Group #140368 — subgroups

Subgroups — #140368

Order 512.

212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 96 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	C₄×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	#11401	C₂	1	1	yes	maximal, characteristic
4	#9608	C₂	1	2	yes	maximal
5	#7176	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	2	yes	—
7	C₄×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₄×#108	C₂²	1	2	yes	—
10	#1565	—	2	2	no	—
11	C₂×#335	C₂²	1	2	yes	—
12	#1450	C₂²	1	1	yes	characteristic
13	#1381	C₄	1	1	yes	characteristic
14	#1381	C₄	1	1	yes	characteristic
15	#1372	—	2	2	no	—
16	#1312	—	2	2	no	—
17	#1283	—	2	2	no	—
18	C₂×M₃₂	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	#431	D₄	1	1	yes	characteristic
22	#431	D₄	1	1	yes	characteristic
23	C₂×#108	C₂×C₄	1	2	yes	—
24	C₂×#108	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₈	—	2	2	no	—
26	C₄×M₁₆	—	2	2	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	—	2	4	no	—
29	#346	Q₈	1	2	yes	—
30	C₄×C₁₆	—	2	1	no	—
31	C₄×C₁₆	D₄	1	2	yes	—
32	C₄×C₁₆	Q₈	1	2	yes	—
33	#341	—	2	2	no	—
34	#335	—	2	2	no	—
35	#335	—	2	2	no	—
36	#335	—	4	2	no	—
37	#334	—	4	2	no	—
38	#322	—	2	1	no	—
39	C₈²	—	2	1	no	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	4	2	no	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
51	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
52	#108	—	2	2	no	—
53	#108	—	2	2	no	—
54	#108	—	2	2	no	—
55	#108	M₁₆	1	4	yes	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	D₈	1	2	yes	—
60	C₂²⋊C₈	—	2	1	no	—
61	C₂²⋊C₈	Q₁₆	1	2	yes	—
62	#97	—	4	2	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₄×C₈	#41	1	1	yes	characteristic
70	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
72	C₂²×C₄	#135	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	M₁₆	—	4	2	no	—
75	M₁₆	—	4	2	no	—
76	M₁₆	—	4	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	16	2	no	—
84	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₈	#125	1	1	yes	characteristic
88	C₂×C₈	C₂×M₁₆	1	2	yes	—
89	C₂×C₈	#131	1	2	yes	—
90	C₄²	#121	1	1	yes	characteristic
91	C₄²	C₄⋊Q₈	1	1	yes	characteristic
92	C₄²	C₂×D₈	1	1	yes	characteristic
93	C₄²	#135	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	—	4	2	no	—
97	C₂³	C₄×Q₁₆	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	#389	1	1	yes	characteristic
101	C₂×C₄	#435	1	2	yes	—
102	C₂×C₄	C₄×D₈	1	1	yes	characteristic
103	C₂×C₄	#444	1	2	yes	—
104	C₂×C₄	#466	1	1	yes	characteristic
105	C₂×C₄	#467	1	1	yes	characteristic
106	C₂×C₄	#499	1	1	yes	characteristic
107	C₂×C₄	#500	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	2	no	—
111	C₈	—	4	4	no	—
112	C₈	—	4	2	no	—
113	C₈	—	8	2	no	—
114	C₈	—	16	2	no	—
115	C₈	#423	1	1	yes	characteristic
116	C₈	#424	1	1	yes	characteristic
117	C₈	#446	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1584	1	2	yes	—
121	C₂²	#1942	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#1581	1	2	yes	—
126	C₄	#1940	1	1	yes	characteristic
127	C₄	#1946	1	1	yes	characteristic
128	C₄	#2223	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	#9598	1	1	yes	characteristic
131	C₂	#13172	1	1	yes	characteristic
132	C₂	#13173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#368

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

#11401

C₂

yes

maximal, characteristic

#9608

C₂

yes

maximal

#7176

C₂

yes

maximal

C₂×#368

C₄

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#108

C₂²

yes

—

#1565

—

C₂×#335

C₂²

yes

—

#1450

C₂²

yes

characteristic

#1381

C₄

yes

characteristic

#1381

C₄

yes

—

—

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

D₄

yes

characteristic

#431

D₄

yes

characteristic

C₂×#108

C₂×C₄

yes

—

C₂×#108

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

#346

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

Q₈

yes

—

—

—

—

—

—

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

—

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Q₁₆

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

#121

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

#135

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₄×Q₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#389

yes

characteristic

101

C₂×C₄

#435

yes

—

102

C₂×C₄

C₄×D₈

yes

characteristic

103

C₂×C₄

#444

yes

—

104

C₂×C₄

#466

yes

characteristic

105

C₂×C₄

#467

yes

characteristic

106

C₂×C₄

#499

yes

characteristic

107

C₂×C₄

#500

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

#423

yes

characteristic

116

C₈

#424

yes

characteristic

117

C₈

#446

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1584

yes

—

121

C₂²

#1942

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1581

yes

—

126

C₄

#1940

yes

characteristic

127

C₄

#1946

yes

characteristic

128

C₄

#2223

yes

—

129

C₂

—

130

C₂

#9598

yes

characteristic

131

C₂

#13172

yes

characteristic

132

C₂

#13173

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂³	C₄×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#466

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

#1942

prime radical

C₂³

C₄×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ 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
C₄×#368	2
C₂×#1372	2
#11401	2
#9608	2
#7176	2

subgroup

index

Subgroups — #140368

Conjugacy classes of subgroups

Maximal subgroups