Group #152974 — subgroups

Subgroups — #152974

Order 512.

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

normal subgroups only

This group has 278 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	C₂²×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1424	C₂	1	1	yes	maximal, characteristic
3	C₂×#1416	C₂	1	2	yes	maximal
4	C₂×#1397	C₂	1	1	yes	maximal, characteristic
5	#13236	C₂	1	2	yes	maximal
6	C₂²×Q₃₂	C₂²	1	2	yes	—
7	C₂×Q₆₄	—	2	2	no	—
8	C₂×Q₆₄	—	2	2	no	—
9	C₂×Q₆₄	C₄	1	4	yes	complemented
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×#366	C₂²	1	2	yes	—
13	C₂×#358	—	2	2	no	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1969	—	2	2	no	—
16	#1424	—	2	2	no	—
17	#1416	—	2	4	no	—
18	#1397	—	2	2	no	—
19	C₂²×Q₁₆	—	2	2	no	—
20	C₂×Q₃₂	—	2	2	no	—
21	C₂×Q₃₂	—	2	2	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	—	2	2	no	—
24	C₂×Q₃₂	—	2	2	no	—
25	C₂×Q₃₂	C₂×C₄	1	4	yes	—
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂³	1	1	yes	characteristic
28	C₂×#107	—	2	2	no	—
29	C₂×#103	—	4	2	no	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	#391	—	4	2	no	—
32	Q₆₄	—	2	8	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	—	2	1	no	—
35	C₂×C₃₂	C₂×C₄	1	2	yes	—
36	C₂×C₃₂	D₄	1	2	yes	—
37	C₂×C₃₂	D₄	1	2	yes	—
38	C₂×C₃₂	D₄	1	2	yes	—
39	#366	—	2	2	no	—
40	#366	—	2	2	no	—
41	#358	—	4	4	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₂²×Q₈	—	4	2	no	—
45	C₂×Q₁₆	—	2	4	no	—
46	C₂×Q₁₆	—	4	2	no	—
47	C₂×Q₁₆	—	4	2	no	—
48	C₂×Q₁₆	—	4	2	no	—
49	C₂×Q₁₆	—	4	2	no	—
50	C₂×Q₁₆	—	4	2	no	—
51	C₂²×C₈	D₈	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	—	4	2	no	—
54	C₂×C₄²	D₈	1	1	yes	characteristic
55	Q₃₂	—	2	4	no	—
56	Q₃₂	—	2	4	no	—
57	Q₃₂	—	4	2	no	—
58	Q₃₂	—	4	2	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	D₈	1	2	yes	—
62	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	#41	1	1	yes	characteristic
67	C₂×C₁₆	#41	1	1	yes	characteristic
68	#107	—	4	2	no	—
69	#107	—	4	2	no	—
70	#103	—	8	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	#95	—	8	4	no	—
74	C₃₂	—	2	2	no	—
75	C₃₂	C₂²⋊C₄	1	4	yes	—
76	C₂×Q₈	—	4	4	no	—
77	C₂×Q₈	—	8	2	no	—
78	C₂×Q₈	—	8	2	no	—
79	C₂×Q₈	—	8	2	no	—
80	C₂×Q₈	—	8	2	no	—
81	C₂×Q₈	—	8	2	no	—
82	C₂²×C₄	—	8	2	no	—
83	C₂²×C₄	—	8	2	no	—
84	C₂²×C₄	D₁₆	1	2	yes	—
85	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
86	Q₁₆	—	4	4	no	—
87	Q₁₆	—	4	4	no	—
88	Q₁₆	—	8	2	no	—
89	Q₁₆	—	8	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	D₁₆	1	2	yes	—
93	C₂×C₈	C₄×D₄	1	1	yes	characteristic
94	C₂×C₈	#121	1	1	yes	characteristic
95	C₂×C₈	#127	1	1	yes	characteristic
96	C₂×C₈	C₂×D₈	1	1	yes	characteristic
97	C₂×C₈	#135	1	1	yes	characteristic
98	C₂×C₈	#135	1	1	yes	characteristic
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	8	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₁₆	C₄×D₄	1	1	yes	characteristic
106	C₁₆	#121	1	1	yes	characteristic
107	C₁₆	#124	1	1	yes	characteristic
108	C₂³	C₂×D₁₆	1	1	yes	characteristic
109	Q₈	—	8	4	no	—
110	Q₈	—	8	4	no	—
111	Q₈	—	16	2	no	—
112	Q₈	—	16	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	16	2	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	—	16	2	no	—
119	C₂×C₄	—	16	4	no	—
120	C₂×C₄	C₄×D₈	1	1	yes	characteristic
121	C₂×C₄	#466	1	1	yes	characteristic
122	C₂×C₄	#493	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
124	C₂×C₄	#508	1	1	yes	characteristic
125	C₂×C₄	#508	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	#390	1	1	yes	characteristic
128	C₈	C₄×D₈	1	1	yes	characteristic
129	C₈	#466	1	1	yes	characteristic
130	C₈	#495	1	1	yes	characteristic
131	C₂²	C₄×D₁₆	1	1	yes	characteristic
132	C₂²	#2215	1	1	yes	characteristic
133	C₂²	#2246	1	1	yes	characteristic
134	C₂²	#2264	1	4	yes	—
135	C₄	—	4	2	no	—
136	C₄	—	16	4	no	—
137	C₄	—	32	2	no	—
138	C₄	#1970	1	1	yes	characteristic
139	C₄	C₄×D₁₆	1	1	yes	characteristic
140	C₄	#2215	1	1	yes	characteristic
141	C₄	#2249	1	1	yes	characteristic
142	C₂	#13235	1	1	yes	characteristic
143	C₂	#13653	1	2	yes	—
144	C₂	#13698	1	2	yes	—
145	C₂	#13730	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₆₄

