Group #353041 — subgroups

Subgroups — #353041

Order 512.

834 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 378 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	#33312	C₂	1	2	yes	maximal
2	#33310	C₂	1	2	yes	maximal
3	#33289	C₂	1	2	yes	maximal
4	#8814	C₂	1	8	yes	maximal
5	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	4	no	—
7	C₂×#498	—	2	4	no	—
8	#2868	C₂²	1	4	yes	—
9	#1990	—	2	8	no	—
10	#1845	—	2	4	no	—
11	#1845	C₂²	1	8	yes	—
12	#1844	—	2	4	no	—
13	#1844	C₂²	1	8	yes	—
14	#1834	C₂²	1	8	yes	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈	—	2	4	no	—
20	C₂×C₄⋊Q₈	—	2	4	no	—
21	C₂×C₄⋊Q₈	—	2	4	no	—
22	#500	—	4	16	no	—
23	#498	—	4	16	no	—
24	C₂×#107	—	2	16	no	—
25	C₂×#106	—	2	16	no	—
26	C₂×C₄×C₈	D₄	1	4	yes	—
27	C₂×C₄×C₈	D₄	1	2	yes	—
28	C₂×C₄×C₈	D₄	1	4	yes	—
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	2	yes	—
31	C₂×C₄×C₈	C₂³	1	2	yes	—
32	C₂×C₄×C₈	C₂³	1	2	yes	—
33	#384	—	2	4	no	—
34	#384	—	2	4	no	—
35	#384	—	2	4	no	—
36	#384	C₂³	1	8	yes	—
37	C₄³	C₂³	1	1	yes	characteristic
38	#340	—	2	16	no	—
39	C₈²	Q₈	1	16	yes	complemented
40	C₂²×Q₈	—	4	4	no	—
41	C₂²×C₈	C₂×D₄	1	4	yes	—
42	C₂²×C₈	C₂×D₄	1	4	yes	—
43	C₂²×C₈	C₂×D₄	1	4	yes	—
44	C₄⋊Q₈	—	4	16	no	—
45	C₄⋊Q₈	—	4	16	no	—
46	C₄⋊Q₈	—	4	16	no	—
47	C₂×C₄⋊C₄	—	2	8	no	—
48	C₂×C₄⋊C₄	—	2	8	no	—
49	C₂×C₄⋊C₄	—	2	8	no	—
50	C₂×C₄⋊C₄	—	4	4	no	—
51	C₂×C₄⋊C₄	—	4	4	no	—
52	C₂×C₄⋊C₄	—	4	4	no	—
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₄	1	2	yes	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₄	1	2	yes	—
57	C₂×C₄²	C₂⁴	1	1	yes	characteristic
58	#107	—	4	32	no	—
59	#106	—	4	32	no	—
60	C₄×C₈	—	2	8	no	—
61	C₄×C₈	D₈	1	8	yes	—
62	C₄×C₈	SD₁₆	1	16	yes	—
63	C₄×C₈	Q₁₆	1	8	yes	—
64	C₄×C₈	C₂×Q₈	1	8	yes	—
65	C₄×C₈	C₂×Q₈	1	8	yes	—
66	C₄×C₈	C₂×Q₈	1	8	yes	—
67	C₂×Q₈	—	8	8	no	—
68	C₂×Q₈	—	8	8	no	—
69	C₂×Q₈	—	8	8	no	—
70	C₂²×C₄	—	4	8	no	—
71	C₂²×C₄	#127	1	2	yes	—
72	C₂²×C₄	#127	1	2	yes	—
73	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	C₄⋊Q₈	1	8	yes	—
78	C₂×C₈	C₄⋊Q₈	1	8	yes	—
79	C₂×C₈	C₄⋊Q₈	1	8	yes	—
80	C₂×C₈	C₂×D₈	1	8	yes	—
81	C₂×C₈	C₂×SD₁₆	1	16	yes	—
82	C₂×C₈	C₂×Q₁₆	1	8	yes	—
83	C₄⋊C₄	—	4	16	no	—
84	C₄⋊C₄	—	4	16	no	—
85	C₄⋊C₄	—	4	16	no	—
86	C₄⋊C₄	—	8	8	no	—
87	C₄⋊C₄	—	8	8	no	—
88	C₄⋊C₄	—	8	8	no	—
89	C₄²	C₂×D₈	1	4	yes	—
90	C₄²	C₂×SD₁₆	1	8	yes	—
91	C₄²	C₂×Q₁₆	1	4	yes	—
92	C₄²	#135	1	8	yes	—
93	C₄²	C₂²×Q₈	1	4	yes	—
94	C₂³	C₂×#127	1	1	yes	characteristic
95	Q₈	—	16	16	no	—
96	C₂×C₄	—	8	8	no	—
97	C₂×C₄	—	8	8	no	—
98	C₂×C₄	—	8	8	no	—
99	C₂×C₄	#492	1	4	yes	—
100	C₂×C₄	#492	1	4	yes	—
101	C₂×C₄	#493	1	4	yes	—
102	C₂×C₄	#494	1	4	yes	—
103	C₂×C₄	#495	1	4	yes	—
104	C₂×C₄	#495	1	4	yes	—
105	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
106	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
108	C₂×C₄	C₂²×D₈	1	2	yes	—
109	C₂×C₄	C₂²×SD₁₆	1	4	yes	—
110	C₂×C₄	C₂²×Q₁₆	1	2	yes	—
111	C₂×C₄	C₂×#135	1	4	yes	—
112	C₈	—	2	8	no	—
113	C₈	#498	1	16	yes	complemented
114	C₈	#500	1	16	yes	complemented
115	C₂²	#2868	1	1	yes	characteristic
116	C₂²	C₂×#492	1	2	yes	—
117	C₂²	C₂×#493	1	1	yes	characteristic
118	C₂²	C₂×#494	1	1	yes	characteristic
119	C₂²	C₂×#495	1	2	yes	—
120	C₄	—	16	8	no	—
121	C₄	#1706	1	8	yes	—
122	C₄	#1707	1	8	yes	—
123	C₄	C₂×#498	1	4	yes	—
124	C₄	C₂×#500	1	4	yes	—
125	C₄	#3160	1	4	yes	—
126	C₂	C₂×#1706	1	2	yes	—
127	C₂	C₂×#1707	1	2	yes	—
128	C₂	#33310	1	1	yes	characteristic
129	C₂	#33312	1	1	yes	characteristic
130	C₂	#33313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal

