Group #491546 — subgroups

Subgroups — #491546

Order 512.

489 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 146 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	#33749	C₂	1	2	yes	maximal
2	#33734	C₂	1	2	yes	maximal
3	#19668	C₂	1	2	yes	maximal
4	#12533	C₂	1	4	yes	maximal, complemented
5	#12528	C₂	1	4	yes	maximal, complemented
6	#11401	C₂	1	1	yes	maximal, characteristic
7	#3231	—	2	4	no	—
8	#3170	C₂²	1	4	yes	—
9	#2254	C₂²	1	2	yes	—
10	#2252	C₂²	1	2	yes	—
11	#2247	C₂²	1	2	yes	—
12	#2247	C₂²	1	2	yes	—
13	#2241	C₂²	1	4	yes	—
14	#2241	C₂²	1	4	yes	—
15	#2219	—	2	4	no	—
16	#2216	—	2	4	no	—
17	#2176	—	2	4	no	—
18	C₄×Q₃₂	—	2	4	no	—
19	C₄×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	#1718	—	2	4	no	—
22	#1713	C₂²	1	2	yes	—
23	#1710	C₂²	1	2	yes	—
24	#1707	C₂²	1	4	yes	—
25	#1450	C₂²	1	1	yes	characteristic
26	#1312	C₂²	1	4	yes	—
27	#533	—	2	4	no	—
28	C₂×Q₃₂	—	2	4	no	—
29	C₂×Q₃₂	—	2	4	no	—
30	C₂×M₃₂	D₄	1	2	yes	—
31	C₂²×C₁₆	D₄	1	2	yes	—
32	#500	—	2	4	no	—
33	#500	C₂³	1	4	yes	—
34	#498	—	2	4	no	—
35	#494	C₂³	1	4	yes	—
36	#487	—	2	4	no	—
37	#487	—	2	4	no	—
38	#487	—	2	4	no	—
39	#485	—	4	8	no	—
40	#479	—	4	8	no	—
41	#467	—	2	4	no	—
42	#467	—	2	4	no	—
43	C₄×Q₁₆	—	2	4	no	—
44	C₄×Q₁₆	—	2	4	no	—
45	#431	D₄	1	2	yes	—
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#367	—	2	4	no	—
48	#366	—	2	4	no	—
49	#358	—	2	8	no	—
50	#358	—	2	8	no	—
51	#346	D₄	1	2	yes	—
52	C₄×C₁₆	D₄	1	2	yes	—
53	C₄×C₁₆	C₂³	1	2	yes	—
54	C₄×C₁₆	C₂³	1	2	yes	—
55	#322	D₄	1	2	yes	—
56	C₈²	C₂³	1	2	yes	—
57	C₂×Q₁₆	—	2	4	no	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	4	no	—
62	C₄⋊Q₈	—	2	4	no	—
63	#125	—	4	4	no	—
64	C₂²⋊Q₈	—	4	4	no	—
65	C₂²⋊Q₈	—	4	4	no	—
66	C₄×Q₈	—	4	4	no	—
67	C₄×Q₈	—	4	4	no	—
68	#117	—	4	4	no	—
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	Q₃₂	—	4	8	no	—
71	Q₃₂	—	4	8	no	—
72	M₃₂	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₂×C₁₆	C₂×D₄	1	4	yes	—
75	C₂×C₁₆	C₂×D₄	1	4	yes	—
76	#107	—	2	4	no	—
77	#107	—	2	4	no	—
78	#107	—	4	8	no	—
79	#106	—	4	8	no	—
80	C₄⋊C₈	D₈	1	4	yes	—
81	#103	—	4	4	no	—
82	#103	—	4	4	no	—
83	#103	—	4	8	no	—
84	#103	—	4	8	no	—
85	C₂²⋊C₈	D₈	1	4	yes	—
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	C₂×D₄	1	1	yes	characteristic
89	C₄×C₈	C₂×D₄	1	4	yes	—
90	C₄×C₈	C₂⁴	1	1	yes	characteristic
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	4	4	no	—
93	C₂²×C₄	#127	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	2	yes	—
95	Q₁₆	—	4	8	no	—
96	Q₁₆	—	4	8	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	#127	1	1	yes	characteristic
100	C₂×C₈	#127	1	1	yes	characteristic
101	C₂×C₈	C₂×D₈	1	8	yes	—
102	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	—	8	4	no	—
109	C₄⋊C₄	—	8	4	no	—
110	C₂²⋊C₄	—	8	4	no	—
111	C₄²	—	8	4	no	—
112	C₄²	#127	1	1	yes	characteristic
113	C₄²	C₂×D₈	1	2	yes	—
114	C₄²	C₂²×D₄	1	1	yes	characteristic
115	C₁₆	—	2	4	no	—
116	C₁₆	—	2	4	no	—
117	C₂³	#493	1	1	yes	characteristic
118	Q₈	—	8	4	no	—
119	Q₈	—	8	4	no	—
120	Q₈	—	8	8	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	8	4	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	#493	1	2	yes	—
127	C₂×C₄	#493	1	1	yes	characteristic
128	C₂×C₄	C₂×#127	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₈	1	2	yes	—
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	2	8	no	—
133	C₈	#575	1	2	yes	—
134	C₈	#575	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×#493	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	16	4	no	—
141	C₄	—	16	4	no	—
142	C₄	#3150	1	1	yes	characteristic
143	C₄	#3150	1	1	yes	characteristic
144	C₄	#3417	1	4	yes	—
145	C₂	—	4	1	no	—
146	C₂	#19616	1	1	yes	characteristic
147	C₂	#33721	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33749

