Group #150792 — subgroups

Subgroups — #150792

Order 512.

209 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 74 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	#33734	C₂	1	1	yes	maximal, characteristic
2	#13639	C₂	1	1	yes	maximal, characteristic
3	#12891	C₂	1	1	yes	maximal, characteristic
4	#7421	C₂	1	4	yes	maximal
5	#3170	—	2	1	no	—
6	#2252	C₄	1	1	yes	characteristic
7	#2247	C₄	1	1	yes	characteristic
8	#2241	—	2	1	no	—
9	#2216	—	2	1	no	—
10	#2216	C₂²	1	2	yes	—
11	C₄×Q₃₂	—	2	1	no	—
12	C₄×Q₃₂	C₄	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1848	—	2	1	no	—
15	C₄⋊C₃₂	—	2	1	no	—
16	C₂²⋊C₃₂	C₂²	1	2	yes	—
17	C₄×C₃₂	—	2	1	no	—
18	#1381	—	2	1	no	—
19	#1381	C₂²	1	2	yes	—
20	#533	—	4	1	no	—
21	C₂×Q₃₂	—	2	1	no	—
22	C₂×Q₃₂	C₂×C₄	1	2	yes	—
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂³	1	1	yes	characteristic
25	#500	—	2	1	no	—
26	#494	—	2	1	no	—
27	#487	—	4	1	no	—
28	#467	—	2	2	no	—
29	#467	—	4	1	no	—
30	C₄×Q₁₆	—	2	2	no	—
31	C₄×Q₁₆	—	4	1	no	—
32	C₂×#108	—	2	2	no	—
33	C₂×#108	—	4	1	no	—
34	#423	—	4	1	no	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃₂	—	2	2	no	—
37	#366	—	2	1	no	—
38	#366	C₂×C₄	1	2	yes	—
39	#358	—	2	2	no	—
40	#358	—	2	2	no	—
41	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	—	4	1	no	—
47	C₂×M₁₆	—	4	2	no	—
48	C₂²×C₈	D₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Q₈	—	4	1	no	—
51	C₄⋊Q₈	—	4	1	no	—
52	#125	—	8	1	no	—
53	C₂²⋊Q₈	—	4	2	no	—
54	C₂²⋊Q₈	—	8	1	no	—
55	C₄×Q₈	—	4	2	no	—
56	C₄×Q₈	—	8	1	no	—
57	#117	—	8	1	no	—
58	C₂×C₄²	D₈	1	1	yes	characteristic
59	Q₃₂	—	4	2	no	—
60	Q₃₂	—	4	2	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	—	4	1	no	—
63	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
67	#108	—	4	2	no	—
68	#108	—	4	2	no	—
69	#108	—	8	2	no	—
70	#107	—	2	2	no	—
71	#107	—	4	1	no	—
72	C₄⋊C₈	—	8	1	no	—
73	C₄⋊C₈	—	8	1	no	—
74	#103	—	4	2	no	—
75	#103	—	4	2	no	—
76	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₈	SD₁₆	1	1	yes	characteristic
79	C₄×C₈	SD₁₆	1	1	yes	characteristic
80	C₃₂	—	4	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	8	1	no	—
83	C₂²×C₄	SD₃₂	1	1	yes	characteristic
84	C₂²×C₄	SD₃₂	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
86	Q₁₆	—	4	2	no	—
87	Q₁₆	—	4	2	no	—
88	Q₁₆	—	8	2	no	—
89	M₁₆	—	8	2	no	—
90	M₁₆	—	8	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	#102	1	1	yes	characteristic
96	C₂×C₈	#102	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄⋊C₄	—	8	2	no	—
103	C₄⋊C₄	—	8	1	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₂²⋊C₄	—	8	2	no	—
106	C₄²	—	8	2	no	—
107	C₄²	#102	1	1	yes	characteristic
108	C₄²	#102	1	1	yes	characteristic
109	C₄²	D₁₆	1	1	yes	characteristic
110	C₄²	D₁₆	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	2	1	no	—
115	C₂³	C₂×SD₃₂	1	1	yes	characteristic
116	Q₈	—	8	2	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	16	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#357	1	1	yes	characteristic
126	C₂×C₄	#357	1	1	yes	characteristic
127	C₂×C₄	#357	1	1	yes	characteristic
128	C₂×C₄	#357	1	1	yes	characteristic
129	C₂×C₄	C₂×#102	1	1	yes	characteristic
130	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	2	1	no	—
133	C₈	—	16	2	no	—
134	C₈	#360	1	2	yes	—
135	C₈	#410	1	1	yes	characteristic
136	C₈	#421	1	1	yes	characteristic
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#357	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	16	2	no	—
145	C₄	—	16	2	no	—
146	C₄	#1417	1	2	yes	—
147	C₄	#1420	1	2	yes	—
148	C₄	#1517	1	1	yes	characteristic
149	C₄	C₂×#360	1	1	yes	characteristic
150	C₂	—	4	1	no	—
151	C₂	#11856	1	1	yes	characteristic
152	C₂	C₂×#1417	1	1	yes	characteristic
153	C₂	C₂×#1420	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33734

C₂

yes

maximal, characteristic

#13639

C₂

yes

maximal, characteristic

#12891

C₂

yes

maximal, characteristic

#7421

C₂

yes

maximal

#3170

—

#2252

C₄

yes

characteristic

#2247

C₄

yes

—

—

C₂²

yes

—

C₄×Q₃₂

—

C₄×Q₃₂

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

C₂²

yes

—

C₄×C₃₂

—

#1381

—

#1381

C₂²

yes

—

#533

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₃₂

—

#366

—

#366

C₂×C₄

yes

—

#358

—

#358

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

#102

yes

characteristic

108

C₄²

#102

yes

characteristic

109

C₄²

D₁₆

yes

characteristic

110

C₄²

D₁₆

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₂³

C₂×SD₃₂

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#357

yes

characteristic

126

C₂×C₄

#357

yes

characteristic

127

C₂×C₄

#357

yes

characteristic

128

C₂×C₄

#357

yes

characteristic

129

C₂×C₄

C₂×#102

yes

characteristic

130

C₂×C₄

C₂×D₁₆

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#360

yes

—

135

C₈

#410

yes

characteristic

136

C₈

#421

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#357

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#1417

yes

—

147

C₄

#1420

yes

—

148

C₄

#1517

yes

characteristic

149

C₄

C₂×#360

yes

characteristic

150

C₂

—

151

C₂

#11856

yes

characteristic

152

C₂

C₂×#1417

yes

characteristic

153

C₂

C₂×#1420

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#357

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

C₂×SD₃₂

LCS

G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33734	2
#13639	2
#12891	2
#7421	2

subgroup

index

Subgroups — #150792

Conjugacy classes of subgroups

Maximal subgroups