Group #146100 — subgroups

Subgroups — #146100

Order 512.

196 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11976	C₂	1	2	yes	maximal
2	#9859	C₂	1	1	yes	maximal, characteristic
3	#8826	C₂	1	1	yes	maximal, characteristic
4	#7223	C₂	1	2	yes	maximal
5	#7196	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	1	yes	characteristic
7	#2163	C₂²	1	2	yes	—
8	#1848	—	2	1	no	—
9	#1837	C₂²	1	1	yes	characteristic
10	#1567	—	2	1	no	—
11	#1567	C₂²	1	2	yes	—
12	#1493	—	2	1	no	—
13	#1493	—	2	1	no	—
14	C₂×#322	C₂²	1	1	yes	characteristic
15	#1365	—	2	2	no	—
16	#1364	—	2	2	no	—
17	#533	—	2	1	no	—
18	#500	D₄	1	1	yes	characteristic
19	#500	D₄	1	1	yes	characteristic
20	#499	—	2	2	no	—
21	#498	—	2	1	no	—
22	C₂×#108	—	4	2	no	—
23	#427	—	2	2	no	—
24	#423	—	2	2	no	—
25	#423	—	2	1	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊C₁₆	—	2	2	no	—
30	C₂²⋊C₁₆	—	2	2	no	—
31	#346	—	2	2	no	—
32	#341	—	2	2	no	—
33	#335	—	2	2	no	—
34	#335	—	2	2	no	—
35	#335	—	4	2	no	—
36	Q₈⋊C₈	—	4	2	no	—
37	Q₈⋊C₈	—	4	2	no	—
38	#322	—	2	2	no	—
39	C₂×M₁₆	—	4	1	no	—
40	C₂×M₁₆	—	4	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	#41	1	2	yes	—
44	C₄⋊Q₈	—	2	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	#125	—	4	1	no	—
47	C₂²⋊Q₈	—	4	1	no	—
48	C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈	—	4	1	no	—
50	C₄×Q₈	—	4	1	no	—
51	#117	—	4	1	no	—
52	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
53	C₂×C₁₆	—	4	2	no	—
54	#108	—	8	4	no	—
55	#107	—	2	2	no	—
56	#107	—	2	2	no	—
57	#106	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	4	1	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	4	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	D₈	1	2	yes	—
65	C₄×C₈	SD₁₆	1	2	yes	—
66	C₄×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄×C₈	#41	1	1	yes	characteristic
68	C₄×C₈	#41	1	2	yes	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂²×C₄	#124	1	1	yes	characteristic
72	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂²×C₄	#137	1	1	yes	characteristic
74	M₁₆	—	8	4	no	—
75	M₁₆	—	8	2	no	—
76	C₂×C₈	—	2	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	—	8	1	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	1	no	—
84	C₂×C₈	#123	1	2	yes	—
85	C₂×C₈	#126	1	2	yes	—
86	C₄⋊C₄	—	4	1	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	8	1	no	—
91	C₄⋊C₄	—	8	1	no	—
92	C₂²⋊C₄	—	8	1	no	—
93	C₄²	—	8	1	no	—
94	C₄²	#121	1	1	yes	characteristic
95	C₄²	#123	1	1	yes	characteristic
96	C₄²	C₂×D₈	1	1	yes	characteristic
97	C₄²	C₂×SD₁₆	1	1	yes	characteristic
98	C₁₆	—	8	2	no	—
99	C₂³	#489	1	1	yes	characteristic
100	Q₈	—	8	2	no	—
101	Q₈	—	8	1	no	—
102	Q₈	—	8	1	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	#397	1	1	yes	characteristic
110	C₂×C₄	#459	1	1	yes	characteristic
111	C₂×C₄	#461	1	1	yes	characteristic
112	C₂×C₄	#480	1	1	yes	characteristic
113	C₂×C₄	#481	1	1	yes	characteristic
114	C₂×C₄	#486	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	2	2	no	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	16	1	no	—
120	C₈	—	16	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#2071	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	16	1	no	—
130	C₄	—	16	1	no	—
131	C₄	#2082	1	2	yes	—
132	C₄	#2213	1	1	yes	characteristic
133	C₄	#2214	1	1	yes	characteristic
134	C₂	—	4	1	no	—
135	C₂	#10632	1	1	yes	characteristic
136	C₂	#13484	1	1	yes	characteristic
137	C₂	#13486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11976

C₂

yes

maximal

#9859

C₂

yes

maximal, characteristic

#8826

C₂

yes

maximal, characteristic

#7223

C₂

yes

maximal

#7196

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

#2163

C₂²

yes

—

#1848

—

#1837

C₂²

yes

characteristic

#1567

—

#1567

C₂²

yes

—

#1493

—

#1493

—

C₂×#322

C₂²

yes

—

—

—

D₄

yes

characteristic

#500

D₄

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#123

yes

—

C₂×C₈

#126

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₁₆

—

C₂³

#489

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#397

yes

characteristic

110

C₂×C₄

#459

yes

characteristic

111

C₂×C₄

#461

yes

characteristic

112

C₂×C₄

#480

yes

characteristic

113

C₂×C₄

#481

yes

characteristic

114

C₂×C₄

#486

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#2071

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#2082

yes

—

132

C₄

#2213

yes

characteristic

133

C₄

#2214

yes

characteristic

134

C₂

—

135

C₂

#10632

yes

characteristic

136

C₂

#13484

yes

characteristic

137

C₂

#13486

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2071
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#489
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2071

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#489

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
#11976	2
#9859	2
#8826	2
#7223	2
#7196	2

subgroup

index

Subgroups — #146100

Conjugacy classes of subgroups

Maximal subgroups