Group #152978 — subgroups

Subgroups — #152978

Order 512.

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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×#1425	C₂	1	1	yes	maximal, characteristic
3	C₂×#1420	C₂	1	2	yes	maximal
4	#13549	C₂	1	1	yes	maximal, characteristic
5	#13244	C₂	1	2	yes	maximal
6	C₂²×Q₃₂	C₂²	1	2	yes	—
7	C₂×Q₆₄	—	2	1	no	—
8	C₂×Q₆₄	—	2	1	no	—
9	C₂×Q₆₄	—	2	2	no	—
10	C₂×Q₆₄	C₄	1	4	yes	complemented
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×#368	C₂²	1	2	yes	—
14	C₂×#362	—	2	2	no	—
15	#2108	C₂²	1	1	yes	characteristic
16	#1979	—	2	2	no	—
17	#1425	—	2	2	no	—
18	#1420	—	2	4	no	—
19	#1397	—	2	1	no	—
20	C₄×C₃₂	—	2	1	no	—
21	C₂²×Q₁₆	—	2	2	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	—	2	2	no	—
24	C₂×Q₃₂	—	2	2	no	—
25	C₂×Q₃₂	—	2	2	no	—
26	C₂×Q₃₂	—	2	2	no	—
27	C₂×Q₃₂	C₂×C₄	1	4	yes	—
28	C₂×M₃₂	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂×#108	—	2	2	no	—
31	#419	—	4	2	no	—
32	C₂×#101	—	4	2	no	—
33	#405	D₄	1	1	yes	characteristic
34	Q₆₄	—	2	4	no	—
35	Q₆₄	—	2	4	no	—
36	M₆₄	—	2	1	no	—
37	M₆₄	D₄	1	2	yes	—
38	C₂×C₃₂	—	2	1	no	—
39	C₂×C₃₂	C₂×C₄	1	2	yes	—
40	C₂×C₃₂	D₄	1	1	yes	characteristic
41	C₂×C₃₂	D₄	1	1	yes	characteristic
42	C₂×C₃₂	D₄	1	2	yes	—
43	#368	—	2	2	no	—
44	#368	—	2	2	no	—
45	#362	—	4	4	no	—
46	#346	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂×Q₁₆	—	2	4	no	—
50	C₂×Q₁₆	—	4	2	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	4	2	no	—
55	C₂×M₁₆	—	4	2	no	—
56	C₂×M₁₆	D₈	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	#117	D₈	1	1	yes	characteristic
59	Q₃₂	—	2	4	no	—
60	Q₃₂	—	2	4	no	—
61	Q₃₂	—	4	2	no	—
62	Q₃₂	—	4	2	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	D₈	1	2	yes	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
70	C₂×C₁₆	#41	1	1	yes	characteristic
71	C₂×C₁₆	#41	1	1	yes	characteristic
72	#108	—	4	2	no	—
73	#108	—	4	2	no	—
74	#103	—	8	4	no	—
75	#101	—	8	4	no	—
76	#97	—	2	1	no	—
77	C₄×C₈	—	2	1	no	—
78	C₃₂	—	2	2	no	—
79	C₃₂	C₂²⋊C₄	1	2	yes	—
80	C₃₂	C₂²⋊C₄	1	2	yes	—
81	C₂×Q₈	—	4	4	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×Q₈	—	8	2	no	—
84	C₂×Q₈	—	8	2	no	—
85	C₂×Q₈	—	8	2	no	—
86	C₂×Q₈	—	8	2	no	—
87	C₂²×C₄	—	8	2	no	—
88	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
89	Q₁₆	—	4	4	no	—
90	Q₁₆	—	4	4	no	—
91	Q₁₆	—	8	2	no	—
92	Q₁₆	—	8	2	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	8	2	no	—
95	M₁₆	—	8	2	no	—
96	M₁₆	D₁₆	1	2	yes	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	C₄×D₄	1	1	yes	characteristic
100	C₂×C₈	#121	1	1	yes	characteristic
101	C₂×C₈	#127	1	1	yes	characteristic
102	C₂×C₈	C₂×D₈	1	1	yes	characteristic
103	C₂×C₈	#135	1	1	yes	characteristic
104	C₂×C₈	#135	1	1	yes	characteristic
105	C₄⋊C₄	D₁₆	1	2	yes	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₁₆	—	2	2	no	—
109	C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₁₆	C₄×D₄	1	1	yes	characteristic
111	C₁₆	#121	1	1	yes	characteristic
112	C₁₆	#124	1	1	yes	characteristic
113	C₂³	#508	1	1	yes	characteristic
114	Q₈	—	8	4	no	—
115	Q₈	—	8	4	no	—
116	Q₈	—	16	2	no	—
117	Q₈	—	16	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	8	4	no	—
120	C₂×C₄	—	16	2	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	C₄×D₈	1	1	yes	characteristic
123	C₂×C₄	#466	1	1	yes	characteristic
124	C₂×C₄	#493	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
126	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
127	C₂×C₄	#508	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	16	2	no	—
130	C₈	#390	1	1	yes	characteristic
131	C₈	C₄×D₈	1	1	yes	characteristic
132	C₈	#466	1	1	yes	characteristic
133	C₈	#495	1	1	yes	characteristic
134	C₂²	—	2	2	no	—
135	C₂²	C₄×D₁₆	1	1	yes	characteristic
136	C₂²	#2215	1	1	yes	characteristic
137	C₂²	#2249	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	16	4	no	—
140	C₄	#1970	1	1	yes	characteristic
141	C₄	C₄×D₁₆	1	1	yes	characteristic
142	C₄	#2215	1	1	yes	characteristic
143	C₄	#2246	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#13235	1	1	yes	characteristic
147	C₂	#13701	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₆₄

