Group #140493 — subgroups

Subgroups — #140493

Order 512.

205 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 86 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	#12036	C₂	1	2	yes	maximal
2	#12012	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	1	yes	maximal, characteristic
4	C₈×#108	C₂	1	1	yes	maximal, characteristic
5	#7298	C₂	1	2	yes	maximal
6	#2162	C₂²	1	2	yes	—
7	#1848	—	2	1	no	—
8	C₄×#108	C₂²	1	1	yes	characteristic
9	#1567	—	2	1	no	—
10	#1567	C₂²	1	2	yes	—
11	#1564	C₂²	1	1	yes	characteristic
12	C₈×M₁₆	—	2	1	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1372	—	2	2	no	—
15	#1371	—	2	2	no	—
16	#1370	C₄	1	2	yes	—
17	#1367	C₄	1	2	yes	—
18	#1278	—	2	1	no	—
19	#431	—	2	1	no	—
20	C₂×#108	—	2	1	no	—
21	C₂×#108	—	4	2	no	—
22	#423	—	2	2	no	—
23	C₄×M₁₆	—	2	1	no	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
28	C₂²⋊C₁₆	—	2	2	no	—
29	C₄×C₁₆	—	2	2	no	—
30	#341	Q₈	1	2	yes	—
31	#335	—	2	2	no	—
32	#335	—	2	2	no	—
33	#335	—	4	1	no	—
34	#335	—	4	1	no	—
35	#335	D₄	1	1	yes	characteristic
36	#335	D₄	1	1	yes	characteristic
37	#334	—	2	1	no	—
38	#322	—	4	1	no	—
39	C₈²	—	4	1	no	—
40	C₈²	C₂×C₄	1	1	yes	characteristic
41	C₈²	C₂×C₄	1	1	yes	characteristic
42	C₈²	Q₈	1	1	yes	characteristic
43	C₈²	Q₈	1	1	yes	characteristic
44	C₂×M₁₆	—	2	2	no	—
45	C₂×M₁₆	—	4	2	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	Q₁₆	1	1	yes	characteristic
48	C₂²×C₈	Q₁₆	1	1	yes	characteristic
49	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	4	no	—
53	#108	—	2	4	no	—
54	#108	—	8	4	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₂²⋊C₈	—	4	1	no	—
60	#97	—	4	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	4	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₈	D₈	1	1	yes	characteristic
69	C₄×C₈	D₈	1	1	yes	characteristic
70	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	M₁₆	—	4	2	no	—
79	M₁₆	—	8	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	#107	1	1	yes	characteristic
93	C₂×C₈	#107	1	1	yes	characteristic
94	C₂×C₈	#107	1	1	yes	characteristic
95	C₂×C₈	#107	1	1	yes	characteristic
96	C₂×C₈	C₄×D₄	1	1	yes	characteristic
97	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₈	#125	1	1	yes	characteristic
99	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
100	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄²	#121	1	1	yes	characteristic
102	C₄²	C₂×D₈	1	1	yes	characteristic
103	C₄²	#137	1	1	yes	characteristic
104	C₁₆	—	4	4	no	—
105	C₂³	#477	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	#389	1	1	yes	characteristic
113	C₂×C₄	C₂×#107	1	1	yes	characteristic
114	C₂×C₄	#428	1	1	yes	characteristic
115	C₂×C₄	#459	1	1	yes	characteristic
116	C₂×C₄	#462	1	1	yes	characteristic
117	C₂×C₄	#474	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	2	2	no	—
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	4	4	no	—
127	C₈	—	8	1	no	—
128	C₈	—	8	1	no	—
129	C₈	—	16	2	no	—
130	C₈	#368	1	2	yes	—
131	C₈	#443	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#1947	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	#1945	1	1	yes	characteristic
140	C₄	#1953	1	1	yes	characteristic
141	C₄	C₂×#368	1	1	yes	characteristic
142	C₄	#2161	1	1	yes	characteristic
143	C₄	#2213	1	1	yes	characteristic
144	C₄	#2214	1	1	yes	characteristic
145	C₂	—	4	1	no	—
146	C₂	#9627	1	1	yes	characteristic
147	C₂	#13179	1	1	yes	characteristic
148	C₂	#13181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12036

C₂

yes

maximal

#12012

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7298

C₂

yes

maximal

#2162

C₂²

yes

—

#1848

—

C₄×#108

C₂²

yes

characteristic

#1567

—

#1567

C₂²

yes

—

#1564

C₂²

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

—

#1367

C₄

yes

—

—

—

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

C₂×C₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

Q₈

yes

—

—

—

—

—

D₄

yes

characteristic

#335

D₄

yes

characteristic

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Q₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

M₁₆

—

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

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

100

C₄²

C₂×C₄⋊C₄

yes

characteristic

101

C₄²

#121

yes

characteristic

102

C₄²

C₂×D₈

yes

characteristic

103

C₄²

#137

yes

characteristic

104

C₁₆

—

105

C₂³

#477

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#389

yes

characteristic

113

C₂×C₄

C₂×#107

yes

characteristic

114

C₂×C₄

#428

yes

characteristic

115

C₂×C₄

#459

yes

characteristic

116

C₂×C₄

#462

yes

characteristic

117

C₂×C₄

#474

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#368

yes

—

131

C₈

#443

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1947

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1945

yes

characteristic

140

C₄

#1953

yes

characteristic

141

C₄

C₂×#368

yes

characteristic

142

C₄

#2161

yes

characteristic

143

C₄

#2213

yes

characteristic

144

C₄

#2214

yes

characteristic

145

C₂

—

146

C₂

#9627

yes

characteristic

147

C₂

#13179

yes

characteristic

148

C₂

#13181

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂³	#477
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#459

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#12036	2
#12012	2
#8823	2
C₈×#108	2
#7298	2

subgroup

index

Subgroups — #140493

Conjugacy classes of subgroups

Maximal subgroups