Group #33828 — subgroups

Subgroups — #33828

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	#3175	C₂	1	2	yes	maximal, complemented
2	#2227	C₂	1	2	yes	maximal, complemented
3	#2224	C₂	1	2	yes	maximal, complemented
4	#2220	C₂	1	2	yes	maximal, complemented
5	#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	#2215	C₂	1	1	yes	maximal, characteristic
7	#2211	C₂	1	2	yes	maximal, complemented
8	#2210	C₂	1	2	yes	maximal, complemented
9	#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#542	—	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	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	2	yes	—
17	#495	C₂²	1	2	yes	—
18	#489	—	2	2	no	—
19	#468	—	2	2	no	—
20	#467	C₂²	1	2	yes	—
21	#466	C₂²	1	2	yes	—
22	#440	—	2	2	no	—
23	C₄×Q₁₆	C₂²	1	2	yes	—
24	C₄×D₈	C₂²	1	2	yes	—
25	#405	C₂²	1	1	yes	characteristic
26	#366	C₂²	1	1	yes	characteristic
27	#366	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	C₂×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈	—	2	2	no	—
41	#126	—	4	2	no	—
42	#124	—	2	2	no	—
43	#124	—	4	2	no	—
44	#123	—	4	2	no	—
45	C₂²⋊Q₈	—	2	2	no	—
46	C₂²⋊Q₈	—	4	2	no	—
47	#121	—	2	2	no	—
48	C₄×Q₈	—	2	2	no	—
49	C₄×D₄	—	2	2	no	—
50	#117	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	C₂×C₁₆	—	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	—	2	2	no	—
60	#107	C₂³	1	2	yes	—
61	#107	C₂³	1	2	yes	—
62	#103	—	2	2	no	—
63	#103	—	2	2	no	—
64	#102	—	2	2	no	—
65	#102	—	2	2	no	—
66	#97	D₄	1	2	yes	—
67	C₄×C₈	C₂³	1	2	yes	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	4	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	4	2	no	—
72	C₂²×C₄	—	4	2	no	—
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	M₁₆	—	4	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	D₈	1	1	yes	characteristic
91	C₄⋊C₄	D₈	1	1	yes	characteristic
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	C₂²⋊C₄	—	4	2	no	—
97	C₂²⋊C₄	D₈	1	2	yes	—
98	C₄²	—	4	2	no	—
99	C₄²	C₂×D₄	1	2	yes	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	2	2	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	Q₈	—	16	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	16	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	16	2	no	—
117	C₂×C₄	C₂×D₈	1	2	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	—	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	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	C₂²×D₈	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	8	2	no	—
136	C₄	—	8	2	no	—
137	C₄	—	8	2	no	—
138	C₄	#537	1	1	yes	characteristic
139	C₄	#575	1	1	yes	characteristic
140	C₂	—	4	1	no	—
141	C₂	—	16	2	no	—
142	C₂	#3245	1	1	yes	characteristic
143	C₂	#3415	1	1	yes	characteristic
144	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3175

C₂

yes

maximal, complemented

#2227

C₂

yes

maximal, complemented

#2224

C₂

yes

maximal, complemented

#2220

C₂

yes

maximal, complemented

#2216

C₂

yes

maximal, characteristic, complemented

#2215

C₂

yes

maximal, characteristic

#2211

C₂

yes

maximal, complemented

#2210

C₂

yes

maximal, complemented

#2151

C₂

yes

maximal, characteristic, complemented

#542

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×SD₃₂

C₂²

yes

—

C₂×D₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#500

C₂²

yes

—

#495

C₂²

yes

—

#489

—

#468

—

#467

C₂²

yes

—

#466

C₂²

yes

—

#440

—

C₄×Q₁₆

C₂²

yes

—

C₄×D₈

C₂²

yes

—

#405

C₂²

yes

characteristic

#366

C₂²

yes

characteristic

#366

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

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

D₄

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

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

D₈

yes

characteristic

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

C₂×D₄

yes

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

—

104

C₂³

C₂×D₈

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

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

—

119

C₂×C₄

C₂×D₈

yes

characteristic

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

—

131

C₂²

C₂²×D₈

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#537

yes

characteristic

139

C₄

#575

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#3245

yes

characteristic

143

C₂

#3415

yes

characteristic

144

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#2215	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

#2215

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. #3175	C₂	semidirect
2. #2227	C₂	semidirect
3. #2224	C₂	semidirect
4. #2220	C₂	semidirect
5. #2216	C₂	semidirect
7. #2211	C₂	semidirect
8. #2210	C₂	semidirect
9. #2151	C₂	semidirect

H ≅ G/N

type

1. #3175

C₂

semidirect

2. #2227

C₂

semidirect

3. #2224

C₂

semidirect

4. #2220

C₂

semidirect

5. #2216

C₂

semidirect

7. #2211

C₂

semidirect

8. #2210

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
#3175	2
#2227	2
#2224	2
#2220	2
#2216	2
#2215	2
#2211	2
#2210	2
#2151	2

subgroup

index

Subgroups — #33828

Conjugacy classes of subgroups

Complements

Maximal subgroups