Group #12600 — subgroups

Subgroups — #12600

Order 256.

257 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#1374	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#117	C₄	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	2	yes	—
11	#405	—	2	2	no	—
12	#405	—	2	2	no	—
13	C₂×#97	C₄	1	1	yes	characteristic
14	C₂×#97	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆	C₄	1	8	yes	complemented
18	#346	—	2	2	no	—
19	C₄×C₁₆	—	2	2	no	—
20	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆	—	2	2	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	C₈	1	4	yes	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	#117	—	2	2	no	—
33	#117	—	2	2	no	—
34	C₂×C₄⋊C₄	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄	C₈	1	2	yes	—
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₁₆	C₂×C₄	1	4	yes	—
44	#97	—	2	2	no	—
45	#97	—	2	2	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	D₄	1	2	yes	—
49	C₄×C₈	D₄	1	2	yes	—
50	C₂⁴	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₄²	1	2	yes	—
56	C₂²×C₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₄	C₂×C₈	1	2	yes	—
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	M₁₆	—	2	8	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	C₄²	1	2	yes	—
69	C₂×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈	C₂×C₈	1	2	yes	—
71	C₂×C₈	C₂×C₈	1	2	yes	—
72	C₂×C₈	M₁₆	1	2	yes	—
73	C₂×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	#41	1	2	yes	—
77	C₄⋊C₄	—	2	4	no	—
78	C₂²⋊C₄	—	2	4	no	—
79	C₄²	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₄²	C₂²⋊C₄	1	2	yes	—
82	C₄²	M₁₆	1	2	yes	—
83	C₁₆	—	2	4	no	—
84	C₁₆	—	2	4	no	—
85	C₂³	—	2	1	no	—
86	C₂³	—	2	1	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₄×C₈	1	2	yes	—
89	C₂³	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₄×C₈	1	2	yes	—
100	C₂×C₄	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄	#117	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	2	no	—
109	C₈	C₂²⋊C₈	1	4	yes	—
110	C₈	C₄×D₄	1	2	yes	—
111	C₈	C₄×D₄	1	2	yes	—
112	C₂²	—	2	4	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	4	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²	#431	1	1	yes	characteristic
119	C₂²	#504	1	4	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	2	2	no	—
122	C₄	—	4	2	no	—
123	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄	C₄×M₁₆	1	1	yes	characteristic
125	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄	#431	1	1	yes	characteristic
127	C₂	—	4	2	no	—
128	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
129	C₂	#2108	1	2	yes	—
130	C₂	#2113	1	2	yes	—
131	C₂	#2164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#405

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₁₆

C₂

yes

maximal

C₂×#346

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

#1374

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₄

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#405

—

#405

—

C₂×#97

C₄

yes

characteristic

C₂×#97

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂²⋊C₁₆

C₄

yes

complemented

#346

—

C₄×C₁₆

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₂²⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂×C₄

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

100

C₂×C₄

C₂²⋊C₈

yes

—

101

C₂×C₄

C₂×C₄²

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₄

#117

yes

characteristic

104

C₂×C₄

C₂²×C₈

yes

characteristic

105

C₂×C₄

C₂×M₁₆

yes

characteristic

106

C₂×C₄

C₂×M₁₆

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

C₂²⋊C₈

yes

—

110

C₈

C₄×D₄

yes

—

111

C₈

C₄×D₄

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂×C₄×C₈

yes

characteristic

117

C₂²

C₂×C₂²⋊C₈

yes

characteristic

118

C₂²

#431

yes

characteristic

119

C₂²

#504

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₄×C₂²⋊C₄

yes

characteristic

124

C₄

C₄×M₁₆

yes

characteristic

125

C₄

C₂×C₂²⋊C₈

yes

characteristic

126

C₄

#431

yes

characteristic

127

C₂

—

128

C₂

C₄×C₂²⋊C₈

yes

characteristic

129

C₂

#2108

yes

—

130

C₂

#2113

yes

—

131

C₂

#2164

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator	C₂²	C₂×C₄×C₈
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator

C₂²

C₂×C₄×C₈

socle

C₂³

C₂²×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. C₂×#346	C₂	semidirect
4. C₂×C₄×C₁₆	C₂	semidirect
5. #1374	C₂	semidirect
17. C₂²⋊C₁₆	C₄	semidirect

H ≅ G/N

type

3. C₂×#346

C₂

semidirect

4. C₂×C₄×C₁₆

C₂

semidirect

5. #1374

C₂

semidirect

17. C₂²⋊C₁₆

C₄

semidirect

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

subgroup	index
C₂×#405	2
C₂×C₂²⋊C₁₆	2
C₂×#346	2
C₂×C₄×C₁₆	2
#1374	2

subgroup

index

C₂×#405

C₂×C₂²⋊C₁₆

C₂×#346

C₂×C₄×C₁₆

#1374

Subgroups — #12600

Conjugacy classes of subgroups

Complements

Maximal subgroups