Group #33822 — subgroups

Subgroups — #33822

Order 256.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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	#3168	C₂	1	2	yes	maximal, complemented
2	#2229	C₂	1	2	yes	maximal, complemented
3	#2228	C₂	1	2	yes	maximal, complemented
4	#2220	C₂	1	2	yes	maximal, complemented
5	#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	#2218	C₂	1	1	yes	maximal, characteristic
7	#2211	C₂	1	2	yes	maximal, complemented
8	#2210	C₂	1	2	yes	maximal, complemented
9	#2150	C₂	1	1	yes	maximal, characteristic, complemented
10	#529	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂	C₂²	1	2	yes	—
13	C₂×D₁₆	C₂²	1	1	yes	characteristic
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	#499	C₂²	1	2	yes	—
17	#495	C₂²	1	2	yes	—
18	#488	—	2	2	no	—
19	#467	C₂²	1	2	yes	—
20	#466	C₂²	1	2	yes	—
21	#465	—	2	2	no	—
22	C₄×Q₁₆	C₂²	1	2	yes	—
23	C₄×SD₁₆	—	2	2	no	—
24	C₄×D₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#367	C₂²	1	1	yes	characteristic
27	#367	C₂²	1	1	yes	characteristic
28	#366	C₂²	1	2	yes	—
29	C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄⋊C₁₆	C₂²	1	2	yes	—
31	#358	C₂²	1	2	yes	—
32	#358	C₂²	1	2	yes	—
33	#357	C₂²	1	2	yes	—
34	#357	C₂²	1	2	yes	—
35	C₂×Q₁₆	C₂³	1	2	yes	—
36	C₂×SD₁₆	—	2	2	no	—
37	C₂×D₈	C₂³	1	2	yes	—
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	#126	—	4	2	no	—
41	#125	—	2	2	no	—
42	#124	—	2	2	no	—
43	#123	—	4	2	no	—
44	C₂²⋊Q₈	—	2	2	no	—
45	#121	—	2	2	no	—
46	C₄×Q₈	—	2	2	no	—
47	C₄×D₄	—	2	2	no	—
48	C₄×D₄	—	4	2	no	—
49	#117	—	4	2	no	—
50	C₂×C₄²	D₄	1	1	yes	characteristic
51	Q₃₂	—	4	1	no	—
52	SD₃₂	—	4	2	no	—
53	D₁₆	—	4	1	no	—
54	M₃₂	—	4	1	no	—
55	M₃₂	—	4	1	no	—
56	C₂×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₆	C₂³	1	1	yes	characteristic
59	#107	C₂³	1	2	yes	—
60	#107	C₂³	1	2	yes	—
61	#106	—	2	2	no	—
62	#103	—	2	2	no	—
63	#103	—	2	2	no	—
64	#102	—	2	2	no	—
65	#102	—	2	2	no	—
66	C₄×C₈	D₄	1	2	yes	—
67	C₄×C₈	C₂³	1	2	yes	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	4	2	no	—
71	C₂²×C₄	—	4	2	no	—
72	C₂²×C₄	D₈	1	2	yes	—
73	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
74	Q₁₆	—	2	2	no	—
75	Q₁₆	—	2	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	2	2	no	—
78	D₈	—	2	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₈	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₂²⋊C₄	—	4	2	no	—
92	C₂²⋊C₄	—	4	2	no	—
93	C₂²⋊C₄	—	4	2	no	—
94	C₂²⋊C₄	—	4	2	no	—
95	C₂²⋊C₄	—	4	2	no	—
96	C₄²	—	4	2	no	—
97	C₄²	D₈	1	1	yes	characteristic
98	C₄²	D₈	1	1	yes	characteristic
99	C₄²	C₂×D₄	1	2	yes	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	2	no	—
102	C₁₆	—	2	1	no	—
103	C₂³	—	4	2	no	—
104	C₂³	C₂×D₈	1	1	yes	characteristic
105	Q₈	—	4	2	no	—
106	Q₈	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	8	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	8	4	no	—
117	C₂×C₄	C₂×D₈	1	2	yes	—
118	C₂×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈	1	2	yes	—
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	2	no	—
130	C₂²	C₂²×D₈	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	8	2	no	—
136	C₄	—	8	2	no	—
137	C₄	#537	1	1	yes	characteristic
138	C₄	#575	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	16	2	no	—
141	C₂	#3245	1	1	yes	characteristic
142	C₂	#3414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3168

C₂

yes

maximal, complemented

#2229

C₂

yes

maximal, complemented

#2228

C₂

yes

maximal, complemented

#2220

C₂

yes

maximal, complemented

#2219

C₂

yes

maximal, characteristic, complemented

#2218

C₂

yes

maximal, characteristic

#2211

C₂

yes

maximal, complemented

#2210

C₂

yes

maximal, complemented

#2150

C₂

yes

maximal, characteristic, complemented

#529

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×SD₃₂

C₂²

yes

—

C₂×D₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

#499

C₂²

yes

—

#495

C₂²

yes

—

#488

—

#467

C₂²

yes

—

#466

C₂²

yes

—

#465

—

C₄×Q₁₆

C₂²

yes

—

C₄×SD₁₆

—

C₄×D₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#367

C₂²

yes

characteristic

#367

C₂²

yes

characteristic

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

#357

C₂²

yes

—

#357

C₂²

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂×SD₁₆

—

C₂×D₈

C₂³

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

—

—

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

D₄

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

#107

C₂³

yes

—

—

—

—

—

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₈

yes

characteristic

C₄²

D₈

yes

characteristic

C₄²

C₂×D₄

yes

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

—

104

C₂³

C₂×D₈

yes

characteristic

105

Q₈

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₈

yes

—

118

C₂×C₄

C₂×D₈

yes

characteristic

119

C₂×C₄

C₂×D₈

yes

—

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

H(5,𝔽₂)

yes

characteristic

124

C₈

2₋¹⁺⁴

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂²×D₈

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#537

yes

characteristic

138

C₄

#575

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#3245

yes

characteristic

142

C₂

#3414

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#2218	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

#2218

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3168	C₂	semidirect
2. #2229	C₂	semidirect
3. #2228	C₂	semidirect
4. #2220	C₂	semidirect
5. #2219	C₂	semidirect
7. #2211	C₂	semidirect
8. #2210	C₂	semidirect
9. #2150	C₂	semidirect

H ≅ G/N

type

1. #3168

C₂

semidirect

2. #2229

C₂

semidirect

3. #2228

C₂

semidirect

4. #2220

C₂

semidirect

5. #2219

C₂

semidirect

7. #2211

C₂

semidirect

8. #2210

C₂

semidirect

9. #2150

C₂

semidirect

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

subgroup	index
#3168	2
#2229	2
#2228	2
#2220	2
#2219	2
#2218	2
#2211	2
#2210	2
#2150	2

subgroup

index

Subgroups — #33822

Conjugacy classes of subgroups

Complements

Maximal subgroups