C₂

yes

maximal

#33734

C₂

yes

maximal

#19668

C₂

yes

maximal

#12533

C₂

yes

maximal, complemented

#12528

C₂

yes

maximal, complemented

#11401

C₂

yes

maximal, characteristic

#3231

—

#3170

C₂²

yes

—

#2254

C₂²

yes

—

#2252

C₂²

yes

—

#2247

C₂²

yes

—

#2247

C₂²

yes

—

#2241

C₂²

yes

—

#2241

C₂²

yes

—

#2219

—

#2216

—

#2176

—

C₄×Q₃₂

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1718

—

#1713

C₂²

yes

—

#1710

C₂²

yes

—

#1707

C₂²

yes

—

#1450

C₂²

yes

characteristic

#1312

C₂²

yes

—

#533

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

C₂³

yes

—

#498

—

#494

C₂³

yes

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#431

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₄×C₁₆

C₂³

yes

—

#322

D₄

yes

—

C₈²

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

C₄⋊C₈

D₈

yes

—

—

—

—

—

C₂²⋊C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂⁴

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#127

yes

characteristic

100

C₂×C₈

#127

yes

characteristic

101

C₂×C₈

C₂×D₈

yes

—

102

C₂×C₈

C₂²×D₄

yes

characteristic

103

C₂×C₈

C₂²×D₄

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

#127

yes

characteristic

113

C₄²

C₂×D₈

yes

—

114

C₄²

C₂²×D₄

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₂³

#493

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#493

yes

—

127

C₂×C₄

#493

yes

characteristic

128

C₂×C₄

C₂×#127

yes

characteristic

129

C₂×C₄

C₂²×D₈

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

#575

yes

—

134

C₈

#575

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#493

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#3150

yes

characteristic

143

C₄

#3150

yes

characteristic

144

C₄

#3417

yes

—

145

C₂

—

146

C₂

#19616

yes

characteristic

147

C₂

#33721

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 ◁ #1450 ◁ 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 ◁ #1450 ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #12533	C₂	semidirect
5. #12528	C₂	semidirect

H ≅ G/N

type

4. #12533

C₂

semidirect

5. #12528

C₂

semidirect

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

subgroup	index
#33749	2
#33734	2
#19668	2
#12533	2
#12528	2
#11401	2

subgroup

index

Subgroups — #491546

Conjugacy classes of subgroups

Complements

Maximal subgroups