Group #33416 — subgroups

Subgroups — C₂×#2163 #33416

Order 256.

343 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 223 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	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#2163	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×#346	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²	C₄	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	4	yes	—
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₄	1	4	yes	—
12	C₂×C₄×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	C₂²	1	8	yes	complemented
17	C₂²⋊C₁₆	C₂²	1	8	yes	—
18	#346	C₂²	1	8	yes	complemented
19	C₂³×C₄	C₈	1	1	yes	characteristic
20	C₂³×C₄	C₈	1	1	yes	characteristic
21	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	—	2	4	no	—
23	C₂²×C₈	—	2	4	no	—
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²	C₈	1	1	yes	characteristic
32	C₂×C₄²	C₈	1	4	yes	—
33	C₂×C₄²	C₈	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	4	yes	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆	—	2	8	no	—
38	C₂×C₁₆	C₂³	1	8	yes	—
39	C₄×C₈	C₂×C₄	1	4	yes	—
40	C₄×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈	C₂³	1	2	yes	—
44	C₄×C₈	C₂³	1	2	yes	—
45	C₂⁴	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	—	2	4	no	—
48	C₂²×C₄	—	2	2	no	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄	C₂×C₈	1	2	yes	—
52	C₂²×C₄	C₂×C₈	1	4	yes	—
53	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂×C₈	1	4	yes	—
57	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₄	1	2	yes	—
59	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	8	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	M₁₆	1	4	yes	—
66	C₂×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	#41	1	4	yes	—
71	C₂×C₈	C₂⁴	1	1	yes	characteristic
72	C₄²	C₂×C₈	1	4	yes	—
73	C₄²	C₂×C₈	1	4	yes	—
74	C₄²	C₂×C₈	1	2	yes	—
75	C₄²	C₂×C₈	1	2	yes	—
76	C₄²	C₂²×C₄	1	2	yes	—
77	C₄²	C₂²×C₄	1	2	yes	—
78	C₁₆	—	2	8	no	—
79	C₂³	—	2	2	no	—
80	C₂³	—	2	2	no	—
81	C₂³	M₃₂	1	4	yes	—
82	C₂³	C₂²×C₈	1	2	yes	—
83	C₂³	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	8	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	M₃₂	1	4	yes	—
94	C₂×C₄	#117	1	2	yes	—
95	C₂×C₄	C₂²×C₈	1	2	yes	—
96	C₂×C₄	C₂²×C₈	1	2	yes	—
97	C₂×C₄	C₂²×C₈	1	2	yes	—
98	C₂×C₄	C₂²×C₈	1	2	yes	—
99	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈	1	2	yes	—
101	C₂×C₄	C₂×M₁₆	1	2	yes	—
102	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	#117	1	2	yes	—
106	C₈	C₂×M₁₆	1	4	yes	—
107	C₈	C₂×#41	1	2	yes	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	8	no	—
111	C₂²	#431	1	2	yes	—
112	C₂²	C₂×M₃₂	1	4	yes	—
113	C₂²	C₂×M₃₂	1	2	yes	—
114	C₂²	C₂×M₃₂	1	2	yes	—
115	C₂²	C₂³×C₈	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	#431	1	2	yes	—
121	C₄	C₂×M₃₂	1	4	yes	—
122	C₄	C₂×#117	1	1	yes	characteristic
123	C₄	C₂²×M₁₆	1	1	yes	characteristic
124	C₂	—	2	4	no	—
125	C₂	#2163	1	4	yes	direct product factor
126	C₂	C₂×#431	1	1	yes	characteristic
127	C₂	C₂²×M₃₂	1	1	yes	characteristic
128	C₂	C₂²×M₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

#2163

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆

C₂

yes

maximal

C₂×#346

C₂

yes

maximal, complemented

C₂³×C₈

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂²×C₄²

C₄

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

complemented

C₂²⋊C₁₆

C₂²

yes

—

#346

C₂²

yes

complemented

C₂³×C₄

C₈

yes

characteristic

C₂³×C₄

C₈

yes

characteristic

C₂³×C₄

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

C₈

yes

characteristic

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₈

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄²

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

M₃₂

yes

—

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

#117

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

100

C₂×C₄

C₂²×C₈

yes

—

101

C₂×C₄

C₂×M₁₆

yes

—

102

C₂×C₄

C₂³×C₄

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

#117

yes

—

106

C₈

C₂×M₁₆

yes

—

107

C₈

C₂×#41

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#431

yes

—

112

C₂²

C₂×M₃₂

yes

—

113

C₂²

C₂×M₃₂

yes

—

114

C₂²

C₂×M₃₂

yes

—

115

C₂²

C₂³×C₈

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#431

yes

—

121

C₄

C₂×M₃₂

yes

—

122

C₄

C₂×#117

yes

characteristic

123

C₄

C₂²×M₁₆

yes

characteristic

124

C₂

—

125

C₂

#2163

yes

direct product factor

126

C₂

C₂×#431

yes

characteristic

127

C₂

C₂²×M₃₂

yes

characteristic

128

C₂

C₂²×M₃₂

yes

characteristic

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

subgroup

G/H

center

C₂²×C₈

C₂³

commutator

C₂²

C₂³×C₈

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂³

C₂²×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2163	C₂	direct (N × H)
3. C₂×C₄⋊C₁₆	C₂	semidirect
5. C₂×#346	C₂	semidirect
16. C₄⋊C₁₆	C₂²	semidirect
18. #346	C₂²	semidirect
125. C₂	#2163	direct (N × H)

H ≅ G/N

type

2. #2163

C₂

direct (N × H)

3. C₂×C₄⋊C₁₆

C₂

semidirect

5. C₂×#346

C₂

semidirect

16. C₄⋊C₁₆

C₂²

semidirect

18. #346

C₂²

semidirect

125. C₂

#2163

direct (N × H)

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

subgroup	index
C₂²×C₄×C₈	2
#2163	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
C₂×#346	2

subgroup

index

C₂²×C₄×C₈

#2163

C₂×C₄⋊C₁₆

C₂×C₂²⋊C₁₆

C₂×#346

Subgroups — C₂×#2163 #33416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2163 #33416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2163 #33416