C₂

yes

maximal, characteristic

C₂×#1424

C₂

yes

maximal, characteristic

C₂×#1416

C₂

yes

maximal

C₂×#1397

C₂

yes

maximal, characteristic

#13236

C₂

yes

maximal

C₂²×Q₃₂

C₂²

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

C₄

yes

complemented

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₄

yes

characteristic

#391

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

D₄

yes

—

C₂×C₃₂

D₄

yes

—

C₂×C₃₂

D₄

yes

—

#366

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₃₂

C₂²⋊C₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

C₂×C₂²⋊C₄

yes

characteristic

105

C₁₆

C₄×D₄

yes

characteristic

106

C₁₆

#121

yes

characteristic

107

C₁₆

#124

yes

characteristic

108

C₂³

C₂×D₁₆

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₄×D₈

yes

characteristic

121

C₂×C₄

#466

yes

characteristic

122

C₂×C₄

#493

yes

characteristic

123

C₂×C₄

C₂×D₁₆

yes

characteristic

124

C₂×C₄

#508

yes

characteristic

125

C₂×C₄

#508

yes

characteristic

126

C₈

—

127

C₈

#390

yes

characteristic

128

C₈

C₄×D₈

yes

characteristic

129

C₈

#466

yes

characteristic

130

C₈

#495

yes

characteristic

131

C₂²

C₄×D₁₆

yes

characteristic

132

C₂²

#2215

yes

characteristic

133

C₂²

#2246

yes

characteristic

134

C₂²

#2264

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#1970

yes

characteristic

139

C₄

C₄×D₁₆

yes

characteristic

140

C₄

#2215

yes

characteristic

141

C₄

#2249

yes

characteristic

142

C₂

#13235

yes

characteristic

143

C₂

#13653

yes

—

144

C₂

#13698

yes

—

145

C₂

#13730

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×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

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂×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
9. C₂×Q₆₄	C₄	semidirect

H ≅ G/N

type

9. C₂×Q₆₄

C₄

semidirect

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

subgroup	index
C₂²×Q₆₄	2
C₂×#1424	2
C₂×#1416	2
C₂×#1397	2
#13236	2

subgroup

index

C₂²×Q₆₄

Subgroups — #152974

Conjugacy classes of subgroups

Complements

Maximal subgroups