Group #11605 — subgroups

Subgroups — C₂×Q₈⋊C₁₆ #11605

Order 256.

191 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	—	2	1	no	—
7	C₂²×C₁₆	—	2	1	no	—
8	C₈×Q₈	—	2	1	no	—
9	C₈×Q₈	—	2	1	no	—
10	C₈×Q₈	C₄	1	2	yes	—
11	C₈×Q₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈	—	2	1	no	—
13	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	1	no	—
17	C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	C₂²	1	2	yes	—
20	C₂²×Q₈	C₈	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₄×Q₈	—	2	1	no	—
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	C₈	1	2	yes	—
27	C₄×Q₈	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂×C₄²	—	2	1	no	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆	—	2	4	no	—
33	C₂×C₁₆	—	2	2	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	—	4	2	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	1	yes	characteristic
44	C₄×C₈	D₄	1	1	yes	characteristic
45	C₄×C₈	C₂³	1	1	yes	characteristic
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	C₁₆	1	4	yes	complemented
49	C₂×Q₈	C₂×C₈	1	2	yes	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄	M₁₆	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈	SD₁₆	1	2	yes	—
61	C₂×C₈	Q₁₆	1	2	yes	—
62	C₂×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₄²	—	2	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²	C₂×C₈	1	1	yes	characteristic
72	C₄²	M₁₆	1	1	yes	characteristic
73	C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₁₆	—	2	4	no	—
75	C₁₆	—	4	2	no	—
76	C₂³	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈	—	2	4	no	—
78	Q₈	—	2	2	no	—
79	Q₈	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄	#103	1	2	yes	—
89	C₂×C₄	#104	1	2	yes	—
90	C₂×C₄	C₂×C₁₆	1	2	yes	—
91	C₂×C₄	M₃₂	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	2	1	no	—
97	C₈	—	4	2	no	—
98	C₈	#103	1	2	yes	—
99	C₈	C₂×SD₁₆	1	1	yes	characteristic
100	C₈	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²	Q₈⋊C₈	1	2	yes	—
102	C₂²	C₂²⋊C₁₆	1	2	yes	—
103	C₂²	#350	1	2	yes	—
104	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	Q₈⋊C₈	1	2	yes	—
110	C₄	C₂²⋊C₁₆	1	2	yes	—
111	C₄	C₂×#103	1	1	yes	characteristic
112	C₄	C₂×#104	1	1	yes	characteristic
113	C₄	C₂²×C₁₆	1	1	yes	characteristic
114	C₄	C₂×M₃₂	1	1	yes	characteristic
115	C₂	Q₈⋊C₁₆	1	4	yes	direct product factor
116	C₂	C₂×Q₈⋊C₈	1	1	yes	characteristic
117	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
118	C₂	C₂×#350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic

Q₈⋊C₁₆

C₂

yes

maximal, direct product factor

C₂×C₄×Q₈

C₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₄

yes

—

C₈×Q₈

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂²

yes

—

C₂²×Q₈

C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₈

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₁₆

yes

complemented

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

M₁₆

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂²⋊C₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

C₂×C₁₆

yes

complemented

C₂×C₄

—

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

#103

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#103

yes

—

C₈

C₂×SD₁₆

yes

characteristic

100

C₈

C₂×Q₁₆

yes

characteristic

101

C₂²

Q₈⋊C₈

yes

—

102

C₂²

C₂²⋊C₁₆

yes

—

103

C₂²

#350

yes

—

104

C₂²

C₂×C₂²⋊C₈

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

Q₈⋊C₈

yes

—

110

C₄

C₂²⋊C₁₆

yes

—

111

C₄

C₂×#103

yes

characteristic

112

C₄

C₂×#104

yes

characteristic

113

C₄

C₂²×C₁₆

yes

characteristic

114

C₄

C₂×M₃₂

yes

characteristic

115

C₂

Q₈⋊C₁₆

yes

direct product factor

116

C₂

C₂×Q₈⋊C₈

yes

characteristic

117

C₂

C₂×C₂²⋊C₁₆

yes

characteristic

118

C₂

C₂×#350

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

D₄

commutator

C₄

C₂²×C₁₆

Frattini, cosocle

C₄×C₈

C₂³

socle, prime radical

C₂³

C₂²⋊C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. Q₈⋊C₁₆	C₂	direct (N × H)
48. C₂×Q₈	C₁₆	semidirect
79. Q₈	C₂×C₁₆	semidirect
115. C₂	Q₈⋊C₁₆	direct (N × H)

H ≅ G/N

type

4. Q₈⋊C₁₆

C₂

direct (N × H)

48. C₂×Q₈

C₁₆

semidirect

79. Q₈

C₂×C₁₆

semidirect

115. C₂

Q₈⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈	2
C₂×C₄⋊C₁₆	2
C₂×C₄×C₁₆	2
Q₈⋊C₁₆	2

subgroup

index

C₂×C₈×Q₈

C₂×C₄⋊C₁₆

C₂×C₄×C₁₆

Q₈⋊C₁₆

Subgroups — C₂×Q₈⋊C₁₆ #11605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Q8⋊C16 #11605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊C₁₆ #11605