Group #153321 — subgroups

Subgroups — #153321

Order 512.

241 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 88 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	C₂×#2252	C₂	1	1	yes	maximal, characteristic
2	C₂×#1423	C₂	1	2	yes	maximal
3	#13165	C₂	1	1	yes	maximal, characteristic
4	#7471	C₂	1	2	yes	maximal
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	#2252	—	2	4	no	—
9	C₂×#366	—	2	1	no	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#366	C₂²	1	2	yes	—
12	C₂×#366	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1942	—	2	1	no	—
15	#1423	—	2	4	no	—
16	#1380	—	2	1	no	—
17	C₂×C₄⋊Q₈	—	4	1	no	—
18	C₂²×C₁₆	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂³	1	1	yes	characteristic
20	#500	—	2	4	no	—
21	#500	—	4	2	no	—
22	C₂×#107	—	2	2	no	—
23	C₂×#107	—	2	1	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	4	1	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	#389	—	4	1	no	—
28	C₂×C₃₂	—	2	2	no	—
29	C₂×C₃₂	—	2	2	no	—
30	C₂×C₃₂	Q₈	1	4	yes	complemented
31	#366	—	2	4	no	—
32	#366	—	2	2	no	—
33	#366	—	2	1	no	—
34	#366	—	2	2	no	—
35	#366	D₄	1	4	yes	—
36	#366	Q₈	1	2	yes	complemented
37	C₄×C₁₆	—	2	2	no	—
38	#340	—	4	1	no	—
39	C₂²×Q₈	—	8	1	no	—
40	C₂²×C₈	D₈	1	1	yes	characteristic
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Q₈	—	4	4	no	—
43	C₄⋊Q₈	—	8	2	no	—
44	C₂×C₄⋊C₄	—	4	2	no	—
45	C₂×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄⋊C₄	—	8	1	no	—
48	C₂×C₄⋊C₄	—	8	1	no	—
49	C₂×C₄²	D₈	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	Q₁₆	1	2	yes	—
53	C₂×C₁₆	C₂×D₄	1	2	yes	—
54	C₂×C₁₆	C₂×Q₈	1	2	yes	—
55	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₁₆	#41	1	1	yes	characteristic
57	#107	—	2	4	no	—
58	#107	—	2	2	no	—
59	#107	—	4	4	no	—
60	#107	—	4	2	no	—
61	#107	—	4	1	no	—
62	#107	—	4	1	no	—
63	#107	—	4	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	#95	—	8	1	no	—
67	C₃₂	—	2	4	no	—
68	C₂×Q₈	—	8	4	no	—
69	C₂×Q₈	—	16	2	no	—
70	C₂×Q₈	—	16	2	no	—
71	C₂²×C₄	—	8	1	no	—
72	C₂²×C₄	—	8	2	no	—
73	C₂²×C₄	—	8	1	no	—
74	C₂²×C₄	D₁₆	1	1	yes	characteristic
75	C₂²×C₄	D₁₆	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	Q₃₂	1	2	yes	—
80	C₂×C₈	#120	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₈	#137	1	1	yes	characteristic
86	C₄⋊C₄	—	4	4	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	8	4	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₄⋊C₄	—	8	1	no	—
91	C₄⋊C₄	—	8	1	no	—
92	C₄⋊C₄	—	8	1	no	—
93	C₄⋊C₄	—	16	2	no	—
94	C₄²	—	2	1	no	—
95	C₄²	—	2	1	no	—
96	C₁₆	—	2	2	no	—
97	C₁₆	C₂²⋊Q₈	1	2	yes	—
98	C₁₆	C₄⋊Q₈	1	2	yes	—
99	C₂³	C₂×D₁₆	1	1	yes	characteristic
100	Q₈	—	16	8	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	8	4	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	16	1	no	—
108	C₂×C₄	—	16	2	no	—
109	C₂×C₄	—	16	2	no	—
110	C₂×C₄	—	16	1	no	—
111	C₂×C₄	—	16	1	no	—
112	C₂×C₄	—	16	1	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	D₃₂	1	2	yes	—
115	C₂×C₄	Q₆₄	1	2	yes	—
116	C₂×C₄	#447	1	1	yes	characteristic
117	C₂×C₄	#474	1	1	yes	characteristic
118	C₂×C₄	#500	1	1	yes	characteristic
119	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
120	C₂×C₄	#509	1	1	yes	characteristic
121	C₂×C₄	#510	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	#395	1	1	yes	characteristic
124	C₈	#451	1	1	yes	characteristic
125	C₈	#477	1	1	yes	characteristic
126	C₈	#501	1	1	yes	characteristic
127	C₂²	#2184	1	1	yes	characteristic
128	C₂²	#2224	1	1	yes	characteristic
129	C₂²	#2252	1	1	yes	characteristic
130	C₂²	C₂×D₃₂	1	1	yes	characteristic
131	C₂²	C₂×Q₆₄	1	1	yes	characteristic
132	C₂²	#2263	1	1	yes	characteristic
133	C₂²	#2264	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	16	4	no	—
137	C₄	—	16	2	no	—
138	C₄	—	32	1	no	—
139	C₄	#2057	1	1	yes	characteristic
140	C₄	#2188	1	1	yes	characteristic
141	C₄	#2227	1	1	yes	characteristic
142	C₄	#2255	1	1	yes	characteristic
143	C₂	#13451	1	1	yes	characteristic
144	C₂	#13661	1	1	yes	characteristic
145	C₂	#13665	1	1	yes	characteristic
146	C₂	#13703	1	1	yes	characteristic
147	C₂	#13706	1	1	yes	characteristic
148	C₂	#13731	1	1	yes	characteristic
149	C₂	#13734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

