Group #146248 — subgroups

Subgroups — #146248

Order 512.

238 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13286	C₂	1	1	yes	maximal, characteristic
2	#13286	C₂	1	1	yes	maximal, characteristic
3	#12910	C₂	1	1	yes	maximal, characteristic
4	#12892	C₂	1	1	yes	maximal, characteristic
5	#9861	C₂	1	1	yes	maximal, characteristic
6	#8825	C₂	1	1	yes	maximal, characteristic
7	#7137	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×#368	—	2	2	no	—
10	C₂×#346	C₂²	1	1	yes	characteristic
11	C₂×#346	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	1	yes	characteristic
15	#1566	—	2	2	no	—
16	#1493	—	2	2	no	—
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1382	—	2	2	no	—
19	#1382	—	2	2	no	—
20	#1381	—	2	2	no	—
21	#533	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	2	yes	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	Q₈	1	2	yes	—
25	#499	—	2	1	no	—
26	#499	—	2	1	no	—
27	#498	—	2	1	no	—
28	C₂×#108	—	2	1	no	—
29	C₂×#108	—	2	1	no	—
30	C₂×#108	—	2	1	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	#427	—	2	1	no	—
35	#427	—	2	1	no	—
36	#423	—	2	1	no	—
37	#423	—	2	1	no	—
38	#423	—	2	1	no	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	#368	—	4	4	no	—
42	#346	—	2	1	no	—
43	#346	—	2	1	no	—
44	#346	—	2	1	no	—
45	#346	—	2	1	no	—
46	#334	—	4	4	no	—
47	Q₈⋊C₈	—	4	4	no	—
48	#322	—	2	1	no	—
49	#322	—	2	1	no	—
50	C₂×M₁₆	—	4	1	no	—
51	C₂×M₁₆	—	4	1	no	—
52	C₂×M₁₆	—	4	1	no	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	2	no	—
57	#125	—	4	1	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	4	2	no	—
60	#117	—	4	1	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	SD₁₆	1	4	yes	—
68	#108	—	4	2	no	—
69	#108	—	4	2	no	—
70	#108	—	4	2	no	—
71	#108	—	4	2	no	—
72	#108	—	4	2	no	—
73	#108	—	4	2	no	—
74	#107	—	2	2	no	—
75	#107	—	2	2	no	—
76	#106	—	4	1	no	—
77	#106	—	4	1	no	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	D₈	1	4	yes	—
83	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	C₄×C₈	#41	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₂×Q₈	—	4	2	no	—
88	C₂²×C₄	#127	1	1	yes	characteristic
89	C₂²×C₄	#137	1	2	yes	—
90	M₁₆	—	8	1	no	—
91	M₁₆	—	8	1	no	—
92	M₁₆	—	8	1	no	—
93	M₁₆	—	8	1	no	—
94	M₁₆	—	8	1	no	—
95	M₁₆	—	8	1	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	—	8	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₈	#123	1	1	yes	characteristic
104	C₂×C₈	#123	1	1	yes	characteristic
105	C₂×C₈	C₂×SD₁₆	1	2	yes	—
106	C₂×C₈	#135	1	2	yes	—
107	C₂×C₈	#137	1	2	yes	—
108	C₂×C₈	#137	1	2	yes	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₂²⋊C₄	—	8	1	no	—
113	C₄²	—	8	1	no	—
114	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
115	C₄²	#123	1	1	yes	characteristic
116	C₄²	C₂×D₈	1	2	yes	—
117	C₁₆	—	2	4	no	—
118	C₁₆	—	4	2	no	—
119	C₂³	#497	1	1	yes	characteristic
120	Q₈	—	8	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	#399	1	1	yes	characteristic
124	C₂×C₄	#474	1	2	yes	—
125	C₂×C₄	#480	1	2	yes	—
126	C₂×C₄	#492	1	1	yes	characteristic
127	C₂×C₄	#493	1	1	yes	characteristic
128	C₂×C₄	#495	1	1	yes	characteristic
129	C₂×C₄	#497	1	1	yes	characteristic
130	C₂×C₄	#497	1	1	yes	characteristic
131	C₈	—	4	4	no	—
132	C₈	—	16	1	no	—
133	C₈	—	16	1	no	—
134	C₈	—	16	1	no	—
135	C₈	#475	1	2	yes	—
136	C₈	#479	1	2	yes	—
137	C₈	#483	1	2	yes	—
138	C₈	#485	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#1716	1	1	yes	characteristic
141	C₂²	#1718	1	1	yes	characteristic
142	C₂²	#2093	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	16	2	no	—
146	C₄	#1706	1	1	yes	characteristic
147	C₄	#1722	1	1	yes	characteristic
148	C₄	#2096	1	1	yes	characteristic
149	C₄	#2097	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	#10675	1	1	yes	characteristic
153	C₂	#13532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13286

C₂

yes

maximal, characteristic

#13286

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#12892

C₂

yes

maximal, characteristic

#9861

C₂

yes

maximal, characteristic

#8825

C₂

yes

maximal, characteristic

#7137

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#368

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₂²×C₁₆

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

#500

Q₈

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

Q₈⋊C₈

—

#322

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#137

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊Q₈

yes

characteristic

102

C₂×C₈

C₂²⋊Q₈

yes

characteristic

103

C₂×C₈

#123

yes

characteristic

104

C₂×C₈

#123

yes

characteristic

105

C₂×C₈

C₂×SD₁₆

yes

—

106

C₂×C₈

#135

yes

—

107

C₂×C₈

#137

yes

—

108

C₂×C₈

#137

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊Q₈

yes

characteristic

115

C₄²

#123

yes

characteristic

116

C₄²

C₂×D₈

yes

—

117

C₁₆

—

118

C₁₆

—

119

C₂³

#497

yes

characteristic

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

#399

yes

characteristic

124

C₂×C₄

#474

yes

—

125

C₂×C₄

#480

yes

—

126

C₂×C₄

#492

yes

characteristic

127

C₂×C₄

#493

yes

characteristic

128

C₂×C₄

#495

yes

characteristic

129

C₂×C₄

#497

yes

characteristic

130

C₂×C₄

#497

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#475

yes

—

136

C₈

#479

yes

—

137

C₈

#483

yes

—

138

C₈

#485

yes

—

139

C₂²

—

140

C₂²

#1716

yes

characteristic

141

C₂²

#1718

yes

characteristic

142

C₂²

#2093

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#1706

yes

characteristic

147

C₄

#1722

yes

characteristic

148

C₄

#2096

yes

characteristic

149

C₄

#2097

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#10675

yes

characteristic

153

C₂

#13532

yes

—

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#497
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ 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
#13286	2
#12910	2
#12892	2
#9861	2
#8825	2
#7137	2

subgroup

index

Subgroups — #146248

Conjugacy classes of subgroups

Maximal subgroups