Group #33495 — subgroups

Subgroups — #33495

Order 256.

282 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 179 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#2169	C₂	1	2	yes	maximal, complemented
3	#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
5	#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	#2113	C₂	1	4	yes	maximal, complemented
8	C₂×#131	C₂²	1	2	yes	—
9	#532	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆	C₂²	1	4	yes	—
12	#436	C₄	1	2	yes	—
13	#436	C₄	1	2	yes	—
14	C₈×D₄	C₂²	1	2	yes	—
15	C₈×D₄	C₂²	1	2	yes	—
16	#423	C₄	1	1	yes	characteristic
17	#408	C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₂²⋊C₁₆	C₂²	1	4	yes	—
24	C₂²⋊C₁₆	C₂²	1	4	yes	—
25	#346	C₂²	1	2	yes	—
26	C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×#41	C₂×C₄	1	2	yes	—
29	#131	—	4	4	no	—
30	C₂×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	4	yes	—
35	C₄⋊Q₈	C₈	1	1	yes	characteristic
36	#127	C₈	1	1	yes	characteristic
37	#124	C₈	1	2	yes	—
38	#121	C₈	1	2	yes	—
39	#121	C₈	1	2	yes	—
40	C₄×D₄	C₂×C₄	1	2	yes	—
41	C₄×D₄	C₂×C₄	1	2	yes	—
42	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
43	M₃₂	—	4	4	no	—
44	C₂×C₁₆	—	4	4	no	—
45	C₂×C₁₆	C₂³	1	2	yes	—
46	C₂×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₆	C₂³	1	4	yes	—
48	C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₈	C₂³	1	1	yes	characteristic
56	#41	—	4	2	no	—
57	#41	—	4	2	no	—
58	C₂×Q₈	C₂×C₈	1	2	yes	—
59	C₂×D₄	C₂×C₈	1	2	yes	—
60	C₂×D₄	C₂×C₈	1	2	yes	—
61	C₂×D₄	C₂×C₈	1	4	yes	—
62	C₂×D₄	C₂×C₈	1	2	yes	—
63	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄	C₂²×C₄	1	4	yes	—
66	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
67	M₁₆	—	4	2	no	—
68	M₁₆	—	4	4	no	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₈	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄	C₂×C₈	1	4	yes	—
81	C₂²⋊C₄	C₂×C₈	1	4	yes	—
82	C₄²	C₂×C₈	1	1	yes	characteristic
83	C₄²	C₂×C₈	1	1	yes	characteristic
84	C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₁₆	—	2	2	no	—
87	C₁₆	—	2	4	no	—
88	C₁₆	—	2	2	no	—
89	C₂³	C₂²×C₈	1	4	yes	—
90	C₂³	C₂²×C₈	1	1	yes	characteristic
91	Q₈	—	4	2	no	—
92	D₄	—	2	4	no	—
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	D₄	—	4	4	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄	C₂²×C₈	1	4	yes	—
107	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	H(5,𝔽₂)	1	2	yes	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	4	no	—
120	C₂²	C₂³×C₈	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	4	no	—
126	C₄	#504	1	2	yes	—
127	C₄	#518	1	1	yes	characteristic
128	C₄	#568	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	—	4	4	no	—
131	C₂	#2972	1	1	yes	characteristic
132	C₂	C₂×#504	1	1	yes	characteristic
133	C₂	#3407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#2169

C₂

yes

maximal, complemented

#2168

C₂

yes

maximal, complemented

D₄×C₁₆

C₂

yes

maximal, complemented

#2163

C₂

yes

maximal, characteristic, complemented

#2162

C₂

yes

maximal, characteristic, complemented

#2113

C₂

yes

maximal, complemented

C₂×#131

C₂²

yes

—

#532

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#436

C₄

yes

—

#436

C₄

yes

—

C₈×D₄

C₂²

yes

—

C₈×D₄

C₂²

yes

—

#423

C₄

yes

characteristic

#408

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

characteristic

C₂×#41

C₂×C₄

yes

—

#131

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

C₈

yes

characteristic

#127

C₈

yes

characteristic

#124

C₈

yes

—

#121

C₈

yes

—

#121

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

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

—

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×C₈

yes

characteristic

103

C₂×C₄

C₂²×C₈

yes

characteristic

104

C₂×C₄

C₂²×C₈

yes

characteristic

105

C₂×C₄

C₂²×C₈

yes

characteristic

106

C₂×C₄

C₂²×C₈

yes

—

107

C₂×C₄

C₂²×C₈

yes

characteristic

108

C₂×C₄

C₂³×C₄

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

H(5,𝔽₂)

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×C₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#504

yes

—

127

C₄

#518

yes

characteristic

128

C₄

#568

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

#2972

yes

characteristic

132

C₂

C₂×#504

yes

characteristic

133

C₂

#3407

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#532	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. #2169	C₂	semidirect
3. #2168	C₂	semidirect
4. D₄×C₁₆	C₂	semidirect
5. #2163	C₂	semidirect
6. #2162	C₂	semidirect
7. #2113	C₂	semidirect

H ≅ G/N

type

2. #2169

C₂

semidirect

3. #2168

C₂

semidirect

4. D₄×C₁₆

C₂

semidirect

5. #2163

C₂

semidirect

6. #2162

C₂

semidirect

7. #2113

C₂

semidirect

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

subgroup	index
#2931	2
#2169	2
#2168	2
D₄×C₁₆	2
#2163	2
#2162	2
#2113	2

subgroup

index

D₄×C₁₆

Subgroups — #33495

Conjugacy classes of subgroups

Complements

Maximal subgroups