Group #146205 — subgroups

Subgroups — #146205

Order 512.

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

normal subgroups only

This group has 235 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, complemented
2	#11654	C₂	1	2	yes	maximal
3	#9852	C₂	1	1	yes	maximal, characteristic
4	#8811	C₂	1	1	yes	maximal, characteristic, complemented
5	#7143	C₂	1	1	yes	maximal, characteristic, complemented
6	#3169	C₂²	1	1	yes	characteristic
7	#2163	C₂²	1	2	yes	—
8	#1962	—	2	1	no	—
9	#1831	—	2	1	no	—
10	#1744	C₂²	1	1	yes	characteristic
11	#1566	C₂²	1	1	yes	characteristic
12	C₂×#335	—	2	1	no	—
13	C₂×#334	C₂²	1	2	yes	—
14	#1492	—	2	1	no	—
15	#1368	—	2	4	no	—
16	#1333	—	2	4	no	—
17	#532	—	2	1	no	—
18	#500	Q₈	1	1	yes	characteristic
19	#493	Q₈	1	1	yes	characteristic
20	#486	—	2	1	no	—
21	#466	—	2	2	no	—
22	C₄×D₈	—	2	2	no	—
23	#423	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×#104	—	4	2	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄³	D₄	1	1	yes	characteristic
29	C₄⋊C₁₆	—	2	2	no	—
30	C₂²⋊C₁₆	—	2	2	no	—
31	#346	D₄	1	4	yes	—
32	#336	—	2	2	no	—
33	#336	—	4	2	no	—
34	#335	—	4	2	no	—
35	#334	—	2	2	no	—
36	#334	—	2	2	no	—
37	#334	—	2	2	no	—
38	D₄⋊C₈	—	4	2	no	—
39	C₂×#41	—	4	1	no	—
40	C₂×D₈	—	2	2	no	—
41	C₂×M₁₆	—	4	1	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	—	4	2	no	—
44	C₂²×C₈	#41	1	2	yes	—
45	C₄⋊Q₈	—	2	1	no	—
46	#127	—	2	1	no	—
47	#124	—	4	1	no	—
48	#121	—	4	1	no	—
49	#121	—	4	1	no	—
50	C₄×D₄	—	4	1	no	—
51	C₄×D₄	—	4	1	no	—
52	C₂×C₄²	—	2	1	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	SD₁₆	1	2	yes	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₁₆	—	2	4	no	—
57	#107	—	2	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	4	1	no	—
60	C₄⋊C₈	—	4	1	no	—
61	C₄⋊C₈	—	4	2	no	—
62	#104	—	8	4	no	—
63	#102	—	4	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	SD₁₆	1	2	yes	—
66	C₄×C₈	C₂×D₄	1	2	yes	—
67	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₄×C₈	#41	1	1	yes	characteristic
69	#41	—	8	2	no	—
70	#41	—	8	2	no	—
71	C₂×Q₈	—	4	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂×D₄	—	4	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	8	1	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	8	1	no	—
78	C₂²×C₄	#123	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	D₈	—	4	4	no	—
82	M₁₆	—	8	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	8	4	no	—
87	C₂×C₈	—	8	1	no	—
88	C₂×C₈	—	8	2	no	—
89	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₈	#123	1	2	yes	—
91	C₄⋊C₄	—	4	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	8	1	no	—
94	C₂²⋊C₄	—	8	1	no	—
95	C₄²	—	4	2	no	—
96	C₄²	—	4	2	no	—
97	C₄²	—	4	2	no	—
98	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
99	C₄²	#127	1	1	yes	characteristic
100	C₄²	C₂×SD₁₆	1	1	yes	characteristic
101	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₁₆	—	4	4	no	—
103	C₂³	—	8	1	no	—
104	C₂³	#481	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	D₄	—	8	2	no	—
107	D₄	—	8	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	16	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	—	16	1	no	—
118	C₂×C₄	—	16	1	no	—
119	C₂×C₄	#399	1	1	yes	characteristic
120	C₂×C₄	#476	1	1	yes	characteristic
121	C₂×C₄	#476	1	1	yes	characteristic
122	C₂×C₄	#481	1	1	yes	characteristic
123	C₂×C₄	#492	1	1	yes	characteristic
124	C₂×C₄	#496	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	—	4	2	no	—
128	C₈	—	16	2	no	—
129	C₈	—	16	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	16	1	no	—
133	C₂²	—	16	1	no	—
134	C₂²	—	16	1	no	—
135	C₂²	#2094	1	1	yes	characteristic
136	C₂²	#2238	1	2	yes	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	8	2	no	—
142	C₄	—	16	1	no	—
143	C₄	#2098	1	2	yes	—
144	C₄	#2231	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	32	1	no	—
147	C₂	#10666	1	1	yes	characteristic
148	C₂	#13534	1	1	yes	characteristic
149	C₂	#13535	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12030

C₂

yes

maximal, complemented

#11654

C₂

yes

maximal

#9852

C₂

yes

maximal, characteristic

#8811

C₂

yes

maximal, characteristic, complemented

#7143

C₂

yes

maximal, characteristic, complemented

#3169

C₂²

yes

characteristic

#2163

C₂²

yes

—

#1962

—

#1831

—

#1744

C₂²

yes

characteristic

#1566

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

C₂²

yes

—

—

—

—

—

Q₈

yes

characteristic

#493

Q₈

yes

characteristic

#486

—

#466

—

C₄×D₈

—

#423

—

C₂×C₄⋊C₈

—

C₂×#104

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

D₄

yes

—

—

—

—

—

—

—

D₄⋊C₈

—

C₂×#41

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

#102

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#123

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

D₈

—

M₁₆

—

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#127

yes

characteristic

100

C₄²

C₂×SD₁₆

yes

characteristic

101

C₄²

AGL(1,ℤ/8)

yes

characteristic

102

C₁₆

—

103

C₂³

—

104

C₂³

#481

yes

characteristic

105

Q₈

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

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

—

119

C₂×C₄

#399

yes

characteristic

120

C₂×C₄

#476

yes

characteristic

121

C₂×C₄

#476

yes

characteristic

122

C₂×C₄

#481

yes

characteristic

123

C₂×C₄

#492

yes

characteristic

124

C₂×C₄

#496

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#2094

yes

characteristic

136

C₂²

#2238

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#2098

yes

—

144

C₄

#2231

yes

—

145

C₂

—

146

C₂

—

147

C₂

#10666

yes

characteristic

148

C₂

#13534

yes

characteristic

149

C₂

#13535

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ 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

#3169

C₂²

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #12030	C₂	semidirect
4. #8811	C₂	semidirect
5. #7143	C₂	semidirect

H ≅ G/N

type

1. #12030

C₂

semidirect

4. #8811

C₂

semidirect

5. #7143

C₂

semidirect

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

subgroup	index
#12030	2
#11654	2
#9852	2
#8811	2
#7143	2

subgroup

index

Subgroups — #146205

Conjugacy classes of subgroups

Complements

Maximal subgroups