Group #146111 — subgroups

Subgroups — #146111

Order 512.

173 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 52 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	#11975	C₂	1	1	yes	maximal, characteristic
2	#11974	C₂	1	1	yes	maximal, characteristic
3	#9778	C₂	1	1	yes	maximal, characteristic
4	#8798	C₂	1	1	yes	maximal, characteristic
5	#7224	C₂	1	1	yes	maximal, characteristic
6	#7224	C₂	1	1	yes	maximal, characteristic
7	#7197	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	#2163	C₂²	1	1	yes	characteristic
10	#2163	C₂²	1	1	yes	characteristic
11	#1837	C₂²	1	1	yes	characteristic
12	#1568	C₂²	1	1	yes	characteristic
13	#1568	C₂²	1	1	yes	characteristic
14	#1549	—	2	1	no	—
15	#1548	—	2	1	no	—
16	#1452	C₂²	1	1	yes	characteristic
17	#1365	—	2	2	no	—
18	#1364	—	2	2	no	—
19	#1287	—	2	1	no	—
20	#1287	—	2	1	no	—
21	C₂×C₄⋊Q₈	—	2	1	no	—
22	#500	—	2	1	no	—
23	#500	—	2	1	no	—
24	#500	—	2	1	no	—
25	#500	D₄	1	2	yes	—
26	#432	—	2	1	no	—
27	#432	—	2	1	no	—
28	#431	D₄	1	1	yes	characteristic
29	#431	D₄	1	1	yes	characteristic
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	#423	—	2	1	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	1	no	—
35	C₄⋊C₁₆	—	2	1	no	—
36	C₂²⋊C₁₆	—	2	1	no	—
37	C₂²⋊C₁₆	—	2	1	no	—
38	#346	—	2	1	no	—
39	#346	—	2	1	no	—
40	#341	—	2	1	no	—
41	#341	—	2	1	no	—
42	#335	—	2	1	no	—
43	#335	—	2	1	no	—
44	#334	—	2	1	no	—
45	#334	—	2	1	no	—
46	#333	—	4	2	no	—
47	#332	—	4	2	no	—
48	#322	—	2	1	no	—
49	#322	—	2	1	no	—
50	C₂²×Q₈	—	4	1	no	—
51	C₂×M₁₆	—	4	1	no	—
52	C₂²×C₈	#41	1	1	yes	characteristic
53	C₂²×C₈	#41	1	1	yes	characteristic
54	C₄⋊Q₈	—	2	2	no	—
55	C₄⋊Q₈	—	4	1	no	—
56	C₄⋊Q₈	—	4	1	no	—
57	C₄⋊Q₈	—	4	1	no	—
58	C₂×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄⋊C₄	—	4	1	no	—
60	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	—	4	1	no	—
63	#107	—	2	2	no	—
64	#107	—	2	2	no	—
65	#107	—	4	1	no	—
66	#107	—	4	1	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	4	1	no	—
70	C₄⋊C₈	—	4	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	C₄⋊C₈	—	4	1	no	—
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	—	2	1	no	—
75	C₂²⋊C₈	—	4	1	no	—
76	C₂²⋊C₈	—	4	1	no	—
77	#97	—	4	1	no	—
78	#97	—	4	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	16	2	no	—
88	C₂²×C₄	—	8	1	no	—
89	C₂²×C₄	#124	1	1	yes	characteristic
90	C₂²×C₄	#135	1	1	yes	characteristic
91	C₂²×C₄	#135	1	1	yes	characteristic
92	M₁₆	—	8	1	no	—
93	M₁₆	—	8	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	8	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	#123	1	1	yes	characteristic
102	C₂×C₈	#123	1	1	yes	characteristic
103	C₂×C₈	#126	1	1	yes	characteristic
104	C₂×C₈	#126	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	8	1	no	—
108	C₄⋊C₄	—	8	1	no	—
109	C₄²	#121	1	1	yes	characteristic
110	C₄²	#123	1	1	yes	characteristic
111	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₁₆	—	8	1	no	—
114	C₁₆	—	8	1	no	—
115	C₂³	#488	1	1	yes	characteristic
116	Q₈	—	16	2	no	—
117	Q₈	—	16	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	16	2	no	—
124	C₂×C₄	#397	1	1	yes	characteristic
125	C₂×C₄	#463	1	1	yes	characteristic
126	C₂×C₄	#463	1	1	yes	characteristic
127	C₂×C₄	#482	1	1	yes	characteristic
128	C₂×C₄	#482	1	1	yes	characteristic
129	C₂×C₄	#490	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	8	1	no	—
135	C₈	—	8	1	no	—
136	C₈	—	16	1	no	—
137	C₈	—	16	1	no	—
138	C₈	—	16	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#2073	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	16	2	no	—
149	C₄	#2082	1	1	yes	characteristic
150	C₄	#2082	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#10635	1	1	yes	characteristic
153	C₂	#13490	1	1	yes	characteristic
154	C₂	#13491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11975

C₂

yes

maximal, characteristic

#11974

C₂

yes

maximal, characteristic

#9778

C₂

yes

maximal, characteristic

#8798

C₂

yes

maximal, characteristic

#7224

C₂

yes

maximal, characteristic

#7224

C₂

yes

maximal, characteristic

#7197

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

#2163

C₂²

yes

characteristic

#2163

C₂²

yes

characteristic

#1837

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1568

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈

—

—

—

—

D₄

yes

—

#432

—

#432

—

#431

D₄

yes

characteristic

#431

D₄

yes

—

—

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

yes

C₂²×C₈

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#123

yes

characteristic

102

C₂×C₈

#123

yes

characteristic

103

C₂×C₈

#126

yes

characteristic

104

C₂×C₈

#126

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

#121

yes

characteristic

110

C₄²

#123

yes

characteristic

111

C₄²

AGL(1,ℤ/8)

yes

characteristic

112

C₄²

AGL(1,ℤ/8)

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₂³

#488

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#397

yes

characteristic

125

C₂×C₄

#463

yes

characteristic

126

C₂×C₄

#463

yes

characteristic

127

C₂×C₄

#482

yes

characteristic

128

C₂×C₄

#482

yes

characteristic

129

C₂×C₄

#490

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#2073

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#2082

yes

characteristic

150

C₄

#2082

yes

characteristic

151

C₂

—

152

C₂

#10635

yes

characteristic

153

C₂

#13490

yes

characteristic

154

C₂

#13491

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#488
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2073

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#488

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ 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
#11975	2
#11974	2
#9778	2
#8798	2
#7224	2
#7197	2

subgroup

index

Subgroups — #146111

Conjugacy classes of subgroups

Maximal subgroups