Group #92921 — subgroups

Subgroups — #92921

Order 512.

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

normal subgroups only

This group has 148 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	#12013	C₂	1	1	yes	maximal, characteristic
2	#11996	C₂	1	1	yes	maximal, characteristic
3	#11402	C₂	1	1	yes	maximal, characteristic
4	#2163	—	2	1	no	—
5	#2150	—	2	1	no	—
6	C₄×M₃₂	C₄	1	1	yes	characteristic
7	C₄×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₈²	C₂²	1	1	yes	characteristic
9	#1370	C₄	1	2	yes	—
10	#1367	—	2	1	no	—
11	#1367	—	2	1	no	—
12	#1367	C₄	1	2	yes	—
13	#1312	—	2	1	no	—
14	#1312	—	2	1	no	—
15	C₂×M₃₂	—	2	1	no	—
16	C₂×M₃₂	—	4	1	no	—
17	C₂×M₃₂	C₈	1	2	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
21	C₄⋊C₁₆	—	2	2	no	—
22	C₄⋊C₁₆	—	2	2	no	—
23	C₄⋊C₁₆	—	4	1	no	—
24	C₂²⋊C₁₆	—	4	1	no	—
25	#346	—	2	1	no	—
26	#346	—	4	1	no	—
27	#346	C₈	1	2	yes	—
28	C₄×C₁₆	—	2	1	no	—
29	C₄×C₁₆	—	2	1	no	—
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	C₂×C₄	1	1	yes	characteristic
32	C₈²	D₄	1	1	yes	characteristic
33	C₈²	D₄	1	1	yes	characteristic
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₈	M₁₆	1	1	yes	characteristic
37	C₂²×C₈	D₈	1	1	yes	characteristic
38	C₂²×C₈	Q₁₆	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	M₃₂	—	4	2	no	—
41	M₃₂	—	4	2	no	—
42	M₃₂	—	8	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	4	2	no	—
46	C₂×C₁₆	—	4	2	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₄²	1	1	yes	characteristic
49	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₂×C₈	1	1	yes	characteristic
55	C₄×C₈	M₁₆	1	1	yes	characteristic
56	C₄×C₈	SD₁₆	1	1	yes	characteristic
57	C₄×C₈	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₄	#107	1	1	yes	characteristic
60	C₂²×C₄	#108	1	1	yes	characteristic
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	C₄×C₈	1	1	yes	characteristic
69	C₂×C₈	#97	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₈	#102	1	1	yes	characteristic
73	C₂×C₈	#102	1	1	yes	characteristic
74	C₂×C₈	#103	1	1	yes	characteristic
75	C₂×C₈	#103	1	1	yes	characteristic
76	C₄²	#95	1	1	yes	characteristic
77	C₄²	C₄⋊C₈	1	1	yes	characteristic
78	C₄²	#106	1	1	yes	characteristic
79	C₄²	#108	1	1	yes	characteristic
80	C₁₆	—	4	2	no	—
81	C₁₆	—	4	2	no	—
82	C₁₆	—	8	2	no	—
83	C₁₆	—	8	2	no	—
84	C₂³	#335	1	1	yes	characteristic
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	#334	1	1	yes	characteristic
91	C₂×C₄	#334	1	1	yes	characteristic
92	C₂×C₄	#335	1	1	yes	characteristic
93	C₂×C₄	#336	1	1	yes	characteristic
94	C₂×C₄	#340	1	1	yes	characteristic
95	C₂×C₄	#341	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	4	1	no	—
103	C₈	#347	1	2	yes	—
104	C₈	#349	1	2	yes	—
105	C₈	#360	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#1277	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#1375	1	1	yes	characteristic
115	C₄	#1377	1	1	yes	characteristic
116	C₄	#1386	1	1	yes	characteristic
117	C₄	#1388	1	1	yes	characteristic
118	C₂	—	4	1	no	—
119	C₂	#7152	1	1	yes	characteristic
120	C₂	#7158	1	1	yes	characteristic
121	C₂	#7162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal, characteristic

#11996

C₂

yes

maximal, characteristic

#11402

C₂

yes

maximal, characteristic

#2163

—

#2150

—

C₄×M₃₂

C₄

yes

characteristic

C₄×M₃₂

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

—

#1367

—

#1367

—

#1367

C₄

yes

—

#1312

—

#1312

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

—

#346

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₂²×C₄

C₄⋊C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

C₄⋊C₈

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#335

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

#347

yes

—

104

C₈

#349

yes

—

105

C₈

#360

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#1277

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#1375

yes

characteristic

115

C₄

#1377

yes

characteristic

116

C₄

#1386

yes

characteristic

117

C₄

#1388

yes

characteristic

118

C₂

—

119

C₂

#7152

yes

characteristic

120

C₂

#7158

yes

characteristic

121

C₂

#7162

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1277
prime radical	C₂³	#335
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#340

commutator

C₂×C₈

C₄×C₈

Frattini, cosocle

C₂×C₈²

C₂²

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#12013	2
#11996	2
#11402	2

subgroup

index

#12013

#11996

#11402

Subgroups — #92921

Conjugacy classes of subgroups

Maximal subgroups