Group #33721 — subgroups

Subgroups — #33721

Order 256.

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

normal subgroups only

This group has 397 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	C₂×#510	C₂	1	2	yes	maximal, complemented
2	C₂×#508	C₂	1	1	yes	maximal, characteristic
3	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#3156	C₂	1	2	yes	maximal, complemented
5	#2251	C₂	1	4	yes	maximal, complemented
6	#2249	C₂	1	2	yes	maximal, complemented
7	#2247	C₂	1	2	yes	maximal, complemented
8	#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137	—	2	2	no	—
10	C₂×#135	C₂²	1	2	yes	—
11	C₂²×Q₁₆	C₂²	1	2	yes	—
12	#536	—	2	2	no	—
13	#510	—	2	8	no	—
14	#508	—	2	4	no	—
15	C₂×Q₃₂	—	2	4	no	—
16	C₂×Q₃₂	C₂²	1	2	yes	—
17	C₂×Q₃₂	C₂²	1	2	yes	—
18	C₂×Q₃₂	C₂²	1	1	yes	characteristic
19	C₂×Q₃₂	C₂²	1	4	yes	—
20	C₂×SD₃₂	C₂²	1	2	yes	—
21	C₂×SD₃₂	C₂²	1	4	yes	—
22	C₂×D₁₆	C₂²	1	1	yes	characteristic
23	C₂×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆	C₂²	1	1	yes	characteristic
25	#497	—	2	4	no	—
26	#495	C₂²	1	4	yes	—
27	#494	C₂²	1	4	yes	—
28	#405	C₂²	1	1	yes	characteristic
29	#346	C₂²	1	2	yes	—
30	C₄×C₁₆	C₂²	1	2	yes	—
31	C₂×#41	—	2	2	no	—
32	C₂²×Q₈	—	2	2	no	—
33	#137	—	4	8	no	—
34	#135	—	2	4	no	—
35	#135	—	2	4	no	—
36	C₂×Q₁₆	—	2	4	no	—
37	C₂×Q₁₆	—	2	4	no	—
38	C₂×Q₁₆	—	2	2	no	—
39	C₂×Q₁₆	—	2	4	no	—
40	C₂×Q₁₆	C₂³	1	4	yes	—
41	C₂×Q₁₆	C₂³	1	2	yes	—
42	C₂×SD₁₆	—	2	2	no	—
43	C₂×SD₁₆	—	2	4	no	—
44	C₂×D₈	C₂³	1	2	yes	—
45	C₂×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	4	no	—
48	#124	—	2	4	no	—
49	#123	—	4	4	no	—
50	C₂²⋊Q₈	—	4	4	no	—
51	#117	D₄	1	1	yes	characteristic
52	Q₃₂	—	2	4	no	—
53	Q₃₂	—	2	4	no	—
54	Q₃₂	—	2	2	no	—
55	Q₃₂	—	2	8	no	—
56	SD₃₂	—	2	4	no	—
57	SD₃₂	—	2	8	no	—
58	D₁₆	—	2	2	no	—
59	M₃₂	D₄	1	4	yes	—
60	C₂×C₁₆	D₄	1	4	yes	—
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	#97	D₄	1	2	yes	—
65	C₄×C₈	C₂³	1	2	yes	—
66	#41	—	4	4	no	—
67	#41	—	4	4	no	—
68	C₂×Q₈	—	2	4	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	—	4	2	no	—
71	C₂×Q₈	—	4	4	no	—
72	C₂×Q₈	—	4	4	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	4	2	no	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	—	4	2	no	—
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	Q₁₆	—	2	4	no	—
79	Q₁₆	—	2	2	no	—
80	Q₁₆	—	2	4	no	—
81	Q₁₆	—	2	2	no	—
82	Q₁₆	—	4	4	no	—
83	Q₁₆	—	4	8	no	—
84	SD₁₆	—	4	4	no	—
85	SD₁₆	—	4	8	no	—
86	D₈	—	2	2	no	—
87	D₈	—	2	2	no	—
88	M₁₆	D₈	1	4	yes	—
89	C₂×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₈	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄	—	4	8	no	—
95	C₄⋊C₄	D₈	1	2	yes	—
96	C₂²⋊C₄	—	4	8	no	—
97	C₂²⋊C₄	D₈	1	2	yes	—
98	C₄²	C₂×D₄	1	2	yes	—
99	C₁₆	C₂×D₄	1	2	yes	—
100	C₁₆	C₂×D₄	1	2	yes	—
101	C₁₆	C₂×D₄	1	4	yes	—
102	C₂³	—	4	2	no	—
103	C₂³	C₂×D₈	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	4	4	no	—
106	Q₈	—	4	4	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	#127	1	2	yes	—
117	C₂×C₄	C₂×D₈	1	4	yes	—
118	C₂×C₄	C₂×D₈	1	2	yes	—
119	C₂×C₄	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₈	#127	1	2	yes	—
122	C₈	C₂×D₈	1	4	yes	—
123	C₈	C₂²×D₄	1	1	yes	characteristic
124	C₈	C₂²×D₄	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	8	2	no	—
127	C₂²	—	8	4	no	—
128	C₂²	#493	1	2	yes	—
129	C₂²	C₂²×D₈	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	8	2	no	—
132	C₄	—	8	4	no	—
133	C₄	#493	1	2	yes	—
134	C₄	C₂×#127	1	1	yes	characteristic
135	C₄	C₂²×D₈	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	—	16	2	no	—
138	C₂	C₂×#493	1	1	yes	characteristic
139	C₂	#3417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#510