#33310

C₂

yes

maximal

#33289

C₂

yes

maximal

#8814

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#498

—

#2868

C₂²

yes

—

#1990

—

#1845

—

#1845

C₂²

yes

—

#1844

—

#1844

C₂²

yes

—

#1834

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₂³

yes

—

C₄³

C₂³

yes

characteristic

#340

—

C₈²

Q₈

yes

complemented

C₂²×Q₈

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂⁴

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

—

C₂²×C₄

#127

yes

—

C₂²×C₄

C₂²×D₄

yes

characteristic

C₂²×C₄

C₂²×D₄

yes

characteristic

C₂²×C₄

C₂²×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₈

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×Q₁₆

yes

—

C₄²

#135

yes

—

C₄²

C₂²×Q₈

yes

—

C₂³

C₂×#127

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#492

yes

—

100

C₂×C₄

#492

yes

—

101

C₂×C₄

#493

yes

—

102

C₂×C₄

#494

yes

—

103

C₂×C₄

#495

yes

—

104

C₂×C₄

#495

yes

—

105

C₂×C₄

C₂×C₄⋊Q₈

yes

—

106

C₂×C₄

C₂×C₄⋊Q₈

yes

—

107

C₂×C₄

C₂×C₄⋊Q₈

yes

—

108

C₂×C₄

C₂²×D₈

yes

—

109

C₂×C₄

C₂²×SD₁₆

yes

—

110

C₂×C₄

C₂²×Q₁₆

yes

—

111

C₂×C₄

C₂×#135

yes

—

112

C₈

—

113

C₈

#498

yes

complemented

114

C₈

#500

yes

complemented

115

C₂²

#2868

yes

characteristic

116

C₂²

C₂×#492

yes

—

117

C₂²

C₂×#493

yes

characteristic

118

C₂²

C₂×#494

yes

characteristic

119

C₂²

C₂×#495

yes

—

120

C₄

—

121

C₄

#1706

yes

—

122

C₄

#1707

yes

—

123

C₄

C₂×#498

yes

—

124

C₄

C₂×#500

yes

—

125

C₄

#3160

yes

—

126

C₂

C₂×#1706

yes

—

127

C₂

C₂×#1707

yes

—

128

C₂

#33310

yes

characteristic

129

C₂

#33312

yes

characteristic

130

C₂

#33313

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#127

commutator, Frattini, cosocle

C₂×C₄²

C₂⁴

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
39. C₈²	Q₈	semidirect
113. C₈	#498	semidirect
114. C₈	#500	semidirect

H ≅ G/N

type

39. C₈²

Q₈

semidirect

113. C₈

#498

semidirect

114. C₈

#500

semidirect

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

subgroup	index
#33312	2
#33310	2
#33289	2
#8814	2
C₄×C₈²	2

subgroup

index

C₄×C₈²

Subgroups — #353041

Conjugacy classes of subgroups

Complements

Maximal subgroups