Group #99302 — subgroups

Subgroups — #99302

Order 512.

346 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 174 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	#8806	C₂	1	2	yes	maximal
2	#8805	C₂	1	2	yes	maximal
3	#8784	C₂	1	2	yes	maximal
4	#8131	C₂	1	1	yes	maximal, characteristic
5	#1842	C₂²	1	4	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	—	2	4	no	—
10	C₂×#334	—	2	4	no	—
11	#1277	—	2	4	no	—
12	#1273	C₄	1	4	yes	—
13	C₂×C₄⋊C₈	—	2	4	no	—
14	C₂×C₄⋊C₈	—	2	4	no	—
15	C₂×C₄⋊C₈	—	2	4	no	—
16	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₂×C₄×C₈	D₄	1	2	yes	—
21	C₂×C₄×C₈	D₄	1	2	yes	—
22	C₂×C₄×C₈	Q₈	1	2	yes	—
23	C₂×C₄×C₈	Q₈	1	2	yes	—
24	C₂×C₄×C₈	Q₈	1	2	yes	—
25	C₄³	C₂³	1	1	yes	characteristic
26	#335	—	4	8	no	—
27	#334	—	4	8	no	—
28	C₂²×C₈	—	4	4	no	—
29	C₂²×C₈	C₄⋊C₄	1	4	yes	—
30	C₂²×C₈	C₄⋊C₄	1	4	yes	—
31	C₂²×C₈	C₄⋊C₄	1	4	yes	—
32	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
39	C₄⋊C₈	—	4	8	no	—
40	C₄⋊C₈	—	4	8	no	—
41	C₄⋊C₈	—	4	8	no	—
42	C₄×C₈	—	2	4	no	—
43	C₄×C₈	—	2	4	no	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	—	2	4	no	—
46	C₄×C₈	—	4	2	no	—
47	C₄×C₈	—	4	2	no	—
48	C₄×C₈	—	4	2	no	—
49	C₄×C₈	D₈	1	4	yes	—
50	C₄×C₈	SD₁₆	1	8	yes	—
51	C₄×C₈	Q₁₆	1	4	yes	—
52	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄	#127	1	1	yes	characteristic
56	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	8	4	no	—
67	C₂×C₈	—	8	4	no	—
68	C₂×C₈	—	8	4	no	—
69	C₂×C₈	#106	1	8	yes	—
70	C₂×C₈	#107	1	8	yes	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	#131	1	4	yes	—
75	C₄²	C₂×D₈	1	2	yes	—
76	C₄²	C₂×SD₁₆	1	4	yes	—
77	C₄²	C₂×Q₁₆	1	2	yes	—
78	C₄²	#135	1	4	yes	—
79	C₂³	#384	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	#424	1	2	yes	—
87	C₂×C₄	#424	1	2	yes	—
88	C₂×C₄	#424	1	2	yes	—
89	C₂×C₄	C₂×#106	1	2	yes	—
90	C₂×C₄	C₂×#107	1	2	yes	—
91	C₂×C₄	#427	1	2	yes	—
92	C₂×C₄	#430	1	2	yes	—
93	C₂×C₄	#430	1	2	yes	—
94	C₂×C₄	#430	1	2	yes	—
95	C₂×C₄	#492	1	2	yes	—
96	C₂×C₄	#493	1	1	yes	characteristic
97	C₂×C₄	#494	1	1	yes	characteristic
98	C₂×C₄	#495	1	2	yes	—
99	C₂×C₄	#498	1	2	yes	—
100	C₂×C₄	#499	1	2	yes	—
101	C₂×C₄	#500	1	2	yes	—
102	C₈	—	2	8	no	—
103	C₈	—	2	8	no	—
104	C₈	—	4	4	no	—
105	C₈	—	16	4	no	—
106	C₂²	#1843	1	1	yes	characteristic
107	C₂²	#1844	1	1	yes	characteristic
108	C₂²	#1845	1	1	yes	characteristic
109	C₂²	#1846	1	1	yes	characteristic
110	C₂²	#1849	1	1	yes	characteristic
111	C₂²	#1849	1	1	yes	characteristic
112	C₂²	#1849	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	2	no	—
115	C₄	—	2	2	no	—
116	C₄	#1570	1	4	yes	—
117	C₄	#1571	1	4	yes	—
118	C₄	#1572	1	4	yes	—
119	C₄	#1706	1	2	yes	—
120	C₄	#1707	1	2	yes	—
121	C₂	#8809	1	1	yes	characteristic
122	C₂	#8810	1	1	yes	characteristic
123	C₂	#8811	1	1	yes	characteristic
124	C₂	#8814	1	1	yes	characteristic
125	C₂	#8825	1	1	yes	characteristic
126	C₂	#8826	1	1	yes	characteristic
127	C₂	#8827	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#8806

C₂

yes

maximal

#8805

C₂

yes

maximal

#8784

C₂

yes

maximal

#8131

C₂

yes

maximal, characteristic

#1842

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

characteristic

#335

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#131

yes

—

C₄²

C₂×D₈

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×Q₁₆

yes

—

C₄²

#135

yes

—

C₂³

#384

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#424

yes

—

C₂×C₄

#424

yes

—

C₂×C₄

#424

yes

—

C₂×C₄

C₂×#106

yes

—

C₂×C₄

C₂×#107

yes

—

C₂×C₄

#427

yes

—

C₂×C₄

#430

yes

—

C₂×C₄

#430

yes

—

C₂×C₄

#430

yes

—

C₂×C₄

#492

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#498

yes

—

100

C₂×C₄

#499

yes

—

101

C₂×C₄

#500

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₂²

#1843

yes

characteristic

107

C₂²

#1844

yes

characteristic

108

C₂²

#1845

yes

characteristic

109

C₂²

#1846

yes

characteristic

110

C₂²

#1849

yes

characteristic

111

C₂²

#1849

yes

characteristic

112

C₂²

#1849

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#1570

yes

—

117

C₄

#1571

yes

—

118

C₄

#1572

yes

—

119

C₄

#1706

yes

—

120

C₄

#1707

yes

—

121

C₂

#8809

yes

characteristic

122

C₂

#8810

yes

characteristic

123

C₂

#8811

yes

characteristic

124

C₂

#8814

yes

characteristic

125

C₂

#8825

yes

characteristic

126

C₂

#8826

yes

characteristic

127

C₂

#8827

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

#384

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
#8806	2
#8805	2
#8784	2
#8131	2

subgroup

index

Subgroups — #99302

Conjugacy classes of subgroups

Maximal subgroups