Group #129147 — subgroups

Subgroups — #129147

Order 512.

290 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₄×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	2	yes	maximal
5	#7171	C₂	1	2	yes	maximal
6	C₂×#366	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₄×#107	C₂²	1	2	yes	—
10	C₈×C₄⋊C₄	—	2	2	no	—
11	C₂×#335	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1380	C₄	1	2	yes	—
14	#1372	C₄	1	4	yes	complemented
15	#1311	C₄	1	4	yes	complemented
16	#1279	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×#107	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₈	—	2	2	no	—
22	C₂×C₄×C₈	—	4	2	no	—
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	C₄×C₄⋊C₄	—	2	2	no	—
27	#366	C₈	1	8	yes	complemented
28	C₄×C₁₆	—	2	1	no	—
29	C₄×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₁₆	C₂×C₄	1	2	yes	—
31	C₄×C₁₆	C₂×C₄	1	2	yes	—
32	#340	—	2	2	no	—
33	#336	—	4	2	no	—
34	#335	—	2	2	no	—
35	#335	C₂×C₄	1	4	yes	—
36	C₈²	C₂×C₄	1	2	yes	—
37	C₈²	Q₈	1	2	yes	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	D₈	1	2	yes	—
41	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₈	#41	1	1	yes	characteristic
43	C₂×C₄⋊C₄	—	2	4	no	—
44	C₂×C₄²	—	4	2	no	—
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	C₄²	1	2	yes	—
50	C₂×C₁₆	C₄²	1	2	yes	—
51	C₂×C₁₆	C₂×C₈	1	4	yes	—
52	#107	—	2	4	no	—
53	#107	C₂×C₈	1	8	yes	—
54	C₄⋊C₈	—	2	4	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	8	4	no	—
58	C₄×C₈	C₄⋊C₄	1	2	yes	—
59	C₄×C₈	C₄⋊C₄	1	2	yes	—
60	C₄×C₈	Q₁₆	1	2	yes	—
61	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₄×C₈	#41	1	1	yes	characteristic
65	#95	—	4	2	no	—
66	C₂²×C₄	—	4	4	no	—
67	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂²×C₄	#135	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	—	8	2	no	—
78	C₂×C₈	#107	1	4	yes	—
79	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₈	#117	1	1	yes	characteristic
81	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
82	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₈	C₂²×C₈	1	2	yes	—
84	C₂×C₈	#131	1	2	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₄²	—	8	2	no	—
88	C₄²	—	8	2	no	—
89	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄²	C₄×D₄	1	1	yes	characteristic
91	C₄²	C₂×Q₁₆	1	1	yes	characteristic
92	C₄²	#135	1	1	yes	characteristic
93	C₁₆	—	2	4	no	—
94	C₁₆	—	2	2	no	—
95	C₁₆	C₄×C₈	1	4	yes	complemented
96	C₂³	C₄×D₈	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	8	no	—
99	C₂×C₄	—	8	4	no	—
100	C₂×C₄	—	8	4	no	—
101	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₂×#107	1	1	yes	characteristic
103	C₂×C₄	#427	1	1	yes	characteristic
104	C₂×C₄	C₈×D₄	1	2	yes	—
105	C₂×C₄	C₄×D₈	1	1	yes	characteristic
106	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
107	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
108	C₂×C₄	#443	1	2	yes	—
109	C₂×C₄	#509	1	2	yes	—
110	C₂×C₄	#510	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	2	1	no	—
113	C₈	—	2	1	no	—
114	C₈	—	4	2	no	—
115	C₈	—	8	4	no	—
116	C₈	C₂×C₄×C₈	1	1	yes	characteristic
117	C₈	#405	1	1	yes	characteristic
118	C₈	C₈×Q₈	1	2	yes	—
119	C₂²	C₈×D₈	1	2	yes	—
120	C₂²	C₄×#107	1	1	yes	characteristic
121	C₂²	#2176	1	1	yes	characteristic
122	C₂²	#2177	1	1	yes	characteristic
123	C₂²	#2179	1	2	yes	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	8	8	no	—
127	C₄	C₈×Q₁₆	1	2	yes	—
128	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
129	C₄	#1772	1	1	yes	characteristic
130	C₄	#2159	1	2	yes	—
131	C₄	#2175	1	2	yes	—
132	C₂	C₈×#107	1	1	yes	characteristic
133	C₂	#12044	1	2	yes	—
134	C₂	#12045	1	2	yes	—
135	C₂	#12671	1	1	yes	characteristic
136	C₂	#12674	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#366

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

C₂×#1311

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal

#7171

C₂

yes

maximal

C₂×#366

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#107

C₂²

yes

—

C₈×C₄⋊C₄

—

C₂×#335

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1380

C₄

yes

—

#1372

C₄

yes

complemented

#1311

C₄

yes

complemented

#1279

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

—

#366

C₈

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₂×C₈

yes

—

#107

—

#107

C₂×C₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

#131

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₄²

#135

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

complemented

C₂³

C₄×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

102

C₂×C₄

C₂×#107

yes

characteristic

103

C₂×C₄

#427

yes

characteristic

104

C₂×C₄

C₈×D₄

yes

—

105

C₂×C₄

C₄×D₈

yes

characteristic

106

C₂×C₄

C₄×Q₁₆

yes

characteristic

107

C₂×C₄

C₄×Q₁₆

yes

characteristic

108

C₂×C₄

#443

yes

—

109

C₂×C₄

#509

yes

—

110

C₂×C₄

#510

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

C₂×C₄×C₈

yes

characteristic

117

C₈

#405

yes

characteristic

118

C₈

C₈×Q₈

yes

—

119

C₂²

C₈×D₈

yes

—

120

C₂²

C₄×#107

yes

characteristic

121

C₂²

#2176

yes

characteristic

122

C₂²

#2177

yes

characteristic

123

C₂²

#2179

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₈×Q₁₆

yes

—

128

C₄

C₈×C₄⋊C₄

yes

characteristic

129

C₄

#1772

yes

characteristic

130

C₄

#2159

yes

—

131

C₄

#2175

yes

—

132

C₂

C₈×#107

yes

characteristic

133

C₂

#12044

yes

—

134

C₂

#12045

yes

—

135

C₂

#12671

yes

characteristic

136

C₂

#12674

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₄×D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1372	C₄	semidirect
15. #1311	C₄	semidirect
27. #366	C₈	semidirect
95. C₁₆	C₄×C₈	semidirect

H ≅ G/N

type

14. #1372

C₄

semidirect

15. #1311

C₄

semidirect

27. #366

C₈

semidirect

95. C₁₆

C₄×C₈

semidirect

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

subgroup	index
C₄×#366	2
C₂×#1372	2
C₂×#1311	2
C₈×#107	2
#7171	2

subgroup

index

Subgroups — #129147

Conjugacy classes of subgroups

Complements

Maximal subgroups