Group #33826 — subgroups

Subgroups — #33826

Order 256.

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

normal subgroups only

This group has 244 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	#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	#3176	C₂	1	1	yes	maximal, characteristic
3	#2229	C₂	1	2	yes	maximal, complemented
4	#2228	C₂	1	2	yes	maximal, complemented
5	#2220	C₂	1	2	yes	maximal, complemented
6	#2217	C₂	1	2	yes	maximal, complemented
7	#2208	C₂	1	2	yes	maximal, complemented
8	#2207	C₂	1	2	yes	maximal, complemented
9	#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#541	—	2	1	no	—
11	#540	—	2	1	no	—
12	C₂×SD₃₂	C₂²	1	2	yes	—
13	C₂×SD₃₂	C₂²	1	2	yes	—
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	#499	C₂²	1	1	yes	characteristic
17	#499	C₂²	1	1	yes	characteristic
18	#494	C₂²	1	1	yes	characteristic
19	#493	C₂²	1	1	yes	characteristic
20	#491	—	2	1	no	—
21	#490	—	2	1	no	—
22	#470	—	2	1	no	—
23	#469	—	2	1	no	—
24	#467	C₂²	1	2	yes	—
25	#466	C₂²	1	2	yes	—
26	#442	—	2	1	no	—
27	#441	—	2	1	no	—
28	C₄×Q₁₆	C₂²	1	2	yes	—
29	C₄×D₈	C₂²	1	2	yes	—
30	#405	C₂²	1	1	yes	characteristic
31	#367	C₂²	1	2	yes	—
32	#366	C₂²	1	2	yes	—
33	C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₄⋊C₁₆	C₂²	1	2	yes	—
35	#358	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	C₂×Q₁₆	—	2	1	no	—
40	C₂×Q₁₆	C₂³	1	2	yes	—
41	C₂×D₈	—	2	1	no	—
42	C₂×D₈	C₂³	1	2	yes	—
43	C₂×M₁₆	D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈	—	2	1	no	—
46	#127	—	2	1	no	—
47	#126	—	4	2	no	—
48	#125	—	2	1	no	—
49	#125	—	2	1	no	—
50	#125	—	4	2	no	—
51	#123	—	4	2	no	—
52	C₂²⋊Q₈	—	2	2	no	—
53	#121	—	2	2	no	—
54	#121	—	4	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×D₄	—	2	2	no	—
57	#117	D₄	1	1	yes	characteristic
58	SD₃₂	—	4	2	no	—
59	SD₃₂	—	4	2	no	—
60	M₃₂	—	4	1	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	2	yes	—
64	#107	C₂³	1	2	yes	—
65	#107	C₂³	1	2	yes	—
66	#106	—	2	1	no	—
67	#106	—	2	1	no	—
68	#103	—	2	2	no	—
69	#103	—	2	2	no	—
70	#102	—	2	2	no	—
71	#102	—	2	2	no	—
72	#97	D₄	1	1	yes	characteristic
73	#97	D₄	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈	C₂³	1	1	yes	characteristic
76	C₂×Q₈	—	2	2	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	Q₁₆	—	2	2	no	—
83	Q₁₆	—	2	2	no	—
84	Q₁₆	—	4	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	2	2	no	—
87	D₈	—	4	2	no	—
88	M₁₆	—	4	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	D₈	1	2	yes	—
103	C₂²⋊C₄	—	4	2	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	D₈	1	2	yes	—
107	C₄²	—	4	2	no	—
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×D₈	1	1	yes	characteristic
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	16	2	no	—
119	D₄	—	16	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	16	2	no	—
126	C₂×C₄	C₂×D₈	1	2	yes	—
127	C₂×C₄	C₂×D₈	1	2	yes	—
128	C₂×C₄	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	2	1	no	—
133	C₈	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	8	2	no	—
138	C₂²	—	8	2	no	—
139	C₂²	—	8	2	no	—
140	C₂²	C₂²×D₈	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	2	no	—
146	C₄	—	8	2	no	—
147	C₄	#537	1	1	yes	characteristic
148	C₄	#575	1	1	yes	characteristic
149	C₂	—	4	1	no	—
150	C₂	—	16	2	no	—
151	C₂	#3245	1	1	yes	characteristic
152	C₂	#3416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3177

C₂

yes

maximal, characteristic, complemented

#3176

C₂

yes

maximal, characteristic

#2229

C₂

yes

maximal, complemented

#2228

C₂

yes

maximal, complemented

#2220

C₂

yes

maximal, complemented

#2217

C₂

yes

maximal, complemented

#2208

C₂

yes

maximal, complemented

#2207

C₂

yes

maximal, complemented

#2151

C₂

yes

maximal, characteristic, complemented

#541

—

#540

—

C₂×SD₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#499

C₂²

yes

characteristic

#499

C₂²

yes

characteristic

#494

C₂²

yes

characteristic

#493

C₂²

yes

—

—

—

—

C₂²

yes

—

#466

C₂²

yes

—

#442

—

#441

—

C₄×Q₁₆

C₂²

yes

—

C₄×D₈

C₂²

yes

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#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

—

C₂×Q₁₆

C₂³

yes

—

C₂×D₈

—

C₂×D₈

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

—

—

—

—

—

—

C₂²⋊Q₈

—

#121

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

D₄

yes

characteristic

SD₃₂

—

SD₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

#97

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

D₈

yes

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

D₈

yes

—

107

C₄²

—

108

C₄²

C₂×D₄

yes

characteristic

109

C₄²

C₂×D₄

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₂³

—

113

C₂³

C₂×D₈

yes

characteristic

114

Q₈

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×D₈

yes

—

127

C₂×C₄

C₂×D₈

yes

—

128

C₂×C₄

C₂×D₈

yes

characteristic

129

C₂×C₄

C₂²×D₄

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

H(5,𝔽₂)

yes

characteristic

134

C₈

2₋¹⁺⁴

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×D₈

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#537

yes

characteristic

148

C₄

#575

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#3245

yes

characteristic

152

C₂

#3416

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3176	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

#3176

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. #3177	C₂	semidirect
3. #2229	C₂	semidirect
4. #2228	C₂	semidirect
5. #2220	C₂	semidirect
6. #2217	C₂	semidirect
7. #2208	C₂	semidirect
8. #2207	C₂	semidirect
9. #2151	C₂	semidirect

H ≅ G/N

type

1. #3177

C₂

semidirect

3. #2229

C₂

semidirect

4. #2228

C₂

semidirect

5. #2220

C₂

semidirect

6. #2217

C₂

semidirect

7. #2208

C₂

semidirect

8. #2207

C₂

semidirect

9. #2151

C₂

semidirect

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

subgroup	index
#3177	2
#3176	2
#2229	2
#2228	2
#2220	2
#2217	2
#2208	2
#2207	2
#2151	2

subgroup

index

Subgroups — #33826

Conjugacy classes of subgroups

Complements

Maximal subgroups