	C₄×C₈	D₈	1	2	yes	—
73	C₄×C₈	SD₁₆	1	2	yes	—
74	C₄×C₈	SD₁₆	1	2	yes	—
75	C₄×C₈	SD₁₆	1	1	yes	characteristic
76	C₄×C₈	SD₁₆	1	1	yes	characteristic
77	C₄×C₈	Q₁₆	1	2	yes	—
78	C₄×C₈	Q₁₆	1	1	yes	characteristic
79	C₄×C₈	Q₁₆	1	1	yes	characteristic
80	C₄×C₈	C₂×Q₈	1	2	yes	—
81	C₄×C₈	#41	1	2	yes	—
82	#95	—	4	4	no	—
83	#95	—	4	4	no	—
84	C₂×Q₈	—	8	4	no	—
85	C₂×Q₈	—	8	8	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	4	4	no	—
89	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	#100	1	2	yes	—
95	C₂×C₈	#101	1	2	yes	—
96	C₂×C₈	#102	1	2	yes	—
97	C₂×C₈	#103	1	1	yes	characteristic
98	C₂×C₈	#103	1	1	yes	characteristic
99	C₂×C₈	#103	1	1	yes	characteristic
100	C₂×C₈	#103	1	1	yes	characteristic
101	C₂×C₈	#103	1	2	yes	—
102	C₂×C₈	C₄×Q₈	1	2	yes	—
103	C₂×C₈	#124	1	1	yes	characteristic
104	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₈	C₂×D₈	1	1	yes	characteristic
106	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	8	4	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	4	4	no	—
115	C₄²	#102	1	2	yes	—
116	C₄²	#103	1	2	yes	—
117	C₄²	C₂²⋊Q₈	1	2	yes	—
118	C₄²	C₂×SD₁₆	1	1	yes	characteristic
119	C₄²	C₂×Q₁₆	1	1	yes	characteristic
120	C₁₆	—	8	2	no	—
121	C₂³	#418	1	1	yes	characteristic
122	Q₈	—	16	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	8	8	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	#391	1	1	yes	characteristic
131	C₂×C₄	C₂×#100	1	1	yes	characteristic
132	C₂×C₄	C₂×#101	1	1	yes	characteristic
133	C₂×C₄	C₂×#102	1	1	yes	characteristic
134	C₂×C₄	C₂×#103	1	1	yes	characteristic
135	C₂×C₄	C₂×#103	1	1	yes	characteristic
136	C₂×C₄	#476	1	2	yes	—
137	C₂×C₄	#477	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	8	4	no	—
140	C₈	#332	1	2	yes	—
141	C₈	#332	1	2	yes	—
142	C₈	#333	1	2	yes	—
143	C₈	#333	1	2	yes	—
144	C₈	#486	1	1	yes	characteristic
145	C₈	#487	1	1	yes	characteristic
146	C₈	#498	1	1	yes	characteristic
147	C₈	#500	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	#1545	1	1	yes	characteristic
150	C₂²	#1549	1	1	yes	characteristic
151	C₂²	#1987	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	16	4	no	—
157	C₄	C₂×#332	1	1	yes	characteristic
158	C₄	C₂×#333	1	1	yes	characteristic
159	C₄	#1990	1	1	yes	characteristic
160	C₄	#1997	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	#9896	1	1	yes	characteristic
164	C₂	#13271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

#8266

C₂

yes

maximal, characteristic

#7279

C₂

yes

maximal

C₂×#500

—

C₂×#498

C₄

yes

—

—

—

—

C₂²

yes

—

#1846

—

#1846

—

#1846

C₄

yes

—

#1846

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

#1450

—

#1276

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

D₄

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

#335

—

#334

Q₈

yes

complemented

#322

—

C₈²

—

C₂²×Q₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#100

yes

—

C₂×C₈

#101

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

100

C₂×C₈

#103

yes

characteristic

101

C₂×C₈

#103

yes

—

102

C₂×C₈

C₄×Q₈

yes

—

103

C₂×C₈

#124

yes

characteristic

104

C₂×C₈

C₄⋊Q₈

yes

characteristic

105

C₂×C₈

C₂×D₈

yes

characteristic

106

C₂×C₈

C₂×SD₁₆

yes

characteristic

107

C₂×C₈

C₂×SD₁₆

yes

characteristic

108

C₂×C₈

C₂×Q₁₆

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

#102

yes

—

116

C₄²

#103

yes

—

117

C₄²

C₂²⋊Q₈

yes

—

118

C₄²

C₂×SD₁₆

yes

characteristic

119

C₄²

C₂×Q₁₆

yes

characteristic

120

C₁₆

—

121

C₂³

#418

yes

characteristic

122

Q₈

—

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×#100

yes

characteristic

132

C₂×C₄

C₂×#101

yes

characteristic

133

C₂×C₄

C₂×#102

yes

characteristic

134

C₂×C₄

C₂×#103

yes

characteristic

135

C₂×C₄

C₂×#103

yes

characteristic

136

C₂×C₄

#476

yes

—

137

C₂×C₄

#477

yes

—

138

C₈

—

139

C₈

—

140

C₈

#332

yes

—

141

C₈

#332

yes

—

142

C₈

#333

yes

—

143

C₈

#333

yes

—

144

C₈

#486

yes

characteristic

145

C₈

#487

yes

characteristic

146

C₈

#498

yes

characteristic

147

C₈

#500

yes

characteristic

148

C₂²

—

149

C₂²

#1545

yes

characteristic

150

C₂²

#1549

yes

characteristic

151

C₂²

#1987

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×#332

yes

characteristic

158

C₄

C₂×#333

yes

characteristic

159

C₄

#1990

yes

characteristic

160

C₄

#1997

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

#9896

yes

characteristic

164

C₂

#13271

yes

—

	subgroup	G/H
center, socle	C₂²	#1987
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1987

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
43. #334	Q₈	semidirect

H ≅ G/N

type

43. #334

Q₈

semidirect

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

subgroup	index
#33313	2
#12900	2
#8266	2
#7279	2

subgroup

index

Subgroups — #142076

Conjugacy classes of subgroups

Complements

Maximal subgroups