Group #33441 — subgroups

Subgroups — #33441

Order 256.

323 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#504	C₂	1	2	yes	maximal, complemented
2	#2955	C₂	1	1	yes	maximal, characteristic
3	#2171	C₂	1	4	yes	maximal, complemented
4	#2170	C₂	1	4	yes	maximal, complemented
5	#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	#2117	C₂	1	2	yes	maximal, complemented
7	#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#575	C₄	1	1	yes	characteristic
9	C₂×#131	C₂²	1	2	yes	—
10	#504	—	2	2	no	—
11	#504	—	2	2	no	—
12	#504	C₂²	1	4	yes	complemented
13	#504	C₂²	1	4	yes	complemented
14	C₂×M₃₂	—	2	2	no	—
15	C₂×M₃₂	C₂²	1	2	yes	—
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	—	2	2	no	—
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	#444	C₄	1	4	yes	—
20	#443	C₂²	1	4	yes	—
21	#430	C₄	1	1	yes	characteristic
22	#421	C₄	1	2	yes	—
23	#405	C₂²	1	1	yes	characteristic
24	#364	C₂²	1	4	yes	complemented
25	#350	C₂²	1	8	yes	complemented
26	#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆	C₂²	1	2	yes	complemented
28	C₂×#41	C₂×C₄	1	2	yes	—
29	#137	C₈	1	4	yes	—
30	AGL(1,ℤ/8)	C₈	1	4	yes	—
31	#135	C₂×C₄	1	4	yes	—
32	#131	—	2	2	no	—
33	#131	—	2	2	no	—
34	#131	C₂×C₄	1	4	yes	—
35	#131	C₂³	1	4	yes	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	#117	C₂×C₄	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	M₃₂	—	2	2	no	—
44	M₃₂	—	2	4	no	—
45	M₃₂	—	2	4	no	—
46	M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	D₄	1	2	yes	—
51	C₂×C₁₆	D₄	1	2	yes	—
52	C₂×C₁₆	C₂³	1	2	yes	—
53	C₂×C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₁₆	C₂³	1	1	yes	characteristic
55	#108	C₂×C₄	1	4	yes	—
56	#104	C₂×C₄	1	8	yes	—
57	#97	C₂×C₄	1	2	yes	—
58	C₄×C₈	C₂³	1	2	yes	—
59	#41	—	2	2	no	—
60	#41	—	2	2	no	—
61	#41	C₂×C₈	1	4	yes	—
62	#41	C₂²×C₄	1	4	yes	—
63	C₂×Q₈	C₂×C₈	1	2	yes	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆	C₂×C₈	1	4	yes	—
69	SD₁₆	C₂×C₈	1	8	yes	—
70	D₈	C₂×C₈	1	4	yes	—
71	M₁₆	—	2	2	no	—
72	M₁₆	—	2	4	no	—
73	M₁₆	—	2	4	no	—
74	M₁₆	C₂×C₈	1	4	yes	—
75	M₁₆	C₂²×C₄	1	4	yes	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₈	#41	1	2	yes	—
83	C₂×C₈	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄	—	2	1	no	—
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	C₂²×C₄	1	2	yes	—
87	C₁₆	—	2	4	no	—
88	C₁₆	—	2	2	no	—
89	C₁₆	C₂×D₄	1	2	yes	—
90	C₁₆	C₂×D₄	1	1	yes	characteristic
91	C₁₆	C₂×D₄	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	C₂³	#131	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	Q₈	C₂²×C₈	1	4	yes	—
96	D₄	—	2	4	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²×C₈	1	4	yes	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	C₄×D₄	1	2	yes	—
105	C₂×C₄	C₂²×C₈	1	2	yes	—
106	C₂×C₄	#131	1	1	yes	characteristic
107	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
108	C₈	—	2	4	no	—
109	C₈	C₄×D₄	1	2	yes	—
110	C₈	C₂²×C₈	1	4	yes	—
111	C₈	C₂²×D₄	1	1	yes	characteristic
112	C₈	C₂×#41	1	1	yes	characteristic
113	C₂²	—	2	4	no	—
114	C₂²	—	4	1	no	—
115	C₂²	—	4	4	no	—
116	C₂²	C₈×D₄	1	2	yes	—
117	C₂²	C₂×#131	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	4	2	no	—
120	C₄	C₈×D₄	1	2	yes	—
121	C₄	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄	C₂³×C₈	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	2	no	—
125	C₂	—	4	4	no	—
126	C₂	C₂×C₈×D₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504

