Group #146286 — subgroups

Subgroups — #146286

Order 512.

177 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 60 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	#9600	C₂	1	1	yes	maximal, characteristic
2	#8827	C₂	1	1	yes	maximal, characteristic
3	#7302	C₂	1	2	yes	maximal
4	#7292	C₂	1	2	yes	maximal
5	#7146	C₂	1	1	yes	maximal, characteristic
6	#2163	C₂²	1	2	yes	—
7	#1939	—	2	1	no	—
8	#1837	—	2	1	no	—
9	C₄×#106	C₂²	1	1	yes	characteristic
10	#1567	—	2	1	no	—
11	#1566	C₂²	1	2	yes	—
12	C₂×#335	C₂²	1	1	yes	characteristic
13	C₂×#322	C₂²	1	1	yes	characteristic
14	#1279	—	2	1	no	—
15	C₂×#106	—	2	1	no	—
16	#423	—	2	2	no	—
17	C₂×C₄⋊C₈	—	2	1	no	—
18	C₂×C₄⋊C₈	—	4	1	no	—
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
22	C₄×C₄⋊C₄	—	2	1	no	—
23	C₄⋊C₁₆	—	2	2	no	—
24	C₂²⋊C₁₆	—	2	2	no	—
25	#346	—	2	2	no	—
26	#341	—	4	2	no	—
27	#340	Q₈	1	1	yes	characteristic
28	#340	Q₈	1	1	yes	characteristic
29	#336	—	4	1	no	—
30	#335	—	2	2	no	—
31	#335	—	4	2	no	—
32	#334	—	2	4	no	—
33	#322	—	2	2	no	—
34	C₂×M₁₆	—	4	2	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	4	1	no	—
37	C₂²×C₈	SD₁₆	1	1	yes	characteristic
38	C₂²×C₈	SD₁₆	1	1	yes	characteristic
39	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₈	#41	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄²	—	4	1	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	C₂×C₁₆	—	4	2	no	—
46	#106	—	4	2	no	—
47	C₄⋊C₈	—	4	1	no	—
48	C₄⋊C₈	—	4	2	no	—
49	C₄⋊C₈	—	4	2	no	—
50	C₄⋊C₈	—	4	1	no	—
51	C₄⋊C₈	—	8	2	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	Q₁₆	1	2	yes	—
56	C₄×C₈	#41	1	2	yes	—
57	C₄×C₈	#41	1	2	yes	—
58	#95	—	4	1	no	—
59	C₂²×C₄	—	4	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
64	M₁₆	—	8	2	no	—
65	M₁₆	—	8	2	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	8	1	no	—
76	C₂×C₈	—	8	1	no	—
77	C₂×C₈	—	8	1	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	#125	1	2	yes	—
80	C₂×C₈	#126	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄²	—	8	1	no	—
86	C₄²	—	8	1	no	—
87	C₄²	#123	1	1	yes	characteristic
88	C₄²	#124	1	1	yes	characteristic
89	C₄²	C₂×Q₁₆	1	1	yes	characteristic
90	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₁₆	—	8	2	no	—
92	C₂³	#476	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	8	1	no	—
101	C₂×C₄	—	8	1	no	—
102	C₂×C₄	—	8	1	no	—
103	C₂×C₄	—	8	1	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	#400	1	1	yes	characteristic
106	C₂×C₄	#477	1	1	yes	characteristic
107	C₂×C₄	#481	1	1	yes	characteristic
108	C₂×C₄	#484	1	1	yes	characteristic
109	C₂×C₄	#487	1	1	yes	characteristic
110	C₂×C₄	#490	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	16	2	no	—
117	C₈	—	16	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#2100	1	1	yes	characteristic
121	C₂²	#2231	1	1	yes	characteristic
122	C₂²	#2231	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	8	2	no	—
128	C₄	—	16	1	no	—
129	C₄	#2102	1	1	yes	characteristic
130	C₄	#2103	1	1	yes	characteristic
131	C₄	#2238	1	1	yes	characteristic
132	C₄	#2239	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#10688	1	1	yes	characteristic
136	C₂	#13541	1	1	yes	characteristic
137	C₂	#13542	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#9600

C₂

yes

maximal, characteristic

#8827

C₂

yes

maximal, characteristic

#7302

C₂

yes

maximal

#7292

C₂

yes

maximal

#7146

C₂

yes

maximal, characteristic

#2163

C₂²

yes

—

#1939

—

#1837

—

C₄×#106

C₂²

yes

characteristic

#1567

—

#1566

C₂²

yes

—

C₂×#335

C₂²

yes

characteristic

C₂×#322

C₂²

yes

—

—

—

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

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#341

—

#340

Q₈

yes

characteristic

#340

Q₈

yes

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

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

—

C₂×C₈

#125

yes

—

C₂×C₈

#126

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

C₂×Q₁₆

yes

characteristic

C₄²

AGL(1,ℤ/8)

yes

characteristic

C₁₆

—

C₂³

#476

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#400

yes

characteristic

106

C₂×C₄

#477

yes

characteristic

107

C₂×C₄

#481

yes

characteristic

108

C₂×C₄

#484

yes

characteristic

109

C₂×C₄

#487

yes

characteristic

110

C₂×C₄

#490

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

#2100

yes

characteristic

121

C₂²

#2231

yes

characteristic

122

C₂²

#2231

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#2102

yes

characteristic

130

C₄

#2103

yes

characteristic

131

C₄

#2238

yes

characteristic

132

C₄

#2239

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#10688

yes

characteristic

136

C₂

#13541

yes

characteristic

137

C₂

#13542

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2100
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#476
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

#2100

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#476

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
#9600	2
#8827	2
#7302	2
#7292	2
#7146	2

subgroup

index

Subgroups — #146286

Conjugacy classes of subgroups

Maximal subgroups