Group #145850 — subgroups

Subgroups — #145850

Order 512.

307 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 84 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	#28825	C₂	1	1	yes	maximal, characteristic
2	#13181	C₂	1	2	yes	maximal
3	#8818	C₂	1	1	yes	maximal, characteristic
4	#7269	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	#3164	—	2	2	no	—
7	#2159	—	2	2	no	—
8	C₂×#364	C₂²	1	2	yes	—
9	#2060	—	2	2	no	—
10	#1952	—	2	2	no	—
11	#1845	—	2	2	no	—
12	#1845	C₂²	1	1	yes	characteristic
13	#1845	C₂²	1	1	yes	characteristic
14	#1772	C₂²	1	2	yes	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1387	—	2	2	no	—
17	#544	—	2	2	no	—
18	C₂×M₃₂	—	2	2	no	—
19	#501	—	4	4	no	—
20	#500	—	4	2	no	—
21	#499	—	4	2	no	—
22	C₂×#108	D₄	1	2	yes	—
23	#428	—	2	4	no	—
24	#427	D₄	1	2	yes	—
25	C₂×#107	—	2	2	no	—
26	C₂×#107	—	2	2	no	—
27	C₂×#107	—	4	4	no	—
28	#405	—	2	2	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#384	—	2	1	no	—
32	#384	D₄	1	2	yes	—
33	#367	—	4	2	no	—
34	#366	—	4	2	no	—
35	#364	—	2	2	no	—
36	#364	Q₈	1	4	yes	complemented
37	#343	—	2	2	no	—
38	#339	—	2	2	no	—
39	C₈²	—	2	1	no	—
40	C₈²	Q₈	1	2	yes	complemented
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×D₄	1	2	yes	—
44	C₄⋊Q₈	—	4	4	no	—
45	#125	—	4	4	no	—
46	C₂²⋊Q₈	—	8	2	no	—
47	#117	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄⋊C₄	—	4	2	no	—
51	C₂×C₄⋊C₄	—	4	1	no	—
52	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
53	M₃₂	—	2	4	no	—
54	M₃₂	—	4	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	#108	—	2	2	no	—
57	#108	Q₁₆	1	4	yes	—
58	#107	—	4	4	no	—
59	#107	—	4	2	no	—
60	#107	—	4	2	no	—
61	#107	—	4	2	no	—
62	#107	—	4	2	no	—
63	#107	—	8	8	no	—
64	#107	D₈	1	2	yes	—
65	#107	D₈	1	2	yes	—
66	#106	—	2	2	no	—
67	#106	—	4	4	no	—
68	#97	—	4	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	4	2	no	—
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	2	yes	—
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂×Q₈	—	8	2	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	4	2	no	—
78	C₂²×C₄	—	8	1	no	—
79	C₂²×C₄	#120	1	1	yes	characteristic
80	M₁₆	—	2	4	no	—
81	M₁₆	—	4	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₈	C₄⋊Q₈	1	2	yes	—
88	C₂×C₈	C₂×D₈	1	2	yes	—
89	C₂×C₈	C₂×Q₁₆	1	2	yes	—
90	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₈	#137	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	—	8	4	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₂²⋊C₄	—	4	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	4	2	no	—
103	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
104	C₄²	C₄⋊Q₈	1	1	yes	characteristic
105	C₁₆	—	4	4	no	—
106	C₂³	#453	1	1	yes	characteristic
107	Q₈	—	32	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	2	no	—
117	C₂×C₄	—	16	1	no	—
118	C₂×C₄	#395	1	1	yes	characteristic
119	C₂×C₄	#447	1	2	yes	—
120	C₂×C₄	#451	1	2	yes	—
121	C₂×C₄	#456	1	1	yes	characteristic
122	C₈	—	2	8	no	—
123	C₈	—	4	4	no	—
124	C₈	#474	1	2	yes	—
125	C₈	#477	1	2	yes	—
126	C₈	#500	1	2	yes	—
127	C₈	#501	1	2	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#1619	1	1	yes	characteristic
131	C₂²	#1633	1	1	yes	characteristic
132	C₂²	#2060	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	8	2	no	—
136	C₄	—	8	2	no	—
137	C₄	—	16	2	no	—
138	C₄	—	32	1	no	—
139	C₄	#1649	1	2	yes	—
140	C₄	#2057	1	2	yes	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#10573	1	1	yes	characteristic
144	C₂	#12310	1	1	yes	characteristic
145	C₂	#12313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28825

C₂

yes

maximal, characteristic

#13181

C₂

yes

maximal

#8818

C₂

yes

maximal, characteristic

#7269

C₂

yes

maximal, characteristic

#7179

C₂

yes

—

—

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#1845

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1387

—

#544

—

C₂×M₃₂

—

—

—

—

D₄

yes

—

#428

—

#427

D₄

yes

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#384

D₄

yes

—

—

—

—

Q₈

yes

complemented

#343

—

#339

—

C₈²

—

C₈²

Q₈

yes

complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

Q₁₆

yes

—

—

—

—

—

—

—

D₈

yes

—

#107

D₈

yes

—

#106

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#120

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊Q₈

yes

characteristic

104

C₄²

C₄⋊Q₈

yes

characteristic

105

C₁₆

—

106

C₂³

#453

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#395

yes

characteristic

119

C₂×C₄

#447

yes

—

120

C₂×C₄

#451

yes

—

121

C₂×C₄

#456

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#474

yes

—

125

C₈

#477

yes

—

126

C₈

#500

yes

—

127

C₈

#501

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

#1619

yes

characteristic

131

C₂²

#1633

yes

characteristic

132

C₂²

#2060

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1649

yes

—

140

C₄

#2057

yes

—

141

C₂

—

142

C₂

—

143

C₂

#10573

yes

characteristic

144

C₂

#12310

yes

characteristic

145

C₂

#12313

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#453
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1619

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#453

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
36. #364	Q₈	semidirect
40. C₈²	Q₈	semidirect

H ≅ G/N

type

36. #364

Q₈

semidirect

40. C₈²

Q₈

semidirect

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

subgroup	index
#28825	2
#13181	2
#8818	2
#7269	2
#7179	2

subgroup

index

Subgroups — #145850

Conjugacy classes of subgroups

Complements

Maximal subgroups