Group #146277 — subgroups

Subgroups — #146277

Order 512.

271 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13290	C₂	1	2	yes	maximal
2	#12903	C₂	1	2	yes	maximal
3	#9836	C₂	1	1	yes	maximal, characteristic
4	#8825	C₂	1	1	yes	maximal, characteristic
5	#7139	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×#365	—	2	4	no	—
8	C₄×M₃₂	C₂²	1	2	yes	—
9	#1990	—	2	1	no	—
10	#1956	—	2	1	no	—
11	#1848	C₂²	1	2	yes	—
12	#1844	C₂²	1	1	yes	characteristic
13	#1566	—	2	1	no	—
14	#1566	—	2	1	no	—
15	C₂×#322	C₂²	1	1	yes	characteristic
16	#1387	—	2	4	no	—
17	C₂×C₄⋊Q₈	—	2	1	no	—
18	C₂×M₃₂	D₄	1	4	yes	—
19	#500	—	2	2	no	—
20	#500	—	2	1	no	—
21	#500	Q₈	1	2	yes	—
22	C₂×#108	—	2	2	no	—
23	C₂×#108	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×#106	—	2	2	no	—
26	#423	—	2	2	no	—
27	C₂×#103	—	4	2	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#384	—	2	1	no	—
32	#365	—	4	8	no	—
33	#346	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	#340	—	4	2	no	—
36	#334	—	4	2	no	—
37	#334	—	4	2	no	—
38	#322	—	2	2	no	—
39	C₂²×Q₈	—	4	1	no	—
40	C₂×M₁₆	—	4	2	no	—
41	C₂²×C₈	SD₁₆	1	2	yes	—
42	C₂²×C₈	SD₁₆	1	2	yes	—
43	C₂²×C₈	C₂×D₄	1	2	yes	—
44	C₄⋊Q₈	—	2	2	no	—
45	C₄⋊Q₈	—	4	1	no	—
46	C₄⋊Q₈	—	4	1	no	—
47	C₄⋊Q₈	—	4	1	no	—
48	C₂×C₄⋊C₄	—	4	1	no	—
49	C₂×C₄⋊C₄	—	4	1	no	—
50	C₂×C₄⋊C₄	—	4	1	no	—
51	C₂×C₄⋊C₄	—	4	1	no	—
52	C₂×C₄⋊C₄	—	4	1	no	—
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	M₃₂	—	2	4	no	—
55	M₃₂	SD₁₆	1	8	yes	—
56	C₂×C₁₆	—	2	4	no	—
57	#108	—	4	4	no	—
58	#108	—	4	4	no	—
59	#107	—	2	4	no	—
60	#107	—	4	2	no	—
61	#106	—	4	4	no	—
62	C₄⋊C₈	—	4	2	no	—
63	C₄⋊C₈	—	4	2	no	—
64	#103	—	8	4	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₄×C₈	#41	1	1	yes	characteristic
71	C₄×C₈	#41	1	2	yes	—
72	C₂×Q₈	—	4	2	no	—
73	C₂×Q₈	—	8	2	no	—
74	C₂×Q₈	—	8	2	no	—
75	C₂×Q₈	—	8	1	no	—
76	C₂²×C₄	—	8	1	no	—
77	C₂²×C₄	—	8	1	no	—
78	C₂²×C₄	#127	1	1	yes	characteristic
79	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
81	M₁₆	—	8	2	no	—
82	M₁₆	—	8	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₈	#123	1	2	yes	—
90	C₂×C₈	C₂×SD₁₆	1	4	yes	—
91	C₂×C₈	#137	1	4	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	2	no	—
95	C₄⋊C₄	—	8	2	no	—
96	C₄⋊C₄	—	8	1	no	—
97	C₄⋊C₄	—	8	1	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
100	C₄²	#123	1	1	yes	characteristic
101	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₁₆	—	2	8	no	—
104	C₂³	#492	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	Q₈	—	16	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	—	16	1	no	—
111	C₂×C₄	—	16	1	no	—
112	C₂×C₄	—	16	1	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	—	16	1	no	—
115	C₂×C₄	#399	1	1	yes	characteristic
116	C₂×C₄	#476	1	1	yes	characteristic
117	C₂×C₄	#476	1	1	yes	characteristic
118	C₂×C₄	#481	1	1	yes	characteristic
119	C₂×C₄	#481	1	1	yes	characteristic
120	C₂×C₄	#492	1	2	yes	—
121	C₂×C₄	#496	1	1	yes	characteristic
122	C₂×C₄	#497	1	2	yes	—
123	C₈	—	4	2	no	—
124	C₈	—	4	2	no	—
125	C₈	—	16	2	no	—
126	C₈	#475	1	4	yes	—
127	C₈	#483	1	4	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#1705	1	1	yes	characteristic
131	C₂²	#1716	1	1	yes	characteristic
132	C₂²	#2094	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	32	1	no	—
137	C₄	#1714	1	2	yes	—
138	C₄	#2096	1	2	yes	—
139	C₄	#2238	1	2	yes	—
140	C₄	#2239	1	2	yes	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#10685	1	1	yes	characteristic
144	C₂	#13535	1	1	yes	characteristic
145	C₂	#13536	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13290

C₂

yes

maximal

#12903

C₂

yes

maximal

#9836

C₂

yes

maximal, characteristic

#8825

C₂

yes

maximal, characteristic

#7139

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#365

—

C₄×M₃₂

C₂²

yes

—

#1990

—

#1956

—

#1848

C₂²

yes

—

#1844

C₂²

yes

—

—

C₂²

yes

characteristic

#1387

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

#500

—

#500

—

#500

Q₈

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊Q₈

yes

characteristic

100

C₄²

#123

yes

characteristic

101

C₄²

AGL(1,ℤ/8)

yes

characteristic

102

C₄²

AGL(1,ℤ/8)

yes

characteristic

103

C₁₆

—

104

C₂³

#492

yes

characteristic

105

Q₈

—

106

Q₈

—

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

#399

yes

characteristic

116

C₂×C₄

#476

yes

characteristic

117

C₂×C₄

#476

yes

characteristic

118

C₂×C₄

#481

yes

characteristic

119

C₂×C₄

#481

yes

characteristic

120

C₂×C₄

#492

yes

—

121

C₂×C₄

#496

yes

characteristic

122

C₂×C₄

#497

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

#475

yes

—

127

C₈

#483

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

#1705

yes

characteristic

131

C₂²

#1716

yes

characteristic

132

C₂²

#2094

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#1714

yes

—

138

C₄

#2096

yes

—

139

C₄

#2238

yes

—

140

C₄

#2239

yes

—

141

C₂

—

142

C₂

—

143

C₂

#10685

yes

characteristic

144

C₂

#13535

yes

characteristic

145

C₂

#13536

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
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

#2094

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#492

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
#13290	2
#12903	2
#9836	2
#8825	2
#7139	2

subgroup

index

Subgroups — #146277

Conjugacy classes of subgroups

Maximal subgroups