Group #135634 — subgroups

Subgroups — #135634

Order 512.

222 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 94 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	2	yes	maximal
2	#7841	C₂	1	1	yes	maximal, characteristic
3	#7196	C₂	1	2	yes	maximal
4	#7143	C₂	1	2	yes	maximal
5	#2163	C₂²	1	4	yes	—
6	#1744	C₄	1	1	yes	characteristic
7	#1744	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₄	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₂²	1	1	yes	characteristic
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1370	—	2	2	no	—
14	#1367	—	2	2	no	—
15	C₂×C₄×C₈	—	2	1	no	—
16	C₂×C₄×C₈	—	2	1	no	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₄³	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₁₆	—	2	4	no	—
25	C₂²⋊C₁₆	—	2	4	no	—
26	#346	—	2	4	no	—
27	#336	—	2	2	no	—
28	#336	—	2	2	no	—
29	#322	—	2	2	no	—
30	#322	—	2	2	no	—
31	C₈²	C₈	1	2	yes	—
32	C₈²	C₈	1	2	yes	—
33	C₈²	D₄	1	1	yes	characteristic
34	C₈²	D₄	1	1	yes	characteristic
35	C₈²	Q₈	1	1	yes	characteristic
36	C₈²	Q₈	1	1	yes	characteristic
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	M₁₆	1	1	yes	characteristic
42	C₂²×C₈	M₁₆	1	1	yes	characteristic
43	C₂²×C₈	#41	1	2	yes	—
44	C₂×C₄²	—	2	1	no	—
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	M₁₆	1	2	yes	—
47	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₁₆	—	4	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈	C₂×C₈	1	2	yes	—
60	C₄×C₈	C₂×C₈	1	2	yes	—
61	C₄×C₈	C₂×C₈	1	1	yes	characteristic
62	C₄×C₈	C₂×C₈	1	1	yes	characteristic
63	C₄×C₈	M₁₆	1	2	yes	—
64	C₄×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₄×C₈	#41	1	2	yes	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	#117	1	1	yes	characteristic
69	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
85	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
88	C₂×C₈	#124	1	2	yes	—
89	C₂×C₈	#125	1	2	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	4	2	no	—
93	C₄²	—	4	2	no	—
94	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄²	#117	1	1	yes	characteristic
96	C₄²	C₂²×C₈	1	1	yes	characteristic
97	C₄²	C₂×M₁₆	1	1	yes	characteristic
98	C₁₆	—	8	4	no	—
99	C₂³	#432	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	#382	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₄	#423	1	1	yes	characteristic
110	C₂×C₄	#431	1	1	yes	characteristic
111	C₂×C₄	#431	1	1	yes	characteristic
112	C₂×C₄	#432	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	4	2	no	—
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	4	2	no	—
122	C₈	#364	1	2	yes	—
123	C₈	#364	1	2	yes	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	#1831	1	1	yes	characteristic
127	C₂²	#2166	1	2	yes	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	4	no	—
133	C₄	#1833	1	2	yes	—
134	C₄	C₂×#364	1	1	yes	characteristic
135	C₄	C₂×#364	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	#8780	1	1	yes	characteristic
138	C₂	#12881	1	1	yes	characteristic
139	C₂	#12883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal

#7841

C₂

yes

maximal, characteristic

#7196

C₂

yes

maximal

#7143

C₂

yes

maximal

#2163

C₂²

yes

—

#1744

C₄

yes

characteristic

#1744

C₂²

yes

characteristic

C₄²×C₈

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1367

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

C₈²

C₈

yes

—

C₈²

C₈

yes

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

#41

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

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

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

#124

yes

—

C₂×C₈

#125

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#117

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#432

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#382

yes

characteristic

108

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

109

C₂×C₄

#423

yes

characteristic

110

C₂×C₄

#431

yes

characteristic

111

C₂×C₄

#431

yes

characteristic

112

C₂×C₄

#432

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#364

yes

—

123

C₈

#364

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

#1831

yes

characteristic

127

C₂²

#2166

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1833

yes

—

134

C₄

C₂×#364

yes

characteristic

135

C₄

C₂×#364

yes

characteristic

136

C₂

—

137

C₂

#8780

yes

characteristic

138

C₂

#12881

yes

characteristic

139

C₂

#12883

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#382

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ 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
#7841	2
#7196	2
#7143	2

subgroup

index

Subgroups — #135634

Conjugacy classes of subgroups

Maximal subgroups