Group #29155 — subgroups

Subgroups — #29155

Order 256.

304 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 117 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, complemented
2	C₂×#444	C₂	1	1	yes	maximal, characteristic
3	#2031	C₂	1	4	yes	maximal, complemented
4	#1976	C₂	1	2	yes	maximal
5	#1833	C₂	1	2	yes	maximal, complemented
6	#1814	C₂	1	2	yes	maximal, complemented
7	#1812	C₂	1	2	yes	maximal, complemented
8	#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	2	yes	—
11	#533	C₂²	1	2	yes	—
12	#501	—	2	4	no	—
13	#444	—	2	4	no	—
14	C₂×#108	C₂²	1	1	yes	characteristic
15	C₂×#108	C₂²	1	1	yes	characteristic
16	#427	C₂²	1	1	yes	characteristic
17	#427	C₂²	1	2	yes	—
18	#427	C₂²	1	2	yes	—
19	#421	C₂²	1	4	yes	—
20	C₂×#104	C₂²	1	2	yes	—
21	#410	C₂²	1	4	yes	—
22	#405	C₂²	1	2	yes	—
23	C₂×#97	C₂²	1	1	yes	characteristic
24	#344	C₂²	1	4	yes	—
25	#339	C₂²	1	2	yes	—
26	H(3,ℤ/4)	C₂²	1	4	yes	—
27	C₂×#41	C₂³	1	2	yes	—
28	#135	—	2	2	no	—
29	#135	—	2	2	no	—
30	C₂×Q₁₆	Q₈	1	1	yes	characteristic, complemented
31	C₂×SD₁₆	Q₈	1	2	yes	complemented
32	C₂×D₈	Q₈	1	1	yes	characteristic, complemented
33	#131	—	2	2	no	—
34	#131	—	2	2	no	—
35	#131	—	2	2	no	—
36	#131	—	2	2	no	—
37	C₂×M₁₆	—	2	2	no	—
38	C₂×M₁₆	C₂³	1	2	yes	—
39	C₂×M₁₆	C₂³	1	2	yes	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈	—	2	2	no	—
44	#125	—	2	2	no	—
45	C₂²⋊Q₈	—	4	2	no	—
46	C₄×Q₈	—	4	2	no	—
47	#117	C₂³	1	4	yes	—
48	#117	C₂³	1	2	yes	—
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	#108	—	2	2	no	—
51	#108	—	2	2	no	—
52	#107	—	2	2	no	—
53	#107	—	2	2	no	—
54	#107	—	2	1	no	—
55	#106	—	2	2	no	—
56	#106	—	2	2	no	—
57	#106	—	2	1	no	—
58	#104	—	2	4	no	—
59	#104	—	4	4	no	—
60	#99	—	4	4	no	—
61	#97	—	2	2	no	—
62	#97	D₄	1	4	yes	—
63	C₄×C₈	—	2	2	no	—
64	#41	—	2	2	no	—
65	#41	—	2	2	no	—
66	#41	—	2	2	no	—
67	#41	—	2	2	no	—
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	C₂×Q₈	1	2	yes	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	C₂×Q₈	1	2	yes	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂⁴	1	1	yes	characteristic
75	Q₁₆	—	2	2	no	—
76	SD₁₆	—	2	4	no	—
77	D₈	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	4	2	no	—
83	M₁₆	—	4	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈	#41	1	2	yes	—
93	C₂×C₈	#41	1	1	yes	characteristic
94	C₂×C₈	#41	1	1	yes	characteristic
95	C₂×C₈	#41	1	2	yes	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₂²⋊C₄	—	2	4	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	4	no	—
105	C₄²	C₂×D₄	1	2	yes	—
106	C₂³	—	2	2	no	—
107	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈	—	2	2	no	—
109	Q₈	—	2	2	no	—
110	Q₈	—	8	4	no	—
111	D₄	—	2	2	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄	C₂×#41	1	1	yes	characteristic
125	C₂×C₄	C₂×#41	1	1	yes	characteristic
126	C₂×C₄	C₂×#41	1	1	yes	characteristic
127	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	4	2	no	—
130	C₈	C₂²⋊Q₈	1	2	yes	—
131	C₈	C₂²⋊Q₈	1	2	yes	—
132	C₈	#124	1	2	yes	—
133	C₈	#124	1	2	yes	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#535	1	1	yes	characteristic
138	C₂²	#554	1	1	yes	characteristic
139	C₂²	#555	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	—	8	4	no	—
144	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₄	C₂×#124	1	1	yes	characteristic
146	C₄	#554	1	1	yes	characteristic
147	C₄	#555	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	8	2	no	—
152	C₂	#2563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3162

