Group #145576 — subgroups

Subgroups — #145576

Order 512.

229 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 76 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	#33728	C₂	1	1	yes	maximal, characteristic
2	#13271	C₂	1	2	yes	maximal
3	#13162	C₂	1	1	yes	maximal, characteristic
4	#11956	C₂	1	2	yes	maximal
5	#7161	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	2	yes	—
7	#2255	—	2	4	no	—
8	#2158	—	2	1	no	—
9	#2158	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	2	yes	—
12	#2151	—	2	1	no	—
13	C₂×#346	C₂²	1	1	yes	characteristic
14	#1772	C₂²	1	1	yes	characteristic
15	#1389	—	2	2	no	—
16	#1384	—	2	2	no	—
17	#1379	—	2	1	no	—
18	#1366	—	2	4	no	—
19	#533	—	2	2	no	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	#500	—	2	2	no	—
25	#499	—	2	2	no	—
26	#499	D₄	1	4	yes	—
27	#498	—	2	2	no	—
28	C₂×#108	D₄	1	1	yes	characteristic
29	#427	—	2	2	no	—
30	#427	—	2	2	no	—
31	#427	D₄	1	1	yes	characteristic
32	#405	—	2	1	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#367	—	2	2	no	—
35	#367	—	2	2	no	—
36	#366	—	2	2	no	—
37	#366	—	2	2	no	—
38	#364	—	2	2	no	—
39	#364	—	2	2	no	—
40	#364	—	2	2	no	—
41	C₄⋊C₁₆	—	4	2	no	—
42	#346	—	2	2	no	—
43	#339	—	2	1	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Q₈	—	4	2	no	—
48	#125	—	2	4	no	—
49	C₂²⋊Q₈	—	8	2	no	—
50	C₄×Q₈	—	8	2	no	—
51	#117	—	2	1	no	—
52	#117	—	4	2	no	—
53	C₂×C₄²	#41	1	1	yes	characteristic
54	M₃₂	—	4	1	no	—
55	M₃₂	—	4	2	no	—
56	M₃₂	—	4	2	no	—
57	M₃₂	—	4	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	—	4	1	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	SD₁₆	1	2	yes	—
65	C₂×C₁₆	Q₁₆	1	2	yes	—
66	#108	—	2	2	no	—
67	#107	—	2	1	no	—
68	#107	—	2	4	no	—
69	#107	—	4	2	no	—
70	#106	—	2	4	no	—
71	#106	—	4	2	no	—
72	#106	Q₁₆	1	2	yes	—
73	#97	—	4	1	no	—
74	C₄×C₈	—	4	1	no	—
75	C₄×C₈	C₂×D₄	1	2	yes	—
76	C₄×C₈	#41	1	2	yes	—
77	C₂×Q₈	—	8	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	#121	1	1	yes	characteristic
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	#120	1	1	yes	characteristic
84	C₂×C₈	#121	1	2	yes	—
85	C₂×C₈	#124	1	1	yes	characteristic
86	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₈	#135	1	1	yes	characteristic
90	C₂×C₈	#137	1	1	yes	characteristic
91	C₂×C₈	#137	1	1	yes	characteristic
92	C₂×C₈	#137	1	2	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₂²⋊C₄	—	4	1	no	—
98	C₂²⋊C₄	—	8	2	no	—
99	C₄²	—	2	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	—	8	2	no	—
102	C₄²	#123	1	2	yes	—
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	—	8	1	no	—
106	C₁₆	—	8	2	no	—
107	C₂³	#472	1	1	yes	characteristic
108	Q₈	—	16	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	8	4	no	—
113	C₂×C₄	#394	1	1	yes	characteristic
114	C₂×C₄	#461	1	1	yes	characteristic
115	C₂×C₄	#462	1	1	yes	characteristic
116	C₂×C₄	#467	1	1	yes	characteristic
117	C₂×C₄	#470	1	1	yes	characteristic
118	C₂×C₄	#472	1	1	yes	characteristic
119	C₈	—	8	1	no	—
120	C₈	#448	1	1	yes	characteristic
121	C₈	#451	1	1	yes	characteristic
122	C₈	#462	1	2	yes	—
123	C₈	#464	1	2	yes	—
124	C₈	#487	1	1	yes	characteristic
125	C₈	#491	1	1	yes	characteristic
126	C₂²	—	4	1	no	—
127	C₂²	#1675	1	1	yes	characteristic
128	C₂²	#1696	1	1	yes	characteristic
129	C₂²	#2045	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	—	8	2	no	—
132	C₄	—	16	4	no	—
133	C₄	#1676	1	1	yes	characteristic
134	C₄	#1695	1	1	yes	characteristic
135	C₄	#2043	1	1	yes	characteristic
136	C₄	#2056	1	1	yes	characteristic
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#10543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33728

C₂

yes

maximal, characteristic

#13271

C₂

yes

maximal

#13162

C₂

yes

maximal, characteristic

#11956

C₂

yes

maximal

#7161

C₂

yes

maximal, characteristic

#3160

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×#346

C₂²

yes

characteristic

#1772

C₂²

yes

—

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

D₄

yes

—

#498

—

C₂×#108

D₄

yes

—

—

D₄

yes

characteristic

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

C₄⋊C₁₆

—

#346

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

Q₁₆

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

#121

yes

—

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

#123

yes

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

C₂³

#472

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#394

yes

characteristic

114

C₂×C₄

#461

yes

characteristic

115

C₂×C₄

#462

yes

characteristic

116

C₂×C₄

#467

yes

characteristic

117

C₂×C₄

#470

yes

characteristic

118

C₂×C₄

#472

yes

characteristic

119

C₈

—

120

C₈

#448

yes

characteristic

121

C₈

#451

yes

characteristic

122

C₈

#462

yes

—

123

C₈

#464

yes

—

124

C₈

#487

yes

characteristic

125

C₈

#491

yes

characteristic

126

C₂²

—

127

C₂²

#1675

yes

characteristic

128

C₂²

#1696

yes

characteristic

129

C₂²

#2045

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1676

yes

characteristic

134

C₄

#1695

yes

characteristic

135

C₄

#2043

yes

characteristic

136

C₄

#2056

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#10543

yes

characteristic

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10543
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1695

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#33728	2
#13271	2
#13162	2
#11956	2
#7161	2

subgroup

index

Subgroups — #145576

Conjugacy classes of subgroups

Maximal subgroups