Group #145504 — subgroups

Subgroups — #145504

Order 512.

268 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 62 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	#33815	C₂	1	1	yes	maximal, characteristic
2	#12899	C₂	1	1	yes	maximal, characteristic
3	#11984	C₂	1	2	yes	maximal
4	#9763	C₂	1	2	yes	maximal
5	#7242	C₂	1	1	yes	maximal, characteristic
6	#3170	C₂²	1	2	yes	—
7	#2227	—	2	2	no	—
8	#2216	—	2	2	no	—
9	#2209	—	2	2	no	—
10	C₂×C₄⋊C₁₆	—	2	2	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	#1726	C₂²	1	1	yes	characteristic
14	#1568	C₂²	1	2	yes	—
15	#1531	—	2	2	no	—
16	#1506	—	2	2	no	—
17	#1364	—	2	2	no	—
18	#1362	—	2	2	no	—
19	#533	—	2	2	no	—
20	C₂×Q₃₂	—	2	2	no	—
21	C₂²×C₁₆	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	2	yes	—
23	#500	D₄	1	2	yes	—
24	#494	D₄	1	2	yes	—
25	#487	—	2	2	no	—
26	#467	—	2	2	no	—
27	#467	—	2	2	no	—
28	C₄×Q₁₆	—	2	2	no	—
29	C₄×Q₁₆	—	2	2	no	—
30	#432	—	2	2	no	—
31	C₂×#97	—	2	1	no	—
32	C₂×#97	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	#376	D₄	1	1	yes	characteristic
36	#366	—	2	2	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₄⋊C₁₆	—	2	2	no	—
39	C₄⋊C₁₆	—	4	4	no	—
40	#358	—	2	4	no	—
41	#335	—	2	2	no	—
42	#334	—	2	2	no	—
43	#330	—	4	4	no	—
44	#328	—	4	4	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₈	#41	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	2	2	no	—
52	#125	—	4	2	no	—
53	C₂²⋊Q₈	—	4	2	no	—
54	C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈	—	4	2	no	—
56	C₄×Q₈	—	4	2	no	—
57	#117	—	4	2	no	—
58	C₂×C₄²	—	2	1	no	—
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	Q₃₂	—	8	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	4	2	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	—	4	1	no	—
65	#107	—	2	2	no	—
66	#107	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	#103	—	4	2	no	—
69	#103	—	4	2	no	—
70	C₂²⋊C₈	—	4	2	no	—
71	#97	—	2	2	no	—
72	#97	—	4	2	no	—
73	#97	—	4	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	C₂×D₄	1	2	yes	—
77	C₄×C₈	#41	1	2	yes	—
78	#95	—	2	1	no	—
79	#95	D₈	1	2	yes	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	#121	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂²×C₄	#135	1	1	yes	characteristic
86	Q₁₆	—	4	2	no	—
87	Q₁₆	—	4	2	no	—
88	Q₁₆	—	8	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	8	2	no	—
96	C₂×C₈	#120	1	1	yes	characteristic
97	C₂×C₈	#123	1	2	yes	—
98	C₂×C₈	#124	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₂²⋊C₄	—	8	2	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	8	2	no	—
109	C₄²	#121	1	2	yes	—
110	C₄²	AGL(1,ℤ/8)	1	2	yes	—
111	C₁₆	—	4	2	no	—
112	C₁₆	—	8	1	no	—
113	C₂³	#466	1	1	yes	characteristic
114	Q₈	—	8	2	no	—
115	Q₈	—	8	2	no	—
116	Q₈	—	16	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	#394	1	1	yes	characteristic
127	C₂×C₄	#459	1	2	yes	—
128	C₂×C₄	#463	1	2	yes	—
129	C₂×C₄	#469	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	2	no	—
133	C₈	—	16	2	no	—
134	C₈	#453	1	1	yes	characteristic
135	C₈	#456	1	1	yes	characteristic
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#2054	1	1	yes	characteristic
139	C₂²	#2222	1	2	yes	—
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	—	16	2	no	—
145	C₄	—	16	2	no	—
146	C₄	#2039	1	1	yes	characteristic
147	C₄	#2050	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	#10518	1	1	yes	characteristic
150	C₂	#13441	1	1	yes	characteristic
151	C₂	#13447	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33815

C₂

yes

maximal, characteristic

#12899

C₂

yes

maximal, characteristic

#11984

C₂

yes

maximal

#9763

C₂

yes

maximal

#7242

C₂

yes

maximal, characteristic

#3170

C₂²

yes

—

#2227

—

#2216

—

#2209

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1568

C₂²

yes

—

—

—

—

—

—

C₂×Q₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

#500

D₄

yes

—

#494

D₄

yes

—

#487

—

#467

—

#467

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#432

—

C₂×#97

—

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

D₄

yes

characteristic

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

D₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

#123

yes

—

C₂×C₈

#124

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

#121

yes

—

110

C₄²

AGL(1,ℤ/8)

yes

—

111

C₁₆

—

112

C₁₆

—

113

C₂³

#466

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#394

yes

characteristic

127

C₂×C₄

#459

yes

—

128

C₂×C₄

#463

yes

—

129

C₂×C₄

#469

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#453

yes

characteristic

135

C₈

#456

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

#2054

yes

characteristic

139

C₂²

#2222

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#2039

yes

characteristic

147

C₄

#2050

yes

characteristic

148

C₂

—

149

C₂

#10518

yes

characteristic

150

C₂

#13441

yes

characteristic

151

C₂

#13447

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2054

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#466

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ 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
#33815	2
#12899	2
#11984	2
#9763	2
#7242	2

subgroup

index

Subgroups — #145504

Conjugacy classes of subgroups

Maximal subgroups