C₂

yes

maximal, characteristic, complemented

C₂×#444

C₂

yes

maximal, characteristic

#2031

C₂

yes

maximal, complemented

#1976

C₂

yes

maximal

#1833

C₂

yes

maximal, complemented

#1814

C₂

yes

maximal, complemented

#1812

C₂

yes

maximal, complemented

#1773

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₂²

yes

characteristic

C₂×#131

C₂²

yes

—

#533

C₂²

yes

—

#501

—

#444

—

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#427

C₂²

yes

—

#427

C₂²

yes

—

#421

C₂²

yes

—

C₂×#104

C₂²

yes

—

#410

C₂²

yes

—

#405

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

#344

C₂²

yes

—

#339

C₂²

yes

—

H(3,ℤ/4)

C₂²

yes

—

C₂×#41

C₂³

yes

—

#135

—

#135

—

C₂×Q₁₆

Q₈

yes

characteristic, complemented

C₂×SD₁₆

Q₈

yes

complemented

C₂×D₈

Q₈

yes

characteristic, complemented

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂³

yes

—

#117

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₄×C₈

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

yes

—

C₂×D₄

—

C₂×D₄

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₄

yes

—

106

C₂³

—

107

C₂³

H(5,𝔽₂)

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×D₄

yes

characteristic

123

C₂×C₄

C₂²×Q₈

yes

characteristic

124

C₂×C₄

C₂×#41

yes

characteristic

125

C₂×C₄

C₂×#41

yes

characteristic

126

C₂×C₄

C₂×#41

yes

characteristic

127

C₂×C₄

H(5,𝔽₂)

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

C₂²⋊Q₈

yes

—

131

C₈

C₂²⋊Q₈

yes

—

132

C₈

#124

yes

—

133

C₈

#124

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#535

yes

characteristic

138

C₂²

#554

yes

characteristic

139

C₂²

#555

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×C₂²⋊Q₈

yes

characteristic

145

C₄

C₂×#124

yes

characteristic

146

C₄

#554

yes

characteristic

147

C₄

#555

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

—

151

C₂

—

152

C₂

#2563

yes

characteristic

	subgroup	G/H
center	C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2563
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#555

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

C₂

C₂²

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3162	C₂	semidirect
3. #2031	C₂	semidirect
5. #1833	C₂	semidirect
6. #1814	C₂	semidirect
7. #1812	C₂	semidirect
8. #1773	C₂	semidirect
30. C₂×Q₁₆	Q₈	semidirect
31. C₂×SD₁₆	Q₈	semidirect
32. C₂×D₈	Q₈	semidirect

H ≅ G/N

type

1. #3162

C₂

semidirect

3. #2031

C₂

semidirect

5. #1833

C₂

semidirect

6. #1814

C₂

semidirect

7. #1812

C₂

semidirect

8. #1773

C₂

semidirect

30. C₂×Q₁₆

Q₈

semidirect

31. C₂×SD₁₆

Q₈

semidirect

32. C₂×D₈

Q₈

semidirect

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

subgroup	index
#3162	2
C₂×#444	2
#2031	2
#1976	2
#1833	2
#1814	2
#1812	2
#1773	2

subgroup

index

Subgroups — #29155

Conjugacy classes of subgroups

Complements

Maximal subgroups