C₂

yes

maximal, complemented

C₂×#508

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂

yes

maximal, characteristic, complemented

#3156

C₂

yes

maximal, complemented

#2251

C₂

yes

maximal, complemented

#2249

C₂

yes

maximal, complemented

#2247

C₂

yes

maximal, complemented

#2108

C₂

yes

maximal, characteristic, complemented

C₂×#137

—

C₂×#135

C₂²

yes

—

C₂²×Q₁₆

C₂²

yes

—

#536

—

#510

—

#508

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²

yes

—

C₂×Q₃₂

C₂²

yes

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×Q₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×D₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#497

—

#495

C₂²

yes

—

#494

C₂²

yes

—

#405

C₂²

yes

characteristic

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₂×#41

—

C₂²×Q₈

—

#137

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

D₄

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

D₄

yes

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#97

D₄

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

D₈

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

—

100

C₁₆

C₂×D₄

yes

—

101

C₁₆

C₂×D₄

yes

—

102

C₂³

—

103

C₂³

C₂×D₈

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#127

yes

—

117

C₂×C₄

C₂×D₈

yes

—

118

C₂×C₄

C₂×D₈

yes

—

119

C₂×C₄

C₂×D₈

yes

characteristic

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₈

#127

yes

—

122

C₈

C₂×D₈

yes

—

123

C₈

C₂²×D₄

yes

characteristic

124

C₈

C₂²×D₄

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#493

yes

—

129

C₂²

C₂²×D₈

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#493

yes

—

134

C₄

C₂×#127

yes

characteristic

135

C₄

C₂²×D₈

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

C₂×#493

yes

characteristic

139

C₂

#3417

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂×#508

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#510	C₂	semidirect
3. C₂²×Q₃₂	C₂	semidirect
4. #3156	C₂	semidirect
5. #2251	C₂	semidirect
6. #2249	C₂	semidirect
7. #2247	C₂	semidirect
8. #2108	C₂	semidirect

H ≅ G/N

type

1. C₂×#510

C₂

semidirect

3. C₂²×Q₃₂

C₂

semidirect

4. #3156

C₂

semidirect

5. #2251

C₂

semidirect

6. #2249

C₂

semidirect

7. #2247

C₂

semidirect

8. #2108

C₂

semidirect

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

subgroup	index
C₂×#510	2
C₂×#508	2
C₂²×Q₃₂	2
#3156	2
#2251	2
#2249	2
#2247	2
#2108	2

subgroup

index

C₂²×Q₃₂

Subgroups — #33721

Conjugacy classes of subgroups

Complements

Maximal subgroups