Group #33499 — subgroups

Subgroups — #33499

Order 256.

226 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 179 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	#2960	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	2	yes	maximal, complemented
3	#2183	C₂	1	4	yes	maximal, complemented
4	Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
5	#2164	C₂	1	4	yes	maximal
6	#2163	C₂	1	1	yes	maximal, characteristic
7	#2162	C₂	1	1	yes	maximal, characteristic
8	#533	C₄	1	1	yes	characteristic
9	#446	C₄	1	2	yes	—
10	#446	C₄	1	2	yes	—
11	C₈×Q₈	C₂²	1	2	yes	—
12	C₈×Q₈	C₂²	1	2	yes	—
13	#424	C₄	1	2	yes	—
14	#423	C₄	1	1	yes	characteristic
15	#405	C₂²	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₄⋊C₁₆	C₂²	1	4	yes	—
20	C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₂²⋊C₁₆	C₂²	1	2	yes	—
23	#346	C₂²	1	2	yes	—
24	#346	C₂²	1	4	yes	—
25	C₄×C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₆	C₂²	1	4	yes	—
28	C₂×M₁₆	C₂×C₄	1	2	yes	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₄⋊Q₈	C₈	1	1	yes	characteristic
33	#125	C₈	1	2	yes	—
34	C₂²⋊Q₈	C₈	1	2	yes	—
35	C₂²⋊Q₈	C₈	1	2	yes	—
36	C₄×Q₈	C₂×C₄	1	2	yes	—
37	C₄×Q₈	C₂×C₄	1	2	yes	—
38	#117	C₂×C₄	1	2	yes	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂³	1	2	yes	—
41	C₂×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₆	C₂³	1	4	yes	—
43	C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₄⋊C₈	C₂×C₄	1	4	yes	—
47	#97	C₂×C₄	1	4	yes	—
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₈	C₂³	1	4	yes	—
52	C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×Q₈	C₂×C₈	1	2	yes	—
54	C₂×Q₈	C₂×C₈	1	2	yes	—
55	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
58	M₁₆	—	4	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₈	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄	C₂×C₈	1	2	yes	—
67	C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₄⋊C₄	C₂×C₈	1	4	yes	—
69	C₄⋊C₄	C₂×C₈	1	4	yes	—
70	C₄⋊C₄	C₂×C₈	1	2	yes	—
71	C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₂²⋊C₄	C₂×C₈	1	4	yes	—
73	C₄²	C₂×C₈	1	1	yes	characteristic
74	C₄²	C₂×C₈	1	1	yes	characteristic
75	C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₄²	C₂²×C₄	1	4	yes	—
78	C₁₆	—	2	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	2	4	no	—
81	C₂³	C₂²×C₈	1	1	yes	characteristic
82	Q₈	—	2	4	no	—
83	Q₈	—	4	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₈	1	4	yes	—
93	C₂×C₄	C₂²×C₈	1	4	yes	—
94	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	4	no	—
100	C₈	2₋¹⁺⁴	1	2	yes	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₂³×C₈	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	4	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	1	no	—
111	C₄	#504	1	2	yes	—
112	C₄	#519	1	1	yes	characteristic
113	C₄	#568	1	1	yes	characteristic
114	C₂	—	4	1	no	—
115	C₂	#2982	1	1	yes	characteristic
116	C₂	C₂×#504	1	1	yes	characteristic
117	C₂	#3407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#2183

C₂

yes

maximal, complemented

#2183

C₂

yes

maximal, complemented

Q₈×C₁₆

C₂

yes

maximal, complemented

#2164

C₂

yes

maximal

#2163

C₂

yes

maximal, characteristic

#2162

C₂

yes

maximal, characteristic

#533

C₄

yes

characteristic

#446

C₄

yes

—

#446

C₄

yes

—

C₈×Q₈

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

#424

C₄

yes

—

#423

C₄

yes

characteristic

#405

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

#346

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₈

yes

characteristic

C₄⋊Q₈

C₈

yes

characteristic

#125

C₈

yes

—

C₂²⋊Q₈

C₈

yes

—

C₂²⋊Q₈

C₈

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

yes

—

#117

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

C₂×C₈

yes

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

2₋¹⁺⁴

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂³×C₈

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#504

yes

—

112

C₄

#519

yes

characteristic

113

C₄

#568

yes

characteristic

114

C₂

—

115

C₂

#2982

yes

characteristic

116

C₂

C₂×#504

yes

characteristic

117

C₂

#3407

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₈

C₂⁴

commutator, socle

C₂²

C₂³×C₈

prime radical

C₂³

C₂²×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2183	C₂	semidirect
3. #2183	C₂	semidirect
4. Q₈×C₁₆	C₂	semidirect

H ≅ G/N

type

2. #2183

C₂

semidirect

3. #2183

C₂

semidirect

4. Q₈×C₁₆

C₂

semidirect

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

subgroup	index
#2960	2
#2183	2
Q₈×C₁₆	2
#2164	2
#2163	2
#2162	2

subgroup

index

Q₈×C₁₆

Subgroups — #33499

Conjugacy classes of subgroups

Complements

Maximal subgroups