Group #22747 — subgroups

Subgroups — #22747

Order 256.

311 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 189 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	Q₈×M₁₆	C₂	1	2	yes	maximal
2	C₂×#446	C₂	1	1	yes	maximal, characteristic
3	#1950	C₂	1	4	yes	maximal
4	#1950	C₂	1	2	yes	maximal
5	#1838	C₂	1	2	yes	maximal
6	#1837	C₂	1	1	yes	maximal, characteristic
7	#1778	C₂	1	2	yes	maximal
8	C₄×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	#446	—	2	2	no	—
11	#446	—	2	4	no	—
12	#446	C₄	1	4	yes	—
13	#446	C₄	1	4	yes	—
14	C₈×Q₈	—	2	2	no	—
15	C₂×#108	C₂²	1	2	yes	—
16	C₂×#108	C₂²	1	4	yes	—
17	#423	C₂²	1	2	yes	—
18	#423	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈	C₂²	1	2	yes	—
21	C₄×M₁₆	C₂²	1	2	yes	—
22	C₄×M₁₆	C₂²	1	4	yes	—
23	C₂×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#341	C₂²	1	2	yes	—
26	#341	C₂²	1	2	yes	—
27	#341	C₂²	1	1	yes	characteristic
28	#341	C₂²	1	4	yes	—
29	#341	C₂²	1	1	yes	characteristic
30	C₂²×Q₈	Q₈	1	1	yes	characteristic
31	C₂×M₁₆	C₂³	1	4	yes	—
32	C₂×M₁₆	C₂³	1	2	yes	—
33	C₂×M₁₆	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₄×Q₈	C₂×C₄	1	2	yes	—
38	C₄×Q₈	C₂×C₄	1	2	yes	—
39	C₄×Q₈	D₄	1	2	yes	—
40	C₄×Q₈	D₄	1	2	yes	—
41	C₂×C₄⋊C₄	Q₈	1	1	yes	characteristic
42	C₂×C₄⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	2	yes	—
45	#108	—	2	4	no	—
46	#108	—	2	8	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	2	4	no	—
49	C₄⋊C₈	—	2	4	no	—
50	C₄⋊C₈	—	2	2	no	—
51	C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₄⋊C₈	C₂×C₄	1	8	yes	—
54	#97	—	2	4	no	—
55	#97	—	2	2	no	—
56	#97	C₂×C₄	1	8	yes	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	4	no	—
59	C₄×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₈	C₂×C₄	1	2	yes	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	C₄⋊C₄	1	2	yes	—
63	C₂×Q₈	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄	C₂×Q₈	1	2	yes	—
67	C₂²×C₄	C₂×Q₈	1	2	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	4	no	—
72	M₁₆	—	2	4	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	#41	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₈	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
88	C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	4	yes	—
91	C₄²	C₂×D₄	1	1	yes	characteristic
92	C₄²	C₂×D₄	1	4	yes	—
93	C₄²	C₂×D₄	1	1	yes	characteristic
94	C₂³	C₂²×Q₈	1	1	yes	characteristic
95	Q₈	—	2	4	no	—
96	Q₈	—	4	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄	C₂×#41	1	2	yes	—
110	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	2	4	no	—
113	C₈	—	2	2	no	—
114	C₈	—	2	4	no	—
115	C₈	—	4	2	no	—
116	C₈	—	4	4	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²	#557	1	2	yes	—
121	C₄	—	2	1	no	—
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	4	no	—
126	C₄	#430	1	2	yes	—
127	C₄	C₄×#41	1	1	yes	characteristic
128	C₄	#519	1	1	yes	characteristic
129	C₄	#548	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	#2351	1	1	yes	characteristic
132	C₂	C₂×#430	1	1	yes	characteristic
133	C₂	#2916	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×M₁₆

C₂

yes

maximal

C₂×#446

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal

#1950

C₂

yes

maximal

#1838

C₂

yes

maximal

#1837

C₂

yes

maximal, characteristic

#1778

C₂

yes

maximal

C₄×#108

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

—

—

C₄

yes

—

#446

C₄

yes

—

C₈×Q₈

—

C₂×#108

C₂²

yes

—

C₂×#108

C₂²

yes

—

#423

C₂²

yes

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₄×M₁₆

C₂²

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#341

C₂²

yes

—

#341

C₂²

yes

—

#341

C₂²

yes

characteristic

#341

C₂²

yes

—

#341

C₂²

yes

characteristic

C₂²×Q₈

Q₈

yes

characteristic

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

D₄

yes

—

C₄×Q₈

D₄

yes

—

C₂×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#97

—

#97

—

#97

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

#41

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₄⋊C₄

yes

—

C₄⋊C₄

yes

—

C₄⋊C₄

yes

—

C₄⋊C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₂³

C₂²×Q₈

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

102

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

103

C₂×C₄

C₂×C₄⋊C₄

yes

—

104

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

105

C₂×C₄

C₂×C₄⋊C₄

yes

—

106

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

107

C₂×C₄

C₂³×C₄

yes

characteristic

108

C₂×C₄

C₂²×D₄

yes

characteristic

109

C₂×C₄

C₂×#41

yes

—

110

C₂×C₄

2₋¹⁺⁴

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂²×C₄⋊C₄

yes

characteristic

120

C₂²

#557

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#430

yes

—

127

C₄

C₄×#41

yes

characteristic

128

C₄

#519

yes

characteristic

129

C₄

#548

yes

—

130

C₂

—

131

C₂

#2351

yes

characteristic

132

C₂

C₂×#430

yes

characteristic

133

C₂

#2916

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂³	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ 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

C₂²×C₄⋊C₄

prime radical

C₂³

C₂²×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
Q₈×M₁₆	2
C₂×#446	2
#1950	2
#1838	2
#1837	2
#1778	2
C₄×#108	2

subgroup

index

Q₈×M₁₆

Subgroups — #22747

Conjugacy classes of subgroups

Maximal subgroups