Group #491484 — subgroups

Subgroups — #491484

Order 512.

516 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 198 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	#33731	C₂	1	4	yes	maximal
2	#19604	C₂	1	1	yes	maximal, characteristic
3	C₂×#1710	C₂	1	1	yes	maximal, characteristic
4	#12524	C₂	1	4	yes	maximal, complemented
5	#12522	C₂	1	4	yes	maximal, complemented
6	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
7	#3170	—	2	2	no	—
8	#3160	C₂²	1	2	yes	—
9	C₂×#500	C₂²	1	2	yes	—
10	C₂×#487	—	2	2	no	—
11	#2255	—	2	8	no	—
12	#2254	C₂²	1	8	yes	—
13	#2252	C₂²	1	8	yes	—
14	#2159	—	2	8	no	—
15	C₄×M₃₂	C₂²	1	2	yes	—
16	#1713	C₂²	1	2	yes	—
17	#1710	—	2	2	no	—
18	#1710	C₂²	1	2	yes	—
19	#1710	C₂²	1	2	yes	—
20	#1710	C₂²	1	2	yes	—
21	#1707	—	2	2	no	—
22	C₂×C₈²	C₂²	1	1	yes	characteristic
23	#1311	C₂²	1	2	yes	—
24	C₈×C₁₆	C₂²	1	2	yes	—
25	#533	—	2	2	no	—
26	C₂×C₄⋊Q₈	—	2	2	no	—
27	C₂×M₃₂	D₄	1	4	yes	—
28	#500	—	2	4	no	—
29	#500	—	2	2	no	—
30	#500	—	2	2	no	—
31	#500	C₂³	1	4	yes	—
32	#500	C₂³	1	4	yes	—
33	#499	—	2	4	no	—
34	#498	—	2	2	no	—
35	#494	—	2	2	no	—
36	#487	—	2	4	no	—
37	#487	—	2	4	no	—
38	#487	—	2	4	no	—
39	#487	—	4	4	no	—
40	#467	—	4	4	no	—
41	C₄×Q₁₆	—	4	4	no	—
42	#427	—	2	4	no	—
43	C₂×#107	—	2	4	no	—
44	C₂×#103	—	4	4	no	—
45	C₂×C₄×C₈	D₄	1	2	yes	—
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#367	—	2	16	no	—
48	#366	—	2	16	no	—
49	#346	D₄	1	4	yes	—
50	C₄×C₁₆	C₂³	1	4	yes	—
51	C₈²	D₄	1	2	yes	—
52	C₈²	C₂³	1	1	yes	characteristic
53	C₈²	C₂³	1	1	yes	characteristic
54	C₂²×Q₈	—	4	2	no	—
55	C₂×Q₁₆	—	4	4	no	—
56	C₂²×C₈	D₈	1	4	yes	—
57	C₂²×C₈	C₂×D₄	1	2	yes	—
58	C₄⋊Q₈	—	2	4	no	—
59	C₄⋊Q₈	—	2	4	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	4	2	no	—
62	C₄⋊Q₈	—	4	2	no	—
63	#125	—	4	2	no	—
64	C₂²⋊Q₈	—	4	4	no	—
65	C₄×Q₈	—	4	4	no	—
66	#117	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	M₃₂	Q₁₆	1	16	yes	complemented
71	C₂×C₁₆	C₂×D₄	1	8	yes	—
72	#107	—	2	4	no	—
73	#107	—	2	8	no	—
74	#107	—	2	4	no	—
75	#107	—	4	4	no	—
76	#107	—	4	4	no	—
77	#106	—	4	4	no	—
78	#103	—	4	8	no	—
79	#103	—	4	8	no	—
80	#103	—	4	8	no	—
81	C₄×C₈	D₈	1	4	yes	—
82	C₄×C₈	C₂×D₄	1	2	yes	—
83	C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	C₄×C₈	C₂×D₄	1	2	yes	—
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	C₄×C₈	C₂⁴	1	1	yes	characteristic
87	C₂×Q₈	—	4	4	no	—
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	8	8	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂²×C₄	—	8	2	no	—
92	C₂²×C₄	#127	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	2	yes	—
94	Q₁₆	—	8	8	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	#127	1	2	yes	—
97	C₂×C₈	C₂×D₈	1	4	yes	—
98	C₂×C₈	C₂×D₈	1	4	yes	—
99	C₂×C₈	C₂×Q₁₆	1	8	yes	—
100	C₂×C₈	C₂²×D₄	1	2	yes	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	4	no	—
108	C₂²⋊C₄	—	8	2	no	—
109	C₄²	—	8	2	no	—
110	C₄²	#127	1	1	yes	characteristic
111	C₄²	C₂×D₈	1	2	yes	—
112	C₄²	C₂²×D₄	1	1	yes	characteristic
113	C₁₆	C₂×Q₁₆	1	16	yes	—
114	C₂³	#493	1	1	yes	characteristic
115	Q₈	—	8	8	no	—
116	Q₈	—	8	8	no	—
117	Q₈	—	16	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	8	4	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	16	4	no	—
122	C₂×C₄	#493	1	2	yes	—
123	C₂×C₄	#493	1	1	yes	characteristic
124	C₂×C₄	#494	1	4	yes	—
125	C₂×C₄	C₂×#127	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₈	1	2	yes	—
127	C₈	—	2	4	no	—
128	C₈	—	2	4	no	—
129	C₈	#494	1	4	yes	—
130	C₈	C₂²×Q₁₆	1	4	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	#1713	1	2	yes	—
133	C₂²	C₂×#493	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	16	4	no	—
136	C₄	—	16	4	no	—
137	C₄	#1713	1	2	yes	—
138	C₄	C₂×#494	1	2	yes	—
139	C₄	#3414	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×#1713	1	1	yes	characteristic
142	C₂	#33719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33731

