Group #33577 — subgroups

Subgroups — C₂×#2209 #33577

Order 256.

351 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 135 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×Q₃₂	C₂	1	2	yes	maximal
2	C₂×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#2209	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#358	C₂	1	2	yes	maximal
7	C₂²×Q₁₆	—	2	1	no	—
8	C₂²×Q₁₆	C₂²	1	2	yes	—
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×Q₈	—	2	1	no	—
12	C₂×Q₃₂	—	2	8	no	—
13	C₂×Q₃₂	C₂²	1	8	yes	—
14	C₂²×C₁₆	C₂²	1	2	yes	—
15	#494	—	2	2	no	—
16	#494	C₂²	1	4	yes	—
17	C₄×Q₁₆	—	2	2	no	—
18	C₄×Q₁₆	C₂²	1	4	yes	—
19	C₂×#107	C₂²	1	1	yes	characteristic
20	C₂×#103	—	2	1	no	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆	C₂²	1	4	yes	—
23	#358	C₂²	1	8	yes	—
24	C₂²×Q₈	—	2	1	no	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	4	no	—
28	C₂×Q₁₆	—	2	4	no	—
29	C₂×Q₁₆	—	2	2	no	—
30	C₂×Q₁₆	—	2	2	no	—
31	C₂×Q₁₆	—	2	4	no	—
32	C₂×Q₁₆	—	4	4	no	—
33	C₂×Q₁₆	D₄	1	4	yes	—
34	C₂×Q₁₆	C₂³	1	4	yes	—
35	C₂×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈	—	2	4	no	—
39	C₄⋊Q₈	—	4	2	no	—
40	C₄×Q₈	—	2	4	no	—
41	C₄×Q₈	—	4	2	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄⋊C₄	—	4	1	no	—
45	C₂×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄²	—	4	1	no	—
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	Q₃₂	—	2	16	no	—
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	C₂³	1	4	yes	—
51	#107	—	2	1	no	—
52	#107	C₂³	1	2	yes	—
53	#103	—	2	4	no	—
54	C₄×C₈	D₄	1	2	yes	—
55	C₄×C₈	C₂³	1	2	yes	—
56	C₂×Q₈	—	2	4	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂×Q₈	—	2	4	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	2	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	4	no	—
63	C₂×Q₈	—	4	4	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	2	no	—
67	C₂²×C₄	D₈	1	1	yes	characteristic
68	C₂²×C₄	D₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	Q₁₆	—	2	4	no	—
71	Q₁₆	—	2	4	no	—
72	Q₁₆	—	2	2	no	—
73	Q₁₆	—	2	4	no	—
74	Q₁₆	—	4	8	no	—
75	Q₁₆	—	4	4	no	—
76	Q₁₆	C₂×D₄	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₈	#41	1	2	yes	—
82	C₂×C₈	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	—	4	4	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄²	—	4	4	no	—
89	C₄²	D₈	1	2	yes	—
90	C₄²	C₂×D₄	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₈	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	4	no	—
96	Q₈	—	4	4	no	—
97	Q₈	—	4	4	no	—
98	Q₈	—	8	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	Q₃₂	1	4	yes	complemented
107	C₂×C₄	#121	1	2	yes	—
108	C₂×C₄	C₂×D₈	1	2	yes	—
109	C₂×C₄	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄	C₂×D₈	1	2	yes	—
111	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	#121	1	2	yes	—
115	C₈	C₂²×D₄	1	1	yes	characteristic
116	C₈	C₂×#41	1	1	yes	characteristic
117	C₂²	#459	1	2	yes	—
118	C₂²	C₂×Q₃₂	1	2	yes	—
119	C₂²	#510	1	2	yes	—
120	C₂²	C₂²×D₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	4	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	8	4	no	—
125	C₄	#459	1	2	yes	—
126	C₄	C₂×Q₃₂	1	4	yes	complemented
127	C₄	C₂×#121	1	1	yes	characteristic
128	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂	#2209	1	4	yes	direct product factor
130	C₂	C₂×#459	1	1	yes	characteristic
131	C₂	C₂²×Q₃₂	1	1	yes	characteristic
132	C₂	C₂×#510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal

C₂×#494

C₂

yes

maximal, characteristic

C₂×C₄×Q₁₆

C₂

yes

maximal, characteristic

#2209

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#494

—

#494

C₂²

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂²

yes

—

C₂×#107

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

#358

C₂²

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

#107

—

#107

C₂³

yes

—

#103

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₄

yes

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₈

yes

—

C₄²

C₂×D₄

yes

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

Q₃₂

yes

complemented

107

C₂×C₄

#121

yes

—

108

C₂×C₄

C₂×D₈

yes

—

109

C₂×C₄

C₂×D₈

yes

characteristic

110

C₂×C₄

C₂×D₈

yes

—

111

C₂×C₄

AGL(1,ℤ/8)

yes

—

112

C₂×C₄

C₂²×D₄

yes

characteristic

113

C₈

—

114

C₈

#121

yes

—

115

C₈

C₂²×D₄

yes

characteristic

116

C₈

C₂×#41

yes

characteristic

117

C₂²

#459

yes

—

118

C₂²

C₂×Q₃₂

yes

—

119

C₂²

#510

yes

—

120

C₂²

C₂²×D₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#459

yes

—

126

C₄

C₂×Q₃₂

yes

complemented

127

C₄

C₂×#121

yes

characteristic

128

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

129

C₂

#2209

yes

direct product factor

130

C₂

C₂×#459

yes

characteristic

131

C₂

C₂²×Q₃₂

yes

characteristic

132

C₂

C₂×#510

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #2209	C₂	direct (N × H)
106. C₂×C₄	Q₃₂	semidirect
126. C₄	C₂×Q₃₂	semidirect
129. C₂	#2209	direct (N × H)

H ≅ G/N

type

4. #2209

C₂

direct (N × H)

106. C₂×C₄

Q₃₂

semidirect

126. C₄

C₂×Q₃₂

semidirect

129. C₂

#2209

direct (N × H)

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

subgroup	index
C₂²×Q₃₂	2
C₂×#494	2
C₂×C₄×Q₁₆	2
#2209	2
C₂×C₄⋊C₁₆	2
C₂×#358	2

subgroup

index

C₂²×Q₃₂

C₂×#494

C₂×C₄×Q₁₆

#2209

C₂×C₄⋊C₁₆

C₂×#358

Subgroups — C₂×#2209 #33577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2209 #33577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2209 #33577