Group #146213 — subgroups

Subgroups — #146213

Order 512.

229 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 74 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	#12032	C₂	1	2	yes	maximal
2	#11646	C₂	1	2	yes	maximal
3	#9848	C₂	1	1	yes	maximal, characteristic
4	#8810	C₂	1	1	yes	maximal, characteristic
5	#7142	C₂	1	1	yes	maximal, characteristic
6	#3170	C₂²	1	1	yes	characteristic
7	#2162	C₂²	1	2	yes	—
8	#1994	—	2	1	no	—
9	#1842	—	2	1	no	—
10	#1744	C₂²	1	1	yes	characteristic
11	#1567	C₂²	1	1	yes	characteristic
12	C₂×#335	—	2	1	no	—
13	C₂×#335	C₂²	1	2	yes	—
14	#1493	—	2	1	no	—
15	#1371	—	2	4	no	—
16	#1332	—	2	4	no	—
17	#533	—	2	1	no	—
18	#500	Q₈	1	1	yes	characteristic
19	#494	Q₈	1	1	yes	characteristic
20	#487	—	2	1	no	—
21	#467	—	2	2	no	—
22	C₄×Q₁₆	—	2	2	no	—
23	#423	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₈	—	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	C₄×C₁₆	D₄	1	4	yes	—
32	#339	—	4	2	no	—
33	#336	—	2	2	no	—
34	#335	—	2	2	no	—
35	#335	—	2	2	no	—
36	#335	—	2	2	no	—
37	#335	—	4	2	no	—
38	Q₈⋊C₈	—	4	2	no	—
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×M₁₆	—	4	1	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	4	2	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	—	2	1	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	D₈	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	C₄⋊C₈	—	8	4	no	—
63	#103	—	4	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	D₈	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	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	#123	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂²×C₄	#137	1	1	yes	characteristic
75	Q₁₆	—	4	4	no	—
76	M₁₆	—	8	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	8	2	no	—
81	C₂×C₈	—	8	1	no	—
82	C₂×C₈	—	8	4	no	—
83	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₈	#123	1	2	yes	—
85	C₄⋊C₄	—	4	1	no	—
86	C₄⋊C₄	—	4	1	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	8	1	no	—
90	C₄⋊C₄	—	8	1	no	—
91	C₂²⋊C₄	—	8	1	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	4	2	no	—
94	C₄²	—	8	1	no	—
95	C₄²	—	8	1	no	—
96	C₄²	Q₃₂	1	4	yes	—
97	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
98	C₄²	#127	1	1	yes	characteristic
99	C₄²	C₂×D₈	1	1	yes	characteristic
100	C₄²	#137	1	1	yes	characteristic
101	C₁₆	—	4	4	no	—
102	C₂³	#480	1	1	yes	characteristic
103	Q₈	—	8	2	no	—
104	Q₈	—	8	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	#399	1	1	yes	characteristic
113	C₂×C₄	#474	1	1	yes	characteristic
114	C₂×C₄	#474	1	1	yes	characteristic
115	C₂×C₄	#480	1	1	yes	characteristic
116	C₂×C₄	#493	1	1	yes	characteristic
117	C₂×C₄	#497	1	1	yes	characteristic
118	C₂×C₄	C₂×Q₃₂	1	2	yes	—
119	C₂×C₄	#509	1	2	yes	—
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	4	2	no	—
123	C₈	—	16	2	no	—
124	C₈	—	16	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#2093	1	1	yes	characteristic
128	C₂²	#2236	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	4	no	—
133	C₄	—	16	1	no	—
134	C₄	—	16	1	no	—
135	C₄	#2098	1	2	yes	—
136	C₄	#2227	1	2	yes	—
137	C₄	#2247	1	1	yes	characteristic
138	C₄	#2250	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	#10667	1	1	yes	characteristic
141	C₂	#13530	1	1	yes	characteristic
142	C₂	#13532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12032

C₂

yes

maximal

#11646

C₂

yes

maximal

#9848

C₂

yes

maximal, characteristic

#8810

C₂

yes

maximal, characteristic

#7142

C₂

yes

maximal, characteristic

#3170

C₂²

yes

characteristic

#2162

C₂²

yes

—

#1994

—

#1842

—

#1744

C₂²

yes

characteristic

#1567

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

—

—

—

—

Q₈

yes

characteristic

#494

Q₈

yes

characteristic

#487

—

#467

—

C₄×Q₁₆

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

—

Q₈⋊C₈

—

C₂×Q₁₆

—

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×C₄²

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#123

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#137

yes

characteristic

Q₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Q₃₂

yes

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#127

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

100

C₄²

#137

yes

characteristic

101

C₁₆

—

102

C₂³

#480

yes

characteristic

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#399

yes

characteristic

113

C₂×C₄

#474

yes

characteristic

114

C₂×C₄

#474

yes

characteristic

115

C₂×C₄

#480

yes

characteristic

116

C₂×C₄

#493

yes

characteristic

117

C₂×C₄

#497

yes

characteristic

118

C₂×C₄

C₂×Q₃₂

yes

—

119

C₂×C₄

#509

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

#2093

yes

characteristic

128

C₂²

#2236

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#2098

yes

—

136

C₄

#2227

yes

—

137

C₄

#2247

yes

characteristic

138

C₄

#2250

yes

characteristic

139

C₂

—

140

C₂

#10667

yes

characteristic

141

C₂

#13530

yes

characteristic

142

C₂

#13532

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#480
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ 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
#12032	2
#11646	2
#9848	2
#8810	2
#7142	2

subgroup

index

Subgroups — #146213

Conjugacy classes of subgroups

Maximal subgroups