C₂

yes

maximal, complemented

#2955

C₂

yes

maximal, characteristic

#2171

C₂

yes

maximal, complemented

#2170

C₂

yes

maximal, complemented

#2153

C₂

yes

maximal, characteristic, complemented

#2117

C₂

yes

maximal, complemented

#2108

C₂

yes

maximal, characteristic, complemented

#575

C₄

yes

characteristic

C₂×#131

C₂²

yes

—

#504

—

#504

—

#504

C₂²

yes

complemented

#504

C₂²

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂²

yes

characteristic

#444

C₄

yes

—

#443

C₂²

yes

—

#430

C₄

yes

characteristic

#421

C₄

yes

—

#405

C₂²

yes

characteristic

#364

C₂²

yes

complemented

#350

C₂²

yes

complemented

#346

C₂²

yes

complemented

C₄×C₁₆

C₂²

yes

complemented

C₂×#41

C₂×C₄

yes

—

#137

C₈

yes

—

AGL(1,ℤ/8)

C₈

yes

—

#135

C₂×C₄

yes

—

#131

—

#131

—

#131

C₂×C₄

yes

—

#131

C₂³

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂³

yes

characteristic

#117

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂³

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

#108

C₂×C₄

yes

—

#104

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

#41

C₂×C₈

yes

—

#41

C₂²×C₄

yes

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

Q₁₆

C₂×C₈

yes

—

SD₁₆

C₂×C₈

yes

—

D₈

C₂×C₈

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₈

yes

—

M₁₆

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

characteristic

C₁₆

C₂×D₄

yes

characteristic

C₂³

—

C₂³

#131

yes

characteristic

Q₈

—

Q₈

C₂²×C₈

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂²×C₈

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₄×D₄

yes

—

105

C₂×C₄

C₂²×C₈

yes

—

106

C₂×C₄

#131

yes

characteristic

107

C₂×C₄

C₂³×C₄

yes

characteristic

108

C₈

—

109

C₈

C₄×D₄

yes

—

110

C₈

C₂²×C₈

yes

—

111

C₈

C₂²×D₄

yes

characteristic

112

C₈

C₂×#41

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₈×D₄

yes

—

117

C₂²

C₂×#131

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

C₈×D₄

yes

—

121

C₄

C₂×C₄×D₄

yes

characteristic

122

C₄

C₂³×C₈

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×C₈×D₄

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×C₈×D₄
prime radical	#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₁₆

C₂×D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂

C₂×C₈×D₄

C₄

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
3. #2171	C₂	semidirect
4. #2170	C₂	semidirect
5. #2153	C₂	semidirect
6. #2117	C₂	semidirect
7. #2108	C₂	semidirect
12. #504	C₂²	semidirect
13. #504	C₂²	semidirect
24. #364	C₂²	semidirect
25. #350	C₂²	semidirect
26. #346	C₂²	semidirect
27. C₄×C₁₆	C₂²	semidirect

H ≅ G/N

type

1. C₂×#504

C₂

semidirect

3. #2171

C₂

semidirect

4. #2170

C₂

semidirect

5. #2153

C₂

semidirect

6. #2117

C₂

semidirect

7. #2108

C₂

semidirect

12. #504

C₂²

semidirect

13. #504

C₂²

semidirect

24. #364

C₂²

semidirect

25. #350

C₂²

semidirect

26. #346

C₂²

semidirect

27. C₄×C₁₆

C₂²

semidirect

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

subgroup	index
C₂×#504	2
#2955	2
#2171	2
#2170	2
#2153	2
#2117	2
#2108	2

subgroup

index

Subgroups — #33441

Conjugacy classes of subgroups

Complements

Maximal subgroups