Group #21705 — subgroups

Subgroups — #21705

Order 256.

523 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 191 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	#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#443	C₂	1	4	yes	maximal, complemented
3	#2931	C₂	1	2	yes	maximal, complemented
4	#1974	C₂	1	4	yes	maximal, complemented
5	#1962	C₂	1	2	yes	maximal, complemented
6	#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	4	yes	—
9	C₂×#131	C₂²	1	4	yes	—
10	#532	C₂²	1	2	yes	—
11	#495	C₄	1	4	yes	complemented
12	#494	C₄	1	1	yes	characteristic, complemented
13	#493	C₄	1	1	yes	characteristic, complemented
14	#492	C₄	1	2	yes	complemented
15	#443	—	2	16	no	—
16	#436	—	2	4	no	—
17	C₈×D₄	—	2	4	no	—
18	C₂×#108	C₂²	1	4	yes	—
19	#423	C₂²	1	2	yes	—
20	C₂×#104	C₂²	1	8	yes	—
21	#408	C₂²	1	4	yes	—
22	C₂×C₄×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄³	C₂²	1	1	yes	characteristic
27	C₂×#41	C₂³	1	4	yes	—
28	#135	—	2	8	no	—
29	#135	—	2	8	no	—
30	C₂×Q₁₆	C₂×C₄	1	4	yes	—
31	C₂×SD₁₆	C₂×C₄	1	8	yes	—
32	C₂×D₈	C₂×C₄	1	4	yes	—
33	#131	—	2	16	no	—
34	C₂×M₁₆	—	2	4	no	—
35	C₂×M₁₆	C₂³	1	4	yes	—
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	2	yes	—
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	#127	C₂×C₄	1	2	yes	—
42	#124	C₂×C₄	1	4	yes	—
43	#121	—	2	4	no	—
44	C₄×D₄	—	2	4	no	—
45	C₂×C₄²	—	2	2	no	—
46	C₂×C₄²	C₂³	1	2	yes	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#108	—	2	8	no	—
49	C₄⋊C₈	—	2	4	no	—
50	#104	—	2	16	no	—
51	C₂²⋊C₈	—	2	8	no	—
52	C₄×C₈	—	2	4	no	—
53	C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈	C₂×C₄	1	2	yes	—
56	C₄×C₈	D₄	1	2	yes	—
57	C₄×C₈	D₄	1	2	yes	—
58	C₄×C₈	D₄	1	8	yes	complemented
59	#41	—	2	8	no	—
60	#41	—	2	8	no	—
61	C₂×Q₈	C₂²×C₄	1	4	yes	—
62	C₂×D₄	—	2	4	no	—
63	C₂×D₄	—	2	4	no	—
64	C₂×D₄	C₂²×C₄	1	4	yes	—
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	—	2	4	no	—
67	C₂²×C₄	#41	1	2	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	Q₁₆	—	2	8	no	—
70	SD₁₆	—	2	16	no	—
71	D₈	—	2	8	no	—
72	M₁₆	—	2	8	no	—
73	M₁₆	—	4	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	4	8	no	—
77	C₂×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₈	C₂×D₄	1	4	yes	—
81	C₂×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈	C₂×D₄	1	4	yes	—
83	C₂×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈	#41	1	4	yes	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	2	8	no	—
87	C₄²	—	2	8	no	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₄²	C₂×D₄	1	4	yes	—
92	C₄²	C₂×D₄	1	2	yes	—
93	C₂³	—	2	4	no	—
94	C₂³	C₂×#41	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	D₄	—	2	8	no	—
97	D₄	—	4	8	no	—
98	D₄	—	4	8	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₄×D₄	1	2	yes	—
106	C₂×C₄	C₄×D₄	1	2	yes	—
107	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₄	1	2	yes	—
109	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄	C₂×#41	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	C₄×D₄	1	4	yes	—
114	C₈	C₄×D₄	1	4	yes	—
115	C₈	#127	1	2	yes	—
116	C₈	#127	1	2	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	4	no	—
119	C₂²	—	4	8	no	—
120	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²	#532	1	2	yes	—
122	C₄	—	2	8	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	#443	1	4	yes	complemented
126	C₄	C₂×C₄×D₄	1	2	yes	—
127	C₄	C₂×#127	1	1	yes	characteristic
128	C₄	#532	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	8	4	no	—
131	C₂	C₄×#127	1	1	yes	characteristic
132	C₂	C₂×#443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3147

C₂

yes

maximal, characteristic

C₂×#443

C₂

yes

maximal, complemented

#2931

C₂

yes

maximal, complemented

#1974

C₂

yes

maximal, complemented

#1962

C₂

yes

maximal, complemented

#1848

C₂

yes

maximal, characteristic, complemented

C₄²×C₈

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₂²

yes

—

C₂×#131

C₂²

yes

—

#532

C₂²

yes

—

#495

C₄

yes

complemented

#494

C₄

yes

characteristic, complemented

#493

C₄

yes

characteristic, complemented

#492

C₄

yes

complemented

#443

—

#436

—

C₈×D₄

—

C₂×#108

C₂²

yes

—

#423

C₂²

yes

—

C₂×#104

C₂²

yes

—

#408

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

C₂×#41

C₂³

yes

—

#135

—

#135

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×SD₁₆

C₂×C₄

yes

—

C₂×D₈

C₂×C₄

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

C₂×C₄

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

complemented

#41

—

#41

—

C₂×Q₈

C₂²×C₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

—

C₂³

—

C₂³

C₂×#41

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×D₄

yes

—

106

C₂×C₄

C₄×D₄

yes

—

107

C₂×C₄

C₂³×C₄

yes

characteristic

108

C₂×C₄

C₂²×D₄

yes

—

109

C₂×C₄

C₂²×D₄

yes

characteristic

110

C₂×C₄

C₂×#41

yes

—

111

C₈

—

112

C₈

—

113

C₈

C₄×D₄

yes

—

114

C₈

C₄×D₄

yes

—

115

C₈

#127

yes

—

116

C₈

#127

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄×D₄

yes

characteristic

121

C₂²

#532

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#443

yes

complemented

126

C₄

C₂×C₄×D₄

yes

—

127

C₄

C₂×#127

yes

characteristic

128

C₄

#532

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

C₄×#127

yes

characteristic

132

C₂

C₂×#443

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#443	C₂	semidirect
3. #2931	C₂	semidirect
4. #1974	C₂	semidirect
5. #1962	C₂	semidirect
6. #1848	C₂	semidirect
7. C₄²×C₈	C₂	semidirect
11. #495	C₄	semidirect
12. #494	C₄	semidirect
13. #493	C₄	semidirect
14. #492	C₄	semidirect
58. C₄×C₈	D₄	semidirect
125. C₄	#443	semidirect

H ≅ G/N

type

2. C₂×#443

C₂

semidirect

3. #2931

C₂

semidirect

4. #1974

C₂

semidirect

5. #1962

C₂

semidirect

6. #1848

C₂

semidirect

7. C₄²×C₈

C₂

semidirect

11. #495

C₄

semidirect

12. #494

C₄

semidirect

13. #493

C₄

semidirect

14. #492

C₄

semidirect

58. C₄×C₈

D₄

semidirect

125. C₄

#443

semidirect

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

subgroup	index
#3147	2
C₂×#443	2
#2931	2
#1974	2
#1962	2
#1848	2
C₄²×C₈	2

subgroup

index

C₄²×C₈

Subgroups — #21705

Conjugacy classes of subgroups

Complements

Maximal subgroups