Group #33954 — subgroups

Subgroups — #33954

Order 256.

225 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3383	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#2255	C₂	1	2	yes	maximal
4	#2254	C₂	1	1	yes	maximal, characteristic
5	#2229	C₂	1	2	yes	maximal
6	#2227	C₂	1	1	yes	maximal, characteristic
7	#2225	C₂	1	2	yes	maximal
8	#2225	C₂	1	1	yes	maximal, characteristic
9	#2183	C₂	1	1	yes	maximal, characteristic
10	#2176	C₂	1	2	yes	maximal
11	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
12	Q₈²	—	2	1	no	—
13	#557	—	2	1	no	—
14	C₂×Q₃₂	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	2	yes	—
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#499	C₂²	1	2	yes	—
19	#479	—	2	2	no	—
20	#477	—	2	1	no	—
21	#477	—	2	2	no	—
22	#475	—	2	1	no	—
23	C₈×Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆	C₂²	1	2	yes	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆	C₂²	1	2	yes	—
28	#367	C₂²	1	2	yes	—
29	#367	C₂²	1	2	yes	—
30	#367	C₂²	1	2	yes	—
31	#366	C₂²	1	1	yes	characteristic
32	#366	C₂²	1	2	yes	—
33	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆	C₂²	1	2	yes	—
35	#358	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	#346	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	—	2	2	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	4	1	no	—
47	C₄⋊Q₈	—	4	1	no	—
48	C₄⋊Q₈	—	4	2	no	—
49	#125	—	2	2	no	—
50	#125	—	4	2	no	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×Q₈	—	4	1	no	—
56	C₄×Q₈	—	4	1	no	—
57	C₄×Q₈	D₄	1	1	yes	characteristic
58	Q₃₂	Q₈	1	4	yes	complemented
59	C₂×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	C₂³	1	1	yes	characteristic
66	#107	C₂³	1	1	yes	characteristic
67	#107	C₂³	1	2	yes	—
68	#107	C₂³	1	2	yes	—
69	#106	—	2	2	no	—
70	C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₄⋊C₈	D₄	1	2	yes	—
72	#103	—	2	2	no	—
73	#103	—	2	2	no	—
74	#103	—	2	1	no	—
75	#103	—	2	1	no	—
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	2	yes	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	D₈	1	1	yes	characteristic
84	Q₁₆	—	2	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	C₂×Q₈	1	2	yes	—
87	Q₁₆	C₂×Q₈	1	2	yes	—
88	C₂×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	D₈	1	1	yes	characteristic
105	C₄⋊C₄	D₈	1	2	yes	—
106	C₄²	—	4	2	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	4	2	no	—
109	C₄²	—	4	1	no	—
110	C₄²	C₂×D₄	1	2	yes	—
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	2	no	—
114	C₁₆	C₂×Q₈	1	2	yes	—
115	Q₈	—	2	2	no	—
116	Q₈	—	2	2	no	—
117	Q₈	—	2	2	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	8	2	no	—
121	Q₈	—	16	2	no	—
122	Q₈	—	16	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×D₈	1	2	yes	—
130	C₂×C₄	C₂×D₈	1	2	yes	—
131	C₂×C₄	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	2	1	no	—
136	C₈	C₂²×Q₈	1	1	yes	characteristic
137	C₈	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂²	C₂²×D₈	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	2	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	1	no	—
146	C₄	—	8	2	no	—
147	C₄	—	8	1	no	—
148	C₄	#510	1	2	yes	—
149	C₄	D₄×Q₈	1	1	yes	characteristic
150	C₄	#578	1	1	yes	characteristic
151	C₂	Q₈×D₈	1	1	yes	characteristic
152	C₂	C₂×#510	1	1	yes	characteristic
153	C₂	#3416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3383

C₂

yes

maximal, characteristic

Q₈×Q₁₆

C₂

yes

maximal, characteristic

#2255

C₂

yes

maximal

#2254

C₂

yes

maximal, characteristic

#2229

C₂

yes

maximal

#2227

C₂

yes

maximal, characteristic

#2225

C₂

yes

maximal

#2225

C₂

yes

maximal, characteristic

#2183

C₂

yes

maximal, characteristic

#2176

C₂

yes

maximal

C₄×Q₃₂

C₂

yes

maximal, characteristic

Q₈²

—

#557

—

C₂×Q₃₂

C₂²

yes

characteristic

#500

C₂²

yes

—

#500

C₂²

yes

characteristic

#500

C₂²

yes

characteristic

#499

C₂²

yes

—

—

—

—

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

#367

C₂²

yes

—

#367

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

#358

C₂²

yes

characteristic

#358

C₂²

yes

characteristic

#358

C₂²

yes

—

#358

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂³

yes

characteristic

C₂×Q₁₆

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

Q₃₂

Q₈

yes

complemented

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

—

—

C₂³

yes

characteristic

#107

C₂³

yes

characteristic

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

—

—

—

—

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×Q₈

yes

—

Q₁₆

C₂×Q₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

D₈

yes

characteristic

105

C₄⋊C₄

D₈

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×D₄

yes

—

111

C₄²

C₂×D₄

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

C₂×Q₈

yes

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×D₈

yes

—

130

C₂×C₄

C₂×D₈

yes

—

131

C₂×C₄

C₂×D₈

yes

characteristic

132

C₂×C₄

C₂×D₈

yes

characteristic

133

C₂×C₄

C₂²×D₄

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

C₂²×Q₈

yes

characteristic

137

C₈

2₋¹⁺⁴

yes

characteristic

138

C₂²

C₂²×D₈

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#510

yes

—

149

C₄

D₄×Q₈

yes

characteristic

150

C₄

#578

yes

characteristic

151

C₂

Q₈×D₈

yes

characteristic

152

C₂

C₂×#510

yes

characteristic

153

C₂

#3416

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
58. Q₃₂	Q₈	semidirect

H ≅ G/N

type

58. Q₃₂

Q₈

semidirect

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

subgroup	index
#3383	2
Q₈×Q₁₆	2
#2255	2
#2254	2
#2229	2
#2227	2
#2225	2
#2183	2
#2176	2
C₄×Q₃₂	2

subgroup

index

Q₈×Q₁₆

C₄×Q₃₂

Subgroups — #33954

Conjugacy classes of subgroups

Complements

Maximal subgroups