Group #146153 — subgroups

Subgroups — #146153

Order 512.

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

normal subgroups only

This group has 211 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	#13185	C₂	1	2	yes	maximal
2	#9596	C₂	1	1	yes	maximal, characteristic
3	#8826	C₂	1	1	yes	maximal, characteristic
4	#7281	C₂	1	2	yes	maximal
5	#7152	C₂	1	1	yes	maximal, characteristic
6	C₂×#365	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#1939	—	2	1	no	—
9	#1848	C₂²	1	2	yes	—
10	C₄×#106	C₂²	1	1	yes	characteristic
11	#1567	—	2	1	no	—
12	#1567	—	2	1	no	—
13	C₂×#335	C₂²	1	1	yes	characteristic
14	C₂×#322	C₂²	1	1	yes	characteristic
15	#1383	—	2	2	no	—
16	#1279	—	2	1	no	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂×#108	—	2	2	no	—
19	C₂×#108	—	2	2	no	—
20	C₂×#106	D₄	1	1	yes	characteristic
21	C₂×#106	D₄	1	1	yes	characteristic
22	#423	—	2	2	no	—
23	C₂×C₄⋊C₈	—	2	1	no	—
24	C₂×C₄⋊C₈	—	4	1	no	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₄⋊C₄	—	2	1	no	—
29	#365	—	4	4	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	Q₈	1	4	yes	—
32	#340	—	2	1	no	—
33	#336	—	4	1	no	—
34	#335	—	2	2	no	—
35	#335	—	4	2	no	—
36	#335	—	4	2	no	—
37	#322	—	2	2	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	4	1	no	—
41	C₂²×C₈	SD₁₆	1	1	yes	characteristic
42	C₂²×C₈	SD₁₆	1	1	yes	characteristic
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄²	—	4	1	no	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	M₃₂	—	4	2	no	—
50	M₃₂	—	4	2	no	—
51	C₂×C₁₆	—	2	4	no	—
52	#108	—	4	2	no	—
53	#108	—	4	2	no	—
54	#108	—	8	2	no	—
55	#106	—	2	1	no	—
56	#106	—	2	2	no	—
57	#106	D₈	1	2	yes	—
58	C₄⋊C₈	—	4	1	no	—
59	C₄⋊C₈	—	4	2	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	4	1	no	—
62	C₄⋊C₈	—	8	2	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	SD₁₆	1	2	yes	—
67	C₄×C₈	C₂×Q₈	1	2	yes	—
68	C₄×C₈	#41	1	2	yes	—
69	#95	—	4	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	—	4	1	no	—
72	C₂²×C₄	#121	1	1	yes	characteristic
73	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
74	C₂²×C₄	#137	1	1	yes	characteristic
75	M₁₆	—	8	2	no	—
76	M₁₆	—	8	2	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	8	1	no	—
85	C₂×C₈	—	8	1	no	—
86	C₂×C₈	—	8	1	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	—	8	1	no	—
89	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₈	#125	1	2	yes	—
91	C₂×C₈	C₂×D₈	1	1	yes	characteristic
92	C₂×C₈	#135	1	1	yes	characteristic
93	C₂×C₈	#137	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄²	—	8	1	no	—
99	C₄²	—	8	1	no	—
100	C₄²	#123	1	1	yes	characteristic
101	C₄²	C₄⋊Q₈	1	1	yes	characteristic
102	C₄²	C₂×SD₁₆	1	1	yes	characteristic
103	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₁₆	—	4	4	no	—
105	C₂³	#461	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	#398	1	1	yes	characteristic
116	C₂×C₄	#460	1	1	yes	characteristic
117	C₂×C₄	#466	1	1	yes	characteristic
118	C₂×C₄	#470	1	1	yes	characteristic
119	C₂×C₄	#472	1	1	yes	characteristic
120	C₂×C₄	#473	1	1	yes	characteristic
121	C₂×C₄	#481	1	1	yes	characteristic
122	C₂×C₄	#483	1	1	yes	characteristic
123	C₂×C₄	#498	1	1	yes	characteristic
124	C₂×C₄	#501	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	4	2	no	—
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	16	2	no	—
130	C₈	—	16	1	no	—
131	C₈	#474	1	2	yes	—
132	C₈	#479	1	2	yes	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#1673	1	1	yes	characteristic
136	C₂²	#1695	1	1	yes	characteristic
137	C₂²	#2084	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	8	2	no	—
142	C₄	—	16	1	no	—
143	C₄	#1668	1	1	yes	characteristic
144	C₄	#1690	1	1	yes	characteristic
145	C₄	#2086	1	1	yes	characteristic
146	C₄	#2089	1	1	yes	characteristic
147	C₄	#2238	1	1	yes	characteristic
148	C₄	#2239	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#10651	1	1	yes	characteristic
152	C₂	#13510	1	1	yes	characteristic
153	C₂	#13511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13185

C₂

yes

maximal

#9596

C₂

yes

maximal, characteristic

#8826

C₂

yes

maximal, characteristic

#7281

C₂

yes

maximal

#7152

C₂

yes

maximal, characteristic

C₂×#365

—

#2150

C₂²

yes

—

#1939

—

#1848

C₂²

yes

—

C₄×#106

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#322

C₂²

yes

characteristic

#1383

—

#1279

—

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#106

D₄

yes

characteristic

C₂×#106

D₄

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

—

#365

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Q₈

yes

—

—

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

D₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

#137

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

#123

yes

characteristic

101

C₄²

C₄⋊Q₈

yes

characteristic

102

C₄²

C₂×SD₁₆

yes

characteristic

103

C₄²

AGL(1,ℤ/8)

yes

characteristic

104

C₁₆

—

105

C₂³

#461

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#398

yes

characteristic

116

C₂×C₄

#460

yes

characteristic

117

C₂×C₄

#466

yes

characteristic

118

C₂×C₄

#470

yes

characteristic

119

C₂×C₄

#472

yes

characteristic

120

C₂×C₄

#473

yes

characteristic

121

C₂×C₄

#481

yes

characteristic

122

C₂×C₄

#483

yes

characteristic

123

C₂×C₄

#498

yes

characteristic

124

C₂×C₄

#501

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

#474

yes

—

132

C₈

#479

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1673

yes

characteristic

136

C₂²

#1695

yes

characteristic

137

C₂²

#2084

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1668

yes

characteristic

144

C₄

#1690

yes

characteristic

145

C₄

#2086

yes

characteristic

146

C₄

#2089

yes

characteristic

147

C₄

#2238

yes

characteristic

148

C₄

#2239

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#10651

yes

characteristic

152

C₂

#13510

yes

characteristic

153

C₂

#13511

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2084
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2084

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#461

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
#13185	2
#9596	2
#8826	2
#7281	2
#7152	2

subgroup

index

Subgroups — #146153

Conjugacy classes of subgroups

Maximal subgroups