Group #129199 — subgroups

Subgroups — #129199

Order 512.

204 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 106 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	#12013	C₂	1	1	yes	maximal, characteristic
2	#11414	C₂	1	2	yes	maximal
3	#11402	C₂	1	1	yes	maximal, characteristic
4	#8293	C₂	1	1	yes	maximal, characteristic
5	#7164	C₂	1	2	yes	maximal
6	#2163	C₂²	1	2	yes	—
7	C₄×M₃₂	C₂²	1	1	yes	characteristic
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₄×#108	C₄	1	1	yes	characteristic
10	#1564	C₄	1	1	yes	characteristic
11	#1451	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1370	C₄	1	2	yes	—
14	#1367	C₄	1	2	yes	—
15	#1313	—	2	4	no	—
16	#1312	C₄	1	2	yes	—
17	#1312	C₄	1	2	yes	—
18	#1278	C₄	1	2	yes	—
19	C₂×M₃₂	—	2	1	no	—
20	C₂×M₃₂	Q₈	1	2	yes	—
21	#431	C₂×C₄	1	2	yes	—
22	C₂×#108	—	2	1	no	—
23	C₄×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	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	#346	—	2	2	no	—
30	#346	—	2	1	no	—
31	#346	D₄	1	2	yes	—
32	C₄×C₁₆	C₂×C₄	1	2	yes	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	#341	—	2	1	no	—
35	#335	—	2	1	no	—
36	#334	—	2	1	no	—
37	#322	—	2	2	no	—
38	#322	—	2	2	no	—
39	C₈²	C₂×C₄	1	1	yes	characteristic
40	C₈²	C₂×C₄	1	1	yes	characteristic
41	C₈²	C₂×C₄	1	1	yes	characteristic
42	C₈²	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆	—	2	2	no	—
44	C₂²×C₈	—	2	1	no	—
45	C₂²×C₈	M₁₆	1	1	yes	characteristic
46	C₂²×C₈	M₁₆	1	1	yes	characteristic
47	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₈	#41	1	1	yes	characteristic
49	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
50	M₃₂	—	4	2	no	—
51	M₃₂	—	4	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	4	no	—
54	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
55	#108	—	4	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	#97	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	C₄²	1	1	yes	characteristic
63	C₄×C₈	C₄²	1	1	yes	characteristic
64	C₄×C₈	C₄²	1	1	yes	characteristic
65	C₄×C₈	C₄²	1	1	yes	characteristic
66	C₄×C₈	M₁₆	1	1	yes	characteristic
67	C₄×C₈	M₁₆	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×D₄	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
74	C₂²×C₄	#131	1	1	yes	characteristic
75	M₁₆	—	4	2	no	—
76	M₁₆	—	4	2	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	#97	1	1	yes	characteristic
87	C₂×C₈	#97	1	1	yes	characteristic
88	C₂×C₈	#97	1	1	yes	characteristic
89	C₂×C₈	#97	1	1	yes	characteristic
90	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₈	#117	1	1	yes	characteristic
92	C₂×C₈	C₄×D₄	1	1	yes	characteristic
93	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₄²	C₂×C₄²	1	1	yes	characteristic
95	C₄²	C₄×D₄	1	1	yes	characteristic
96	C₄²	C₂×M₁₆	1	1	yes	characteristic
97	C₄²	#131	1	1	yes	characteristic
98	C₁₆	—	2	4	no	—
99	C₁₆	—	4	2	no	—
100	C₁₆	—	4	4	no	—
101	C₂³	#446	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₂×#97	1	1	yes	characteristic
109	C₂×C₄	#405	1	1	yes	characteristic
110	C₂×C₄	#436	1	1	yes	characteristic
111	C₂×C₄	#436	1	1	yes	characteristic
112	C₂×C₄	#446	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	—	4	1	no	—
120	C₈	—	8	2	no	—
121	C₈	—	8	2	no	—
122	C₈	#347	1	2	yes	—
123	C₈	#430	1	2	yes	—
124	C₈	#444	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#1770	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	#1773	1	1	yes	characteristic
134	C₄	#1778	1	1	yes	characteristic
135	C₄	C₂×#347	1	1	yes	characteristic
136	C₄	#2110	1	1	yes	characteristic
137	C₂	—	4	1	no	—
138	C₂	#8295	1	1	yes	characteristic
139	C₂	#12666	1	1	yes	characteristic
140	C₂	#12667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal, characteristic

#11414

C₂

yes

maximal

#11402

C₂

yes

maximal, characteristic

#8293

C₂

yes

maximal, characteristic

#7164

C₂

yes

maximal

#2163

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×#108

C₄

yes

characteristic

#1564

C₄

yes

characteristic

#1451

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

—

#1367

C₄

yes

—

#1313

—

#1312

C₄

yes

—

#1312

C₄

yes

—

#1278

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

Q₈

yes

—

#431

C₂×C₄

yes

—

C₂×#108

—

C₄×M₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

C₂×C₄

yes

—

C₂²⋊C₁₆

—

#346

—

#346

—

#346

D₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×Q₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

#131

yes

characteristic

M₁₆

—

M₁₆

—

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

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

#131

yes

characteristic

C₁₆

—

C₁₆

—

100

C₁₆

—

101

C₂³

#446

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

108

C₂×C₄

C₂×#97

yes

characteristic

109

C₂×C₄

#405

yes

characteristic

110

C₂×C₄

#436

yes

characteristic

111

C₂×C₄

#436

yes

characteristic

112

C₂×C₄

#446

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#347

yes

—

123

C₈

#430

yes

—

124

C₈

#444

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#1770

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1773

yes

characteristic

134

C₄

#1778

yes

characteristic

135

C₄

C₂×#347

yes

characteristic

136

C₄

#2110

yes

characteristic

137

C₂

—

138

C₂

#8295

yes

characteristic

139

C₂

#12666

yes

characteristic

140

C₂

#12667

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂³	#446
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₄×C₄⋊C₄

commutator

C₄²

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ 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
#12013	2
#11414	2
#11402	2
#8293	2
#7164	2

subgroup

index

Subgroups — #129199

Conjugacy classes of subgroups

Maximal subgroups