C₂×#1423

C₂

yes

maximal

#13165

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal

#7465

C₂

yes

maximal, characteristic

C₂×#500

—

C₂²×C₃₂

C₂²

yes

—

#2252

—

C₂×#366

—

C₂×#366

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

—

—

—

—

D₄

yes

—

#366

Q₈

yes

complemented

C₄×C₁₆

—

#340

—

C₂²×Q₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊Q₈

yes

—

C₁₆

C₄⋊Q₈

yes

—

C₂³

C₂×D₁₆

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₃₂

yes

—

115

C₂×C₄

Q₆₄

yes

—

116

C₂×C₄

#447

yes

characteristic

117

C₂×C₄

#474

yes

characteristic

118

C₂×C₄

#500

yes

characteristic

119

C₂×C₄

C₂×Q₃₂

yes

characteristic

120

C₂×C₄

#509

yes

characteristic

121

C₂×C₄

#510

yes

characteristic

122

C₈

—

123

C₈

#395

yes

characteristic

124

C₈

#451

yes

characteristic

125

C₈

#477

yes

characteristic

126

C₈

#501

yes

characteristic

127

C₂²

#2184

yes

characteristic

128

C₂²

#2224

yes

characteristic

129

C₂²

#2252

yes

characteristic

130

C₂²

C₂×D₃₂

yes

characteristic

131

C₂²

C₂×Q₆₄

yes

characteristic

132

C₂²

#2263

yes

characteristic

133

C₂²

#2264

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#2057

yes

characteristic

140

C₄

#2188

yes

characteristic

141

C₄

#2227

yes

characteristic

142

C₄

#2255

yes

characteristic

143

C₂

#13451

yes

characteristic

144

C₂

#13661

yes

characteristic

145

C₂

#13665

yes

characteristic

146

C₂

#13703

yes

characteristic

147

C₂

#13706

yes

characteristic

148

C₂

#13731

yes

characteristic

149

C₂

#13734

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
30. C₂×C₃₂	Q₈	semidirect
36. #366	Q₈	semidirect

H ≅ G/N

type

30. C₂×C₃₂

Q₈

semidirect

36. #366

Q₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#2252	2
C₂×#1423	2
#13165	2
#7471	2
#7465	2

subgroup

index

Subgroups — #153321

Conjugacy classes of subgroups

Complements

Maximal subgroups