Group #140504 — subgroups

Subgroups — #140504

Order 512.

188 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 82 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	#12029	C₂	1	2	yes	maximal
2	#12013	C₂	1	1	yes	maximal, characteristic
3	#8824	C₂	1	1	yes	maximal, characteristic
4	#8293	C₂	1	1	yes	maximal, characteristic
5	#7300	C₂	1	2	yes	maximal
6	#2163	C₂²	1	2	yes	—
7	#1837	—	2	1	no	—
8	C₄×#108	C₂²	1	1	yes	characteristic
9	#1567	—	2	1	no	—
10	#1566	C₂²	1	2	yes	—
11	#1564	C₂²	1	1	yes	characteristic
12	#1451	—	2	1	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1370	C₄	1	2	yes	—
15	#1368	—	2	4	no	—
16	#1367	C₄	1	2	yes	—
17	#1278	—	2	1	no	—
18	#431	—	2	1	no	—
19	C₂×#108	—	2	1	no	—
20	#423	—	2	2	no	—
21	C₄×M₁₆	—	2	1	no	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
26	C₂²⋊C₁₆	—	2	2	no	—
27	#346	—	2	2	no	—
28	#341	—	4	2	no	—
29	#341	Q₈	1	2	yes	—
30	#335	—	2	1	no	—
31	#335	—	4	2	no	—
32	#334	—	2	2	no	—
33	#334	—	2	2	no	—
34	#334	D₄	1	2	yes	—
35	#322	—	4	2	no	—
36	C₈²	C₂×C₄	1	1	yes	characteristic
37	C₈²	C₂×C₄	1	1	yes	characteristic
38	C₈²	Q₈	1	1	yes	characteristic
39	C₈²	Q₈	1	1	yes	characteristic
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	—	4	2	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	SD₁₆	1	2	yes	—
44	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₈	#41	1	1	yes	characteristic
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₆	—	2	4	no	—
48	#108	—	4	2	no	—
49	C₄⋊C₈	—	2	2	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₂²⋊C₈	—	4	1	no	—
53	#97	—	4	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	4	1	no	—
56	C₄×C₈	—	4	1	no	—
57	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₈	SD₁₆	1	2	yes	—
62	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₄×C₈	#41	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×C₄	#137	1	1	yes	characteristic
69	M₁₆	—	4	2	no	—
70	M₁₆	—	4	2	no	—
71	M₁₆	—	8	4	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	8	2	no	—
81	C₂×C₈	#106	1	2	yes	—
82	C₂×C₈	#106	1	2	yes	—
83	C₂×C₈	C₄×D₄	1	1	yes	characteristic
84	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₈	#125	1	1	yes	characteristic
86	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
87	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₄²	#121	1	1	yes	characteristic
89	C₄²	C₂×SD₁₆	1	1	yes	characteristic
90	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₁₆	—	4	4	no	—
92	C₂³	#475	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#389	1	1	yes	characteristic
98	C₂×C₄	C₂×#106	1	1	yes	characteristic
99	C₂×C₄	#428	1	1	yes	characteristic
100	C₂×C₄	#460	1	1	yes	characteristic
101	C₂×C₄	#461	1	1	yes	characteristic
102	C₂×C₄	#476	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	—	2	2	no	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	1	no	—
110	C₈	—	8	2	no	—
111	C₈	—	8	2	no	—
112	C₈	—	16	2	no	—
113	C₈	#444	1	2	yes	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	#1948	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	#1946	1	1	yes	characteristic
123	C₄	#1953	1	1	yes	characteristic
124	C₄	#2155	1	2	yes	—
125	C₂	—	4	1	no	—
126	C₂	#9631	1	1	yes	characteristic
127	C₂	#13184	1	1	yes	characteristic
128	C₂	#13188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12029

C₂

yes

maximal

#12013

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

#8293

C₂

yes

maximal, characteristic

#7300

C₂

yes

maximal

#2163

C₂²

yes

—

#1837

—

C₄×#108

C₂²

yes

characteristic

#1567

—

#1566

C₂²

yes

—

#1564

C₂²

yes

characteristic

#1451

—

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

—

#1368

—

#1367

C₄

yes

—

—

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

C₂×C₄

yes

—

C₂²⋊C₁₆

—

#346

—

#341

—

#341

Q₈

yes

—

—

—

—

—

D₄

yes

—

#322

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

#137

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#121

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

AGL(1,ℤ/8)

yes

characteristic

C₁₆

—

C₂³

#475

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#460

yes

characteristic

101

C₂×C₄

#461

yes

characteristic

102

C₂×C₄

#476

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

#444

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#1948

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#1946

yes

characteristic

123

C₄

#1953

yes

characteristic

124

C₄

#2155

yes

—

125

C₂

—

126

C₂

#9631

yes

characteristic

127

C₂

#13184

yes

characteristic

128

C₂

#13188

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1948
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#475
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1948

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#475

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12029	2
#12013	2
#8824	2
#8293	2
#7300	2

subgroup

index

Subgroups — #140504

Conjugacy classes of subgroups

Maximal subgroups