Group #153021 — subgroups

Subgroups — #153021

Order 512.

234 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33726	C₂	1	1	yes	maximal, characteristic
2	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#12907	C₂	1	1	yes	maximal, characteristic
4	#7477	C₂	1	4	yes	maximal
5	#3160	—	2	1	no	—
6	C₂×M₆₄	C₂²	1	2	yes	—
7	#2254	—	2	1	no	—
8	#2253	—	2	1	no	—
9	#2253	C₄	1	2	yes	complemented
10	#2252	C₄	1	2	yes	complemented
11	#2158	—	2	1	no	—
12	#2158	C₂²	1	2	yes	—
13	C₂×#367	C₂²	1	2	yes	—
14	C₂×#366	—	2	1	no	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1846	—	2	1	no	—
17	#1397	—	2	1	no	—
18	C₄×C₃₂	—	2	1	no	—
19	#533	—	4	1	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	#500	—	2	2	no	—
23	#499	—	2	2	no	—
24	#499	—	4	1	no	—
25	#498	—	4	1	no	—
26	#427	—	2	2	no	—
27	#427	—	4	1	no	—
28	C₂×#107	—	2	2	no	—
29	C₂×#106	—	4	1	no	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	#382	—	4	1	no	—
32	M₆₄	—	2	2	no	—
33	M₆₄	Q₈	1	4	yes	complemented
34	C₂×C₃₂	—	2	2	no	—
35	#367	—	2	2	no	—
36	#367	—	2	2	no	—
37	#367	—	2	4	no	—
38	#366	—	2	2	no	—
39	#366	—	4	2	no	—
40	#366	C₂×C₄	1	4	yes	—
41	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₈	D₈	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Q₈	—	4	2	no	—
48	#125	—	4	2	no	—
49	C₂²⋊Q₈	—	8	2	no	—
50	C₄×Q₈	—	8	2	no	—
51	#117	—	4	2	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄²	D₈	1	1	yes	characteristic
54	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₁₆	SD₁₆	1	2	yes	—
57	C₂×C₁₆	Q₁₆	1	2	yes	—
58	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
60	C₂×C₁₆	C₂×Q₈	1	2	yes	—
61	C₂×C₁₆	#41	1	2	yes	—
62	#107	—	2	4	no	—
63	#107	—	4	2	no	—
64	#107	—	4	2	no	—
65	#107	—	4	2	no	—
66	#106	—	4	2	no	—
67	#106	—	4	2	no	—
68	#106	—	8	2	no	—
69	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₈	SD₁₆	1	1	yes	characteristic
72	C₄×C₈	SD₁₆	1	1	yes	characteristic
73	#95	—	8	2	no	—
74	C₃₂	—	2	4	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂²×C₄	—	8	2	no	—
77	C₂²×C₄	D₁₆	1	2	yes	—
78	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
79	C₂×C₈	#102	1	1	yes	characteristic
80	C₂×C₈	#102	1	1	yes	characteristic
81	C₂×C₈	SD₃₂	1	2	yes	—
82	C₂×C₈	Q₃₂	1	2	yes	—
83	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈	#124	1	1	yes	characteristic
85	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	8	4	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₂²⋊C₄	—	8	2	no	—
95	C₄²	—	8	2	no	—
96	C₄²	#102	1	1	yes	characteristic
97	C₄²	#102	1	1	yes	characteristic
98	C₄²	SD₃₂	1	2	yes	—
99	C₁₆	#103	1	2	yes	—
100	C₁₆	#103	1	2	yes	—
101	C₁₆	C₄×Q₈	1	2	yes	—
102	C₁₆	C₂²⋊Q₈	1	2	yes	—
103	C₂³	C₂×D₁₆	1	1	yes	characteristic
104	Q₈	—	16	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	4	no	—
107	C₂×C₄	—	16	4	no	—
108	C₂×C₄	—	16	2	no	—
109	C₂×C₄	#357	1	2	yes	—
110	C₂×C₄	#357	1	2	yes	—
111	C₂×C₄	C₂×#102	1	1	yes	characteristic
112	C₂×C₄	#486	1	1	yes	characteristic
113	C₂×C₄	#500	1	1	yes	characteristic
114	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
116	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
117	C₈	#358	1	2	yes	—
118	C₈	#358	1	2	yes	—
119	C₈	#391	1	1	yes	characteristic
120	C₈	C₂×#103	1	1	yes	characteristic
121	C₈	#487	1	1	yes	characteristic
122	C₈	#498	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#357	1	1	yes	characteristic
125	C₂²	#2240	1	1	yes	characteristic
126	C₂²	#2252	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	16	4	no	—
130	C₄	—	32	2	no	—
131	C₄	#1990	1	1	yes	characteristic
132	C₄	C₂×#358	1	1	yes	characteristic
133	C₄	#2241	1	1	yes	characteristic
134	C₄	#2254	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#13282	1	1	yes	characteristic
138	C₂	#13622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33726

C₂

yes

maximal, characteristic

C₄×M₆₄

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#7477

C₂

yes

maximal

#3160

—

C₂×M₆₄

C₂²

yes

—

#2254

—

#2253

—

#2253

C₄

yes

complemented

#2252

C₄

yes

complemented

#2158

—

#2158

C₂²

yes

—

C₂×#367

C₂²

yes

—

C₂×#366

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1846

—

#1397

—

C₄×C₃₂

—

#533

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#382

—

M₆₄

—

M₆₄

Q₈

yes

complemented

C₂×C₃₂

—

—

—

—

—

—

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

#95

—

C₃₂

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

SD₃₂

yes

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

SD₃₂

yes

—

C₁₆

#103

yes

—

100

C₁₆

#103

yes

—

101

C₁₆

C₄×Q₈

yes

—

102

C₁₆

C₂²⋊Q₈

yes

—

103

C₂³

C₂×D₁₆

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#357

yes

—

110

C₂×C₄

#357

yes

—

111

C₂×C₄

C₂×#102

yes

characteristic

112

C₂×C₄

#486

yes

characteristic

113

C₂×C₄

#500

yes

characteristic

114

C₂×C₄

C₂×SD₃₂

yes

characteristic

115

C₂×C₄

C₂×SD₃₂

yes

characteristic

116

C₂×C₄

C₂×Q₃₂

yes

characteristic

117

C₈

#358

yes

—

118

C₈

#358

yes

—

119

C₈

#391

yes

characteristic

120

C₈

C₂×#103

yes

characteristic

121

C₈

#487

yes

characteristic

122

C₈

#498

yes

characteristic

123

C₂²

—

124

C₂²

C₂×#357

yes

characteristic

125

C₂²

#2240

yes

characteristic

126

C₂²

#2252

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#1990

yes

characteristic

132

C₄

C₂×#358

yes

characteristic

133

C₄

#2241

yes

characteristic

134

C₄

#2254

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#13282

yes

characteristic

138

C₂

#13622

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#357

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

C₂×D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #2253	C₄	semidirect
10. #2252	C₄	semidirect
33. M₆₄	Q₈	semidirect

H ≅ G/N

type

9. #2253

C₄

semidirect

10. #2252

C₄

semidirect

33. M₆₄

Q₈

semidirect

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

subgroup	index
#33726	2
C₄×M₆₄	2
#12907	2
#7477	2

subgroup

index

#33726

C₄×M₆₄

#12907

#7477

Subgroups — #153021

Conjugacy classes of subgroups

Complements

Maximal subgroups