C₂

yes

maximal

#19604

C₂

yes

maximal, characteristic

C₂×#1710

C₂

yes

maximal, characteristic

#12524

C₂

yes

maximal, complemented

#12522

C₂

yes

maximal, complemented

C₈×M₃₂

C₂

yes

maximal, characteristic

#3170

—

#3160

C₂²

yes

—

C₂×#500

C₂²

yes

—

C₂×#487

—

#2255

—

#2254

C₂²

yes

—

#2252

C₂²

yes

—

#2159

—

C₄×M₃₂

C₂²

yes

—

#1713

C₂²

yes

—

#1710

—

#1710

C₂²

yes

—

#1710

C₂²

yes

—

#1710

C₂²

yes

—

#1707

—

C₂×C₈²

C₂²

yes

characteristic

#1311

C₂²

yes

—

C₈×C₁₆

C₂²

yes

—

#533

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

—

—

—

C₂³

yes

—

#500

C₂³

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₈²

D₄

yes

—

C₈²

C₂³

yes

characteristic

C₈²

C₂³

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

Q₁₆

yes

complemented

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

—

—

—

—

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂⁴

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

#127

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

100

C₂×C₈

C₂²×D₄

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

#127

yes

characteristic

111

C₄²

C₂×D₈

yes

—

112

C₄²

C₂²×D₄

yes

characteristic

113

C₁₆

C₂×Q₁₆

yes

—

114

C₂³

#493

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#493

yes

—

123

C₂×C₄

#493

yes

characteristic

124

C₂×C₄

#494

yes

—

125

C₂×C₄

C₂×#127

yes

characteristic

126

C₂×C₄

C₂²×D₈

yes

—

127

C₈

—

128

C₈

—

129

C₈

#494

yes

—

130

C₈

C₂²×Q₁₆

yes

—

131

C₂²

—

132

C₂²

#1713

yes

—

133

C₂²

C₂×#493

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#1713

yes

—

138

C₄

C₂×#494

yes

—

139

C₄

#3414

yes

—

140

C₂

—

141

C₂

C₂×#1713

yes

characteristic

142

C₂

#33719

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#493

commutator, Frattini, cosocle

C₄×C₈

C₂⁴

prime radical

C₂³

#493

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #12524	C₂	semidirect
5. #12522	C₂	semidirect
70. M₃₂	Q₁₆	semidirect

H ≅ G/N

type

4. #12524

C₂

semidirect

5. #12522

C₂

semidirect

70. M₃₂

Q₁₆

semidirect

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

subgroup	index
#33731	2
#19604	2
C₂×#1710	2
#12524	2
#12522	2
C₈×M₃₂	2

subgroup

index

C₈×M₃₂

Subgroups — #491484

Conjugacy classes of subgroups

Complements

Maximal subgroups