Group #128230 — subgroups

Subgroups — #128230

Order 512.

234 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 126 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	#11411	C₂	1	1	yes	maximal, characteristic
2	#11410	C₂	1	1	yes	maximal, characteristic
3	#7845	C₂	1	1	yes	maximal, characteristic
4	#7145	C₂	1	2	yes	maximal
5	#7142	C₂	1	2	yes	maximal
6	#2162	C₂²	1	2	yes	—
7	#2162	C₂²	1	2	yes	—
8	#1744	C₄	1	1	yes	characteristic
9	#1744	C₄	1	1	yes	characteristic
10	#1744	C₂²	1	1	yes	characteristic
11	C₂×#322	C₄	1	1	yes	characteristic
12	C₂×#322	C₄	1	1	yes	characteristic
13	C₂×#322	C₂²	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1312	C₄	1	2	yes	complemented
16	#1312	C₄	1	2	yes	complemented
17	#1311	C₄	1	2	yes	complemented
18	C₈×C₁₆	C₄	1	2	yes	complemented
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄³	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	2	2	no	—
27	C₄⋊C₁₆	—	2	2	no	—
28	C₂²⋊C₁₆	—	2	2	no	—
29	C₂²⋊C₁₆	—	2	2	no	—
30	C₄×C₁₆	C₂×C₄	1	4	yes	—
31	C₄×C₁₆	C₂×C₄	1	4	yes	—
32	#336	—	2	2	no	—
33	#336	—	2	2	no	—
34	#336	C₈	1	4	yes	—
35	#322	—	2	2	no	—
36	#322	—	2	2	no	—
37	#322	C₈	1	4	yes	—
38	C₈²	C₂×C₄	1	1	yes	characteristic
39	C₈²	C₂×C₄	1	1	yes	characteristic
40	C₈²	C₂×C₄	1	1	yes	characteristic
41	C₈²	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	C₂×C₈	1	2	yes	—
45	C₂²×C₈	C₂×C₈	1	2	yes	—
46	C₂²×C₈	M₁₆	1	2	yes	—
47	C₂²×C₈	#41	1	2	yes	—
48	C₂×C₄²	—	2	1	no	—
49	C₂×C₄²	—	2	1	no	—
50	C₂×C₄²	C₂×C₈	1	2	yes	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	2	4	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₄²	1	1	yes	characteristic
59	C₄×C₈	C₄²	1	1	yes	characteristic
60	C₄×C₈	C₄²	1	1	yes	characteristic
61	C₄×C₈	C₄²	1	1	yes	characteristic
62	C₄×C₈	C₂×C₈	1	2	yes	—
63	C₄×C₈	C₂×C₈	1	2	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	M₁₆	1	2	yes	—
66	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₄×C₈	#41	1	2	yes	—
69	C₂²×C₄	—	2	2	no	—
70	C₂²×C₄	#117	1	1	yes	characteristic
71	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	C₄×C₈	1	2	yes	—
83	C₂×C₈	C₄×C₈	1	2	yes	—
84	C₂×C₈	#97	1	2	yes	—
85	C₂×C₈	#97	1	2	yes	—
86	C₂×C₈	#117	1	2	yes	—
87	C₂×C₈	#117	1	2	yes	—
88	C₄²	—	2	4	no	—
89	C₄²	—	4	2	no	—
90	C₄²	—	4	2	no	—
91	C₄²	C₂×C₄²	1	1	yes	characteristic
92	C₄²	#117	1	1	yes	characteristic
93	C₄²	C₂²×C₈	1	1	yes	characteristic
94	C₄²	C₂×M₁₆	1	1	yes	characteristic
95	C₁₆	—	4	4	no	—
96	C₁₆	—	4	4	no	—
97	C₂³	#431	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#376	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
105	C₂×C₄	C₂×#97	1	1	yes	characteristic
106	C₂×C₄	#431	1	1	yes	characteristic
107	C₂×C₄	#431	1	1	yes	characteristic
108	C₂×C₄	#431	1	1	yes	characteristic
109	C₂×C₄	#504	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	4	2	no	—
117	C₈	—	4	2	no	—
118	C₈	—	4	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	#1744	1	1	yes	characteristic
122	C₂²	#2164	1	2	yes	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	8	2	no	—
126	C₄	#1747	1	2	yes	—
127	C₄	#2108	1	2	yes	—
128	C₄	#2110	1	2	yes	—
129	C₄	#2164	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	#8138	1	1	yes	characteristic
132	C₂	#12592	1	1	yes	characteristic
133	C₂	#12593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11411

C₂

yes

maximal, characteristic

#11410

C₂

yes

maximal, characteristic

#7845

C₂

yes

maximal, characteristic

#7145

C₂

yes

maximal

#7142

C₂

yes

maximal

#2162

C₂²

yes

—

#2162

C₂²

yes

—

#1744

C₄

yes

characteristic

#1744

C₄

yes

characteristic

#1744

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1312

C₄

yes

complemented

#1312

C₄

yes

complemented

#1311

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#336

—

#336

—

#336

C₈

yes

—

#322

—

#322

—

#322

C₈

yes

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#117

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₄²

yes

characteristic

C₄²

#117

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

#431

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#376

yes

characteristic

104

C₂×C₄

C₂×C₄×C₈

yes

characteristic

105

C₂×C₄

C₂×#97

yes

characteristic

106

C₂×C₄

#431

yes

characteristic

107

C₂×C₄

#431

yes

characteristic

108

C₂×C₄

#431

yes

characteristic

109

C₂×C₄

#504

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

#1744

yes

characteristic

122

C₂²

#2164

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#1747

yes

—

127

C₄

#2108

yes

—

128

C₄

#2110

yes

—

129

C₄

#2164

yes

—

130

C₂

—

131

C₂

#8138

yes

characteristic

132

C₂

#12592

yes

characteristic

133

C₂

#12593

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1744
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄²

C₂×C₄²

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₄² ◁ G

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. #1312	C₄	semidirect
16. #1312	C₄	semidirect
17. #1311	C₄	semidirect
18. C₈×C₁₆	C₄	semidirect

H ≅ G/N

type

15. #1312

C₄

semidirect

16. #1312

C₄

semidirect

17. #1311

C₄

semidirect

18. C₈×C₁₆

C₄

semidirect

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

subgroup	index
#11411	2
#11410	2
#7845	2
#7145	2
#7142	2

subgroup

index

Subgroups — #128230

Conjugacy classes of subgroups

Complements

Maximal subgroups