C₂

yes

maximal, characteristic

C₂×#1425

C₂

yes

maximal, characteristic

C₂×#1420

C₂

yes

maximal

#13549

C₂

yes

maximal, characteristic

#13244

C₂

yes

maximal

C₂²×Q₃₂

C₂²

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

C₄

yes

complemented

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#362

—

#2108

C₂²

yes

—

—

—

—

C₄×C₃₂

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×C₄

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

Q₆₄

—

Q₆₄

—

M₆₄

—

M₆₄

D₄

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

—

—

—

—

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

D₈

yes

—

C₂×C₁₆

—

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

—

C₃₂

—

C₃₂

C₂²⋊C₄

yes

—

C₃₂

C₂²⋊C₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

D₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

characteristic

100

C₂×C₈

#121

yes

characteristic

101

C₂×C₈

#127

yes

characteristic

102

C₂×C₈

C₂×D₈

yes

characteristic

103

C₂×C₈

#135

yes

characteristic

104

C₂×C₈

#135

yes

characteristic

105

C₄⋊C₄

D₁₆

yes

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

C₂×C₂²⋊C₄

yes

characteristic

110

C₁₆

C₄×D₄

yes

characteristic

111

C₁₆

#121

yes

characteristic

112

C₁₆

#124

yes

characteristic

113

C₂³

#508

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×D₈

yes

characteristic

123

C₂×C₄

#466

yes

characteristic

124

C₂×C₄

#493

yes

characteristic

125

C₂×C₄

C₂×D₁₆

yes

characteristic

126

C₂×C₄

C₂×D₁₆

yes

characteristic

127

C₂×C₄

#508

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

#390

yes

characteristic

131

C₈

C₄×D₈

yes

characteristic

132

C₈

#466

yes

characteristic

133

C₈

#495

yes

characteristic

134

C₂²

—

135

C₂²

C₄×D₁₆

yes

characteristic

136

C₂²

#2215

yes

characteristic

137

C₂²

#2249

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

#1970

yes

characteristic

141

C₄

C₄×D₁₆

yes

characteristic

142

C₄

#2215

yes

characteristic

143

C₄

#2246

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#13235

yes

characteristic

147

C₂

#13701

yes

—

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. C₂×Q₆₄	C₄	semidirect

H ≅ G/N

type

10. 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×#1425	2
C₂×#1420	2
#13549	2
#13244	2

subgroup

index

C₂²×Q₆₄

Subgroups — #152978

Conjugacy classes of subgroups

Complements

Maximal subgroups