Group #153018 — subgroups

Subgroups — #153018

Order 512.

252 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 118 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×#1397	C₂	1	1	yes	maximal, characteristic
3	#12907	C₂	1	1	yes	maximal, characteristic
4	#7471	C₂	1	4	yes	maximal
5	C₂×#500	—	2	1	no	—
6	C₂²×C₃₂	C₂²	1	2	yes	—
7	#2252	—	2	2	no	—
8	#2252	C₄	1	4	yes	complemented
9	C₂×#367	—	2	1	no	—
10	C₂×#366	—	2	1	no	—
11	C₂×#366	C₂²	1	2	yes	—
12	C₂×#366	C₂²	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1846	—	2	1	no	—
15	#1397	—	2	2	no	—
16	C₂×C₄⋊Q₈	—	4	1	no	—
17	C₂²×C₁₆	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂³	1	1	yes	characteristic
19	#500	—	2	4	no	—
20	#500	—	4	2	no	—
21	C₂×#107	—	2	2	no	—
22	C₂×#107	—	2	2	no	—
23	C₂×#107	—	4	1	no	—
24	C₂×#106	—	4	1	no	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	#382	—	4	1	no	—
27	C₂×C₃₂	—	2	2	no	—
28	C₂×C₃₂	—	2	2	no	—
29	C₂×C₃₂	Q₈	1	4	yes	complemented
30	#367	—	4	2	no	—
31	#366	—	2	4	no	—
32	#366	—	2	2	no	—
33	#366	—	2	4	no	—
34	#366	C₂×C₄	1	4	yes	—
35	C₄×C₁₆	C₂×C₄	1	2	yes	—
36	C₄×C₁₆	D₄	1	2	yes	—
37	C₂²×Q₈	—	8	1	no	—
38	C₂²×C₈	D₈	1	1	yes	characteristic
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Q₈	—	4	4	no	—
41	C₄⋊Q₈	—	8	2	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	4	2	no	—
44	C₂×C₄⋊C₄	—	8	1	no	—
45	C₂×C₄²	D₈	1	1	yes	characteristic
46	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
47	C₂×C₁₆	SD₁₆	1	2	yes	—
48	C₂×C₁₆	Q₁₆	1	2	yes	—
49	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	C₂×Q₈	1	2	yes	—
52	C₂×C₁₆	#41	1	2	yes	—
53	#107	—	2	4	no	—
54	#107	—	4	4	no	—
55	#107	—	4	2	no	—
56	#107	—	4	2	no	—
57	#107	—	4	2	no	—
58	#106	—	8	2	no	—
59	C₄×C₈	C₂²⋊C₄	1	2	yes	—
60	C₄×C₈	SD₁₆	1	2	yes	—
61	#95	—	8	1	no	—
62	#95	—	8	1	no	—
63	C₃₂	—	2	4	no	—
64	C₂×Q₈	—	8	4	no	—
65	C₂×Q₈	—	16	4	no	—
66	C₂²×C₄	—	8	2	no	—
67	C₂²×C₄	—	8	2	no	—
68	C₂²×C₄	D₁₆	1	1	yes	characteristic
69	C₂²×C₄	D₁₆	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
71	C₂×C₈	#102	1	2	yes	—
72	C₂×C₈	SD₃₂	1	2	yes	—
73	C₂×C₈	Q₃₂	1	2	yes	—
74	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈	#124	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	8	4	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₄⋊C₄	—	8	2	no	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄²	#102	1	2	yes	—
86	C₄²	SD₃₂	1	2	yes	—
87	C₁₆	#103	1	4	yes	—
88	C₁₆	C₄×Q₈	1	2	yes	—
89	C₁₆	C₂²⋊Q₈	1	2	yes	—
90	C₂³	C₂×D₁₆	1	1	yes	characteristic
91	Q₈	—	16	8	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	8	4	no	—
95	C₂×C₄	—	16	4	no	—
96	C₂×C₄	—	16	2	no	—
97	C₂×C₄	—	16	2	no	—
98	C₂×C₄	—	16	2	no	—
99	C₂×C₄	#357	1	2	yes	—
100	C₂×C₄	#357	1	2	yes	—
101	C₂×C₄	C₂×#102	1	1	yes	characteristic
102	C₂×C₄	#486	1	1	yes	characteristic
103	C₂×C₄	#500	1	1	yes	characteristic
104	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
105	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
106	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
107	C₈	#358	1	4	yes	—
108	C₈	#391	1	1	yes	characteristic
109	C₈	C₂×#103	1	1	yes	characteristic
110	C₈	#487	1	1	yes	characteristic
111	C₈	#498	1	1	yes	characteristic
112	C₂²	C₂×#357	1	1	yes	characteristic
113	C₂²	#2240	1	1	yes	characteristic
114	C₂²	#2252	1	1	yes	characteristic
115	C₂²	#2263	1	2	yes	—
116	C₂²	#2264	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	16	4	no	—
120	C₄	—	32	2	no	—
121	C₄	#1990	1	1	yes	characteristic
122	C₄	C₂×#358	1	1	yes	characteristic
123	C₄	#2241	1	1	yes	characteristic
124	C₄	#2254	1	1	yes	characteristic
125	C₂	#13282	1	1	yes	characteristic
126	C₂	#13617	1	1	yes	characteristic
127	C₂	#13618	1	1	yes	characteristic
128	C₂	#13722	1	2	yes	—
129	C₂	#13734	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

C₂×#1397

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal

C₂×#500

—

C₂²×C₃₂

C₂²

yes

—

#2252

—

#2252

C₄

yes

—

—

C₂²

yes

—

C₂×#366

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1846

—

#1397

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#382

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

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×C₄²

D₈

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

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

—

C₄×C₈

SD₁₆

yes

—

#95

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

#102

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

—

C₄²

#102

yes

—

C₄²

SD₃₂

yes

—

C₁₆

#103

yes

—

C₁₆

C₄×Q₈

yes

—

C₁₆

C₂²⋊Q₈

yes

—

C₂³

C₂×D₁₆

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#357

yes

—

100

C₂×C₄

#357

yes

—

101

C₂×C₄

C₂×#102

yes

characteristic

102

C₂×C₄

#486

yes

characteristic

103

C₂×C₄

#500

yes

characteristic

104

C₂×C₄

C₂×SD₃₂

yes

characteristic

105

C₂×C₄

C₂×SD₃₂

yes

characteristic

106

C₂×C₄

C₂×Q₃₂

yes

characteristic

107

C₈

#358

yes

—

108

C₈

#391

yes

characteristic

109

C₈

C₂×#103

yes

characteristic

110

C₈

#487

yes

characteristic

111

C₈

#498

yes

characteristic

112

C₂²

C₂×#357

yes

characteristic

113

C₂²

#2240

yes

characteristic

114

C₂²

#2252

yes

characteristic

115

C₂²

#2263

yes

—

116

C₂²

#2264

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#1990

yes

characteristic

122

C₄

C₂×#358

yes

characteristic

123

C₄

#2241

yes

characteristic

124

C₄

#2254

yes

characteristic

125

C₂

#13282

yes

characteristic

126

C₂

#13617

yes

characteristic

127

C₂

#13618

yes

characteristic

128

C₂

#13722

yes

—

129

C₂

#13734

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
8. #2252	C₄	semidirect
29. C₂×C₃₂	Q₈	semidirect

H ≅ G/N

type

8. #2252

C₄

semidirect

29. C₂×C₃₂

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×#1397	2
#12907	2
#7471	2

subgroup

index

Subgroups — #153018

Conjugacy classes of subgroups

Complements

Maximal subgroups