Group #92840 — subgroups

Subgroups — #92840

Order 512.

159 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 42 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	#9859	C₂	1	1	yes	maximal, characteristic
2	#7196	C₂	1	1	yes	maximal, characteristic
3	#7137	C₂	1	1	yes	maximal, characteristic
4	#3160	C₄	1	1	yes	characteristic
5	#2163	—	2	1	no	—
6	C₂×#346	—	2	1	no	—
7	#1837	C₄	1	1	yes	characteristic
8	#1493	—	2	1	no	—
9	#1493	—	2	1	no	—
10	C₂×#322	C₂²	1	1	yes	characteristic
11	#533	—	2	1	no	—
12	C₂²×C₁₆	—	2	2	no	—
13	#500	C₈	1	1	yes	characteristic
14	#500	C₈	1	1	yes	characteristic
15	#499	—	4	1	no	—
16	#498	—	2	1	no	—
17	#427	—	4	1	no	—
18	#423	—	2	1	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₄⋊C₁₆	—	4	1	no	—
23	C₂²⋊C₁₆	—	2	2	no	—
24	#346	—	4	1	no	—
25	#346	—	4	1	no	—
26	#346	—	4	1	no	—
27	#341	—	4	1	no	—
28	Q₈⋊C₈	—	4	1	no	—
29	Q₈⋊C₈	—	4	1	no	—
30	Q₈⋊C₈	—	4	1	no	—
31	Q₈⋊C₈	—	4	1	no	—
32	#322	—	4	1	no	—
33	C₂×M₁₆	—	4	1	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	#125	—	4	1	no	—
40	C₂²⋊Q₈	—	4	1	no	—
41	C₂²⋊Q₈	—	4	1	no	—
42	C₄×Q₈	—	4	1	no	—
43	C₄×Q₈	—	4	1	no	—
44	#117	—	4	1	no	—
45	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₂×C₁₆	—	4	2	no	—
49	C₂×C₁₆	—	4	2	no	—
50	#107	—	4	1	no	—
51	#107	—	4	1	no	—
52	#106	—	4	2	no	—
53	C₄⋊C₈	—	4	1	no	—
54	C₄⋊C₈	—	4	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×C₈	1	1	yes	characteristic
59	C₄×C₈	M₁₆	1	1	yes	characteristic
60	C₄×C₈	D₈	1	1	yes	characteristic
61	C₄×C₈	D₈	1	1	yes	characteristic
62	C₄×C₈	SD₁₆	1	1	yes	characteristic
63	C₄×C₈	SD₁₆	1	1	yes	characteristic
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	—	4	1	no	—
66	C₂²×C₄	#100	1	1	yes	characteristic
67	C₂²×C₄	#104	1	1	yes	characteristic
68	C₂²×C₄	#104	1	1	yes	characteristic
69	M₁₆	—	8	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	8	1	no	—
80	C₂×C₈	—	8	1	no	—
81	C₄⋊C₄	—	4	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	—	8	1	no	—
86	C₄⋊C₄	—	8	1	no	—
87	C₂²⋊C₄	—	8	1	no	—
88	C₄²	—	8	1	no	—
89	C₄²	C₂²⋊C₈	1	1	yes	characteristic
90	C₄²	#99	1	1	yes	characteristic
91	C₄²	#102	1	1	yes	characteristic
92	C₄²	#102	1	1	yes	characteristic
93	C₁₆	—	4	4	no	—
94	C₁₆	—	8	2	no	—
95	C₂³	#329	1	1	yes	characteristic
96	Q₈	—	8	2	no	—
97	Q₈	—	8	1	no	—
98	Q₈	—	8	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	8	1	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	#323	1	1	yes	characteristic
106	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
108	C₂×C₄	#327	1	1	yes	characteristic
109	C₂×C₄	#327	1	1	yes	characteristic
110	C₂×C₄	#331	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	4	1	no	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	4	1	no	—
117	C₈	—	16	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1270	1	1	yes	characteristic
121	C₂²	#1356	1	2	yes	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	16	1	no	—
128	C₄	—	16	1	no	—
129	C₄	#1335	1	1	yes	characteristic
130	C₄	#1336	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	#7109	1	1	yes	characteristic
133	C₂	#7114	1	1	yes	characteristic
134	C₂	#7117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#9859

C₂

yes

maximal, characteristic

#7196

C₂

yes

maximal, characteristic

#7137

C₂

yes

maximal, characteristic

#3160

C₄

yes

—

—

C₄

yes

—

—

C₂²

yes

characteristic

#533

—

C₂²×C₁₆

—

#500

C₈

yes

characteristic

#500

C₈

yes

—

—

—

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#329

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#323

yes

characteristic

106

C₂×C₄

D₄⋊C₈

yes

characteristic

107

C₂×C₄

D₄⋊C₈

yes

characteristic

108

C₂×C₄

#327

yes

characteristic

109

C₂×C₄

#327

yes

characteristic

110

C₂×C₄

#331

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1270

yes

characteristic

121

C₂²

#1356

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#1335

yes

characteristic

130

C₄

#1336

yes

characteristic

131

C₂

—

132

C₂

#7109

yes

characteristic

133

C₂

#7114

yes

characteristic

134

C₂

#7117

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1270
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₂×#322	C₂²
prime radical	C₂³	#329
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1270

commutator

C₄×C₈

C₂×C₈

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#9859	2
#7196	2
#7137	2

subgroup

index

#9859

#7196

#7137

Subgroups — #92840

Conjugacy classes of subgroups

Maximal subgroups