Group #146199 — subgroups

Subgroups — #146199

Order 512.

231 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 64 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	#12030	C₂	1	2	yes	maximal
2	#11664	C₂	1	2	yes	maximal
3	#9664	C₂	1	1	yes	maximal, characteristic
4	#8825	C₂	1	1	yes	maximal, characteristic
5	#7146	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
7	#2163	C₂²	1	2	yes	—
8	#1984	—	2	1	no	—
9	#1848	—	2	1	no	—
10	#1566	—	2	1	no	—
11	#1566	C₂²	1	2	yes	—
12	C₂×#334	C₂²	1	1	yes	characteristic
13	C₂×Q₈⋊C₈	—	2	1	no	—
14	C₂×#322	C₂²	1	1	yes	characteristic
15	#1368	—	2	4	no	—
16	#1334	—	2	4	no	—
17	C₂²×Q₁₆	Q₈	1	1	yes	characteristic
18	C₂×C₄×Q₈	—	2	1	no	—
19	C₄×Q₁₆	—	2	2	no	—
20	C₄×Q₁₆	—	2	2	no	—
21	C₂×#108	—	4	2	no	—
22	C₂×#107	Q₈	1	1	yes	characteristic
23	#423	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₂×#103	—	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	D₄	1	4	yes	—
32	#336	—	4	2	no	—
33	#334	—	2	2	no	—
34	#334	—	2	2	no	—
35	#334	—	2	2	no	—
36	#334	—	4	2	no	—
37	Q₈⋊C₈	—	4	2	no	—
38	#322	—	2	2	no	—
39	C₂²×Q₈	—	2	1	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	4	2	no	—
42	C₂×M₁₆	—	4	2	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	—	4	1	no	—
45	C₂²×C₈	SD₁₆	1	2	yes	—
46	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₈	#41	1	1	yes	characteristic
48	C₄×Q₈	—	4	1	no	—
49	C₄×Q₈	—	4	1	no	—
50	C₄×Q₈	—	4	1	no	—
51	C₄×Q₈	—	4	1	no	—
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄⋊C₄	—	4	1	no	—
54	C₂×C₄²	—	4	1	no	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₁₆	—	2	4	no	—
57	#108	—	8	4	no	—
58	#107	—	2	2	no	—
59	C₄⋊C₈	—	4	1	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	4	1	no	—
62	C₄⋊C₈	—	4	2	no	—
63	#103	—	4	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	SD₁₆	1	2	yes	—
68	C₄×C₈	C₂×D₄	1	2	yes	—
69	C₄×C₈	#41	1	2	yes	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	4	1	no	—
72	C₂×Q₈	—	4	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	1	no	—
76	C₂²×C₄	—	4	1	no	—
77	C₂²×C₄	—	4	1	no	—
78	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	4	2	no	—
83	M₁₆	—	8	4	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	—	8	1	no	—
92	C₂×C₈	—	8	2	no	—
93	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₈	#123	1	2	yes	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	8	1	no	—
98	C₄⋊C₄	—	8	1	no	—
99	C₄²	—	8	1	no	—
100	C₄²	—	8	1	no	—
101	C₄²	#123	1	1	yes	characteristic
102	C₄²	#127	1	1	yes	characteristic
103	C₄²	C₂×SD₁₆	1	1	yes	characteristic
104	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₁₆	—	4	4	no	—
106	C₂³	#476	1	1	yes	characteristic
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	8	1	no	—
110	Q₈	—	8	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	#399	1	1	yes	characteristic
121	C₂×C₄	#476	1	1	yes	characteristic
122	C₂×C₄	#481	1	1	yes	characteristic
123	C₂×C₄	#481	1	1	yes	characteristic
124	C₂×C₄	#492	1	1	yes	characteristic
125	C₂×C₄	#496	1	1	yes	characteristic
126	C₈	—	2	4	no	—
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	4	2	no	—
130	C₈	—	16	1	no	—
131	C₈	—	16	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#2094	1	1	yes	characteristic
135	C₂²	#2230	1	2	yes	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	#2091	1	1	yes	characteristic
143	C₄	#2098	1	1	yes	characteristic
144	C₄	#2239	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	#10661	1	1	yes	characteristic
147	C₂	#13533	1	1	yes	characteristic
148	C₂	#13536	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12030

C₂

yes

maximal

#11664

C₂

yes

maximal

#9664

C₂

yes

maximal, characteristic

#8825

C₂

yes

maximal, characteristic

#7146

C₂

yes

maximal, characteristic

C₂×C₄×Q₁₆

C₂²

yes

characteristic

#2163

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#334

C₂²

yes

characteristic

C₂×Q₈⋊C₈

—

C₂×#322

C₂²

yes

characteristic

#1368

—

#1334

—

C₂²×Q₁₆

Q₈

yes

characteristic

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#108

—

C₂×#107

Q₈

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

D₄

yes

—

—

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#108

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

#123

yes

characteristic

102

C₄²

#127

yes

characteristic

103

C₄²

C₂×SD₁₆

yes

characteristic

104

C₄²

AGL(1,ℤ/8)

yes

characteristic

105

C₁₆

—

106

C₂³

#476

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#399

yes

characteristic

121

C₂×C₄

#476

yes

characteristic

122

C₂×C₄

#481

yes

characteristic

123

C₂×C₄

#481

yes

characteristic

124

C₂×C₄

#492

yes

characteristic

125

C₂×C₄

#496

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

#2094

yes

characteristic

135

C₂²

#2230

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#2091

yes

characteristic

143

C₄

#2098

yes

characteristic

144

C₄

#2239

yes

—

145

C₂

—

146

C₂

#10661

yes

characteristic

147

C₂

#13533

yes

characteristic

148

C₂

#13536

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#476
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

#2094

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#476

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
#12030	2
#11664	2
#9664	2
#8825	2
#7146	2

subgroup

index

Subgroups — #146199

Conjugacy classes of subgroups

Maximal subgroups