Group #21736 — subgroups

Subgroups — #21736

Order 256.

335 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 187 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	#3162	C₂	1	1	yes	maximal, characteristic
2	#2961	C₂	1	2	yes	maximal
3	#1994	C₂	1	2	yes	maximal
4	#1946	C₂	1	2	yes	maximal
5	#1945	C₂	1	2	yes	maximal
6	#1849	C₂	1	1	yes	maximal, characteristic
7	#1773	C₂	1	2	yes	maximal
8	#1772	C₂	1	2	yes	maximal
9	C₄×#97	C₂	1	1	yes	maximal, characteristic
10	#533	C₂²	1	2	yes	—
11	#501	C₄	1	8	yes	complemented
12	#446	—	2	8	no	—
13	C₂×#108	C₂²	1	2	yes	—
14	C₂×#108	C₂²	1	2	yes	—
15	#427	C₂²	1	2	yes	—
16	#427	C₂²	1	2	yes	—
17	#424	C₂²	1	4	yes	—
18	#423	C₂²	1	2	yes	—
19	#405	C₂²	1	4	yes	—
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄³	C₂²	1	1	yes	characteristic
27	#339	C₂²	1	4	yes	—
28	#339	C₂²	1	4	yes	—
29	C₂×M₁₆	C₂³	1	4	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	#125	C₂×C₄	1	4	yes	—
36	C₂²⋊Q₈	—	2	4	no	—
37	C₄×Q₈	—	2	4	no	—
38	#117	C₂³	1	4	yes	—
39	C₂×C₄²	—	2	2	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#108	—	2	4	no	—
44	#108	—	2	4	no	—
45	#107	C₂×C₄	1	4	yes	—
46	#107	C₂×C₄	1	4	yes	—
47	#106	C₂×C₄	1	4	yes	—
48	#106	C₂×C₄	1	4	yes	—
49	C₄⋊C₈	—	2	8	no	—
50	C₄⋊C₈	—	2	4	no	—
51	#97	—	2	4	no	—
52	#97	—	2	2	no	—
53	#97	C₂×C₄	1	4	yes	—
54	#97	D₄	1	4	yes	—
55	#97	Q₈	1	4	yes	complemented
56	C₄×C₈	—	2	4	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	Q₈	1	4	yes	complemented
59	C₂×Q₈	—	2	4	no	—
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	#41	1	2	yes	—
63	C₂²×C₄	C₂⁴	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	M₁₆	—	2	4	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₈	C₂×Q₈	1	2	yes	—
75	C₂×C₈	C₂×Q₈	1	2	yes	—
76	C₂×C₈	C₂×Q₈	1	2	yes	—
77	C₂×C₈	#41	1	2	yes	—
78	C₂×C₈	#41	1	2	yes	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	C₂²×C₄	1	8	yes	—
82	C₂²⋊C₄	—	2	4	no	—
83	C₄²	—	2	4	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	4	4	no	—
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₄²	C₂×D₄	1	2	yes	—
89	C₄²	C₂×Q₈	1	2	yes	—
90	C₄²	C₂×Q₈	1	2	yes	—
91	C₂³	C₂×#41	1	1	yes	characteristic
92	Q₈	—	8	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₄×D₄	1	4	yes	—
100	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄	C₂×#41	1	1	yes	characteristic
105	C₂×C₄	C₂×#41	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	4	4	no	—
108	C₈	C₄×Q₈	1	4	yes	—
109	C₈	C₄×Q₈	1	4	yes	—
110	C₈	C₄⋊Q₈	1	4	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²	#533	1	1	yes	characteristic
114	C₂²	#533	1	1	yes	characteristic
115	C₄	—	2	4	no	—
116	C₄	—	2	4	no	—
117	C₄	—	4	2	no	—
118	C₄	—	4	8	no	—
119	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄	#532	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
126	C₂	#2956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3162

C₂

yes

maximal, characteristic

#2961

C₂

yes

maximal

#1994

C₂

yes

maximal

#1946

C₂

yes

maximal

#1945

C₂

yes

maximal

#1849

C₂

yes

maximal, characteristic

#1773

C₂

yes

maximal

#1772

C₂

yes

maximal

C₄×#97

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#501

C₄

yes

complemented

#446

—

C₂×#108

C₂²

yes

—

C₂×#108

C₂²

yes

—

#427

C₂²

yes

—

#427

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

—

#405

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#339

C₂²

yes

—

#339

C₂²

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#125

C₂×C₄

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

—

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

#106

C₂×C₄

yes

—

#106

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

C₂×C₄

yes

—

#97

D₄

yes

—

#97

Q₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₈

yes

complemented

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

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×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂×#41

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

100

C₂×C₄

C₂³×C₄

yes

characteristic

101

C₂×C₄

C₂²×D₄

yes

characteristic

102

C₂×C₄

C₂²×Q₈

yes

characteristic

103

C₂×C₄

C₂²×Q₈

yes

characteristic

104

C₂×C₄

C₂×#41

yes

characteristic

105

C₂×C₄

C₂×#41

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

C₄×Q₈

yes

—

109

C₈

C₄×Q₈

yes

—

110

C₈

C₄⋊Q₈

yes

—

111

C₂²

—

112

C₂²

C₂×C₄×D₄

yes

characteristic

113

C₂²

#533

yes

characteristic

114

C₂²

#533

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×C₄×Q₈

yes

characteristic

120

C₄

C₂×C₄×Q₈

yes

characteristic

121

C₄

C₂×C₄⋊Q₈

yes

characteristic

122

C₄

#532

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₄×C₄⋊Q₈

yes

characteristic

126

C₂

#2956

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

prime radical

C₂³

C₂×#41

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. #501	C₄	semidirect
55. #97	Q₈	semidirect
58. C₄×C₈	Q₈	semidirect

H ≅ G/N

type

11. #501

C₄

semidirect

55. #97

Q₈

semidirect

58. C₄×C₈

Q₈

semidirect

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

subgroup	index
#3162	2
#2961	2
#1994	2
#1946	2
#1945	2
#1849	2
#1773	2
#1772	2
C₄×#97	2

subgroup

index

C₄×#97

Subgroups — #21736

Conjugacy classes of subgroups

Complements

Maximal subgroups