Group #33491 — subgroups

Subgroups — #33491

Order 256.

292 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 181 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	#2929	C₂	1	1	yes	maximal, characteristic
2	#2168	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₆	C₂	1	4	yes	maximal, complemented
4	#2164	C₂	1	2	yes	maximal, complemented
5	#2113	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×#131	C₂²	1	1	yes	characteristic
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	#525	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	2	yes	complemented
11	C₂²×C₁₆	C₂²	1	4	yes	—
12	C₂²×C₁₆	C₂²	1	2	yes	complemented
13	#435	C₄	1	4	yes	—
14	C₈×D₄	C₂²	1	4	yes	—
15	#424	C₄	1	1	yes	characteristic
16	#408	C₄	1	1	yes	characteristic
17	#407	C₄	1	1	yes	characteristic
18	#405	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₂²⋊C₁₆	C₂²	1	4	yes	—
21	C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
24	#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆	C₂²	1	2	yes	complemented
26	C₂×#41	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	#131	—	4	2	no	—
29	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	—	4	2	no	—
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	2	yes	—
38	#123	C₈	1	2	yes	—
39	C₂²⋊Q₈	C₈	1	2	yes	—
40	#121	C₈	1	2	yes	—
41	#120	C₈	1	2	yes	—
42	C₄×D₄	C₂×C₄	1	4	yes	—
43	#117	C₂×C₄	1	1	yes	characteristic
44	M₃₂	—	4	2	no	—
45	C₂×C₁₆	—	4	4	no	—
46	C₂×C₁₆	—	4	2	no	—
47	C₂×C₁₆	C₂³	1	4	yes	—
48	C₂×C₁₆	C₂³	1	4	yes	—
49	C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	#97	C₂×C₄	1	2	yes	—
53	C₄×C₈	C₂³	1	2	yes	—
54	C₂⁴	C₂×C₈	1	1	yes	characteristic
55	#41	—	4	1	no	—
56	#41	—	4	1	no	—
57	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄	C₂×C₈	1	4	yes	—
59	C₂×D₄	C₂×C₈	1	2	yes	—
60	C₂×D₄	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄	—	4	1	no	—
62	C₂²×C₄	—	4	1	no	—
63	C₂²×C₄	C₂×C₈	1	2	yes	—
64	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄	C₂²×C₄	1	2	yes	—
66	C₂²×C₄	C₂²×C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
68	M₁₆	—	4	1	no	—
69	M₁₆	—	4	2	no	—
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₈	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄	C₂×C₈	1	2	yes	—
83	C₄⋊C₄	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄	C₂×C₈	1	4	yes	—
86	C₂²⋊C₄	C₂×C₈	1	4	yes	—
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₁₆	—	2	4	no	—
89	C₁₆	—	2	4	no	—
90	C₂³	—	2	4	no	—
91	C₂³	—	4	1	no	—
92	C₂³	C₂²×C₈	1	1	yes	characteristic
93	C₂³	C₂²×C₈	1	2	yes	—
94	C₂³	C₂²×C₈	1	2	yes	—
95	Q₈	—	4	1	no	—
96	D₄	—	2	8	no	—
97	D₄	—	4	1	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₂²×C₈	1	4	yes	—
108	C₂×C₄	C₂²×C₈	1	2	yes	—
109	C₂×C₄	C₂²×C₈	1	2	yes	—
110	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	H(5,𝔽₂)	1	2	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	#504	1	4	yes	—
125	C₂²	C₂³×C₈	1	1	yes	characteristic
126	C₄	—	2	4	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	#518	1	1	yes	characteristic
131	C₄	#568	1	1	yes	characteristic
132	C₂	—	2	4	no	—
133	C₂	—	4	2	no	—
134	C₂	—	4	1	no	—
135	C₂	#2972	1	1	yes	characteristic
136	C₂	C₂×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2929

C₂

yes

maximal, characteristic

#2168

C₂

yes

maximal, complemented

D₄×C₁₆

C₂

yes

maximal, complemented

#2164

C₂

yes

maximal, complemented

#2113

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆

C₂

yes

maximal, complemented

C₂×#131

C₂²

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

#525

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

complemented

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

complemented

#435

C₄

yes

—

C₈×D₄

C₂²

yes

—

#424

C₄

yes

characteristic

#408

C₄

yes

characteristic

#407

C₄

yes

characteristic

#405

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

complemented

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

complemented

#346

C₂²

yes

complemented

C₄×C₁₆

C₂²

yes

complemented

C₂×#41

C₂×C₄

yes

characteristic

C₂³×C₄

C₂×C₄

yes

characteristic

#131

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

#123

C₈

yes

—

C₂²⋊Q₈

C₈

yes

—

#121

C₈

yes

—

#120

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

#117

C₂×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₂×C₈

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

characteristic

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₂²×C₈

yes

characteristic

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×C₈

yes

—

108

C₂×C₄

C₂²×C₈

yes

—

109

C₂×C₄

C₂²×C₈

yes

—

110

C₂×C₄

C₂²×C₈

yes

characteristic

111

C₂×C₄

C₂³×C₄

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

H(5,𝔽₂)

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#504

yes

—

125

C₂²

C₂³×C₈

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#518

yes

characteristic

131

C₄

#568

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

#2972

yes

characteristic

136

C₂

C₂×#504

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₈

C₂⁴

commutator, socle

C₂²

C₂³×C₈

C₄

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ G

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2168	C₂	semidirect
3. D₄×C₁₆	C₂	semidirect
4. #2164	C₂	semidirect
5. #2113	C₂	semidirect
6. C₂×C₂²⋊C₁₆	C₂	semidirect
10. C₂×M₃₂	C₂²	semidirect
12. C₂²×C₁₆	C₂²	semidirect
19. C₄⋊C₁₆	C₂²	semidirect
23. C₂²⋊C₁₆	C₂²	semidirect
24. #346	C₂²	semidirect
25. C₄×C₁₆	C₂²	semidirect

H ≅ G/N

type

2. #2168

C₂

semidirect

3. D₄×C₁₆

C₂

semidirect

4. #2164

C₂

semidirect

5. #2113

C₂

semidirect

6. C₂×C₂²⋊C₁₆

C₂

semidirect

10. C₂×M₃₂

C₂²

semidirect

12. C₂²×C₁₆

C₂²

semidirect

19. C₄⋊C₁₆

C₂²

semidirect

23. C₂²⋊C₁₆

C₂²

semidirect

24. #346

C₂²

semidirect

25. C₄×C₁₆

C₂²

semidirect

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

subgroup	index
#2929	2
#2168	2
D₄×C₁₆	2
#2164	2
#2113	2
C₂×C₂²⋊C₁₆	2

subgroup

index

D₄×C₁₆

C₂×C₂²⋊C₁₆

Subgroups — #33491

Conjugacy classes of subgroups

Complements

Maximal subgroups