Group #135668 — subgroups

Subgroups — #135668

Order 512.

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

normal subgroups only

This group has 242 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	#12900	C₂	1	1	yes	maximal, characteristic
2	#12883	C₂	1	1	yes	maximal, characteristic
3	#7842	C₂	1	1	yes	maximal, characteristic
4	#7152	C₂	1	2	yes	maximal
5	#7152	C₂	1	2	yes	maximal
6	C₂×#364	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#2150	C₂²	1	2	yes	—
9	#1744	C₄	1	1	yes	characteristic
10	C₄×#97	C₄	1	1	yes	characteristic
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×#322	C₄	1	1	yes	characteristic
13	C₂×#322	C₄	1	1	yes	characteristic
14	C₂×#322	C₂²	1	1	yes	characteristic
15	C₂×#322	C₂²	1	1	yes	characteristic
16	#1376	—	2	2	no	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂×M₃₂	—	2	2	no	—
19	C₂×#97	—	2	2	no	—
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	Q₈	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄³	C₂×C₄	1	1	yes	characteristic
29	#364	—	4	4	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	#336	—	2	2	no	—
35	#322	—	2	2	no	—
36	#322	—	2	1	no	—
37	#322	—	2	1	no	—
38	#322	—	2	1	no	—
39	#322	—	2	1	no	—
40	#322	C₈	1	4	yes	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	C₄⋊C₄	1	2	yes	—
44	C₂²×C₈	M₁₆	1	2	yes	—
45	C₂²×C₈	M₁₆	1	2	yes	—
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄²	—	2	1	no	—
49	C₂×C₄²	—	2	1	no	—
50	C₂×C₄²	C₄⋊C₄	1	2	yes	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	M₃₂	—	4	4	no	—
53	M₃₂	—	4	2	no	—
54	M₃₂	—	4	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	4	2	no	—
57	#97	—	4	4	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	C₂×C₈	1	2	yes	—
69	C₄×C₈	C₂×C₈	1	2	yes	—
70	C₄×C₈	M₁₆	1	2	yes	—
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
78	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	C₄⋊C₈	1	4	yes	—
91	C₂×C₈	#124	1	1	yes	characteristic
92	C₂×C₈	#124	1	1	yes	characteristic
93	C₂×C₈	#125	1	1	yes	characteristic
94	C₂×C₈	#125	1	1	yes	characteristic
95	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₈	#137	1	2	yes	—
97	C₄²	—	2	4	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	4	2	no	—
101	C₄²	#117	1	1	yes	characteristic
102	C₄²	#117	1	1	yes	characteristic
103	C₄²	C₂²×C₈	1	1	yes	characteristic
104	C₄²	C₂×M₁₆	1	1	yes	characteristic
105	C₁₆	—	8	2	no	—
106	C₁₆	—	8	2	no	—
107	C₂³	#423	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	#382	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₄	#428	1	2	yes	—
116	C₂×C₄	#430	1	2	yes	—
117	C₂×C₄	#431	1	1	yes	characteristic
118	C₂×C₄	#431	1	1	yes	characteristic
119	C₂×C₄	#432	1	1	yes	characteristic
120	C₂×C₄	#432	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	2	no	—
126	C₈	—	4	2	no	—
127	C₈	—	4	2	no	—
128	C₈	#489	1	2	yes	—
129	C₈	#490	1	1	yes	characteristic
130	C₈	#491	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#1569	1	2	yes	—
134	C₂²	#1831	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	4	no	—
138	C₄	#1565	1	2	yes	—
139	C₄	#1836	1	1	yes	characteristic
140	C₄	#1838	1	1	yes	characteristic
141	C₄	#2165	1	2	yes	—
142	C₄	#2166	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	#8786	1	1	yes	characteristic
145	C₂	#12881	1	1	yes	characteristic
146	C₂	#12882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal, characteristic

#12883

C₂

yes

maximal, characteristic

#7842

C₂

yes

maximal, characteristic

#7152

C₂

yes

maximal

#7152

C₂

yes

maximal

C₂×#364

—

#2150

C₂²

yes

—

#2150

C₂²

yes

—

#1744

C₄

yes

characteristic

C₄×#97

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×#322

C₂²

yes

characteristic

#1376

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

—

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

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

—

C₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

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

C₄⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#137

yes

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

#117

yes

characteristic

102

C₄²

#117

yes

characteristic

103

C₄²

C₂²×C₈

yes

characteristic

104

C₄²

C₂×M₁₆

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₂³

#423

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#382

yes

characteristic

114

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

115

C₂×C₄

#428

yes

—

116

C₂×C₄

#430

yes

—

117

C₂×C₄

#431

yes

characteristic

118

C₂×C₄

#431

yes

characteristic

119

C₂×C₄

#432

yes

characteristic

120

C₂×C₄

#432

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

#489

yes

—

129

C₈

#490

yes

characteristic

130

C₈

#491

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

#1569

yes

—

134

C₂²

#1831

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#1565

yes

—

139

C₄

#1836

yes

characteristic

140

C₄

#1838

yes

characteristic

141

C₄

#2165

yes

—

142

C₄

#2166

yes

—

143

C₂

—

144

C₂

#8786

yes

characteristic

145

C₂

#12881

yes

characteristic

146

C₂

#12882

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	#423
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
#12900	2
#12883	2
#7842	2
#7152	2

subgroup

index

Subgroups — #135668

Conjugacy classes of subgroups

Maximal subgroups