Group #8169 — subgroups

Subgroups — C₄×Q₈⋊C₈ #8169

Order 256.

327 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	2	yes	maximal
5	#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₄	1	2	yes	—
7	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	—	2	2	no	—
10	C₂×C₄×C₈	—	2	1	no	—
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₄×C₄⋊C₄	—	2	1	no	—
13	C₄×C₄⋊C₄	—	2	2	no	—
14	C₄×C₄⋊C₄	C₄	1	2	yes	—
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₄³	—	2	1	no	—
17	C₄³	C₂²	1	1	yes	characteristic
18	#336	—	2	1	no	—
19	Q₈⋊C₈	C₄	1	8	yes	direct product factor
20	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₄×Q₈	—	2	4	no	—
24	C₄×Q₈	—	2	2	no	—
25	C₄×Q₈	C₈	1	8	yes	complemented
26	C₄×Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²	—	2	2	no	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	—	2	2	no	—
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	C₂×C₄	1	2	yes	—
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₄²	D₄	1	2	yes	—
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₄×C₈	—	2	8	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	4	2	no	—
45	C₄×C₈	C₂×C₄	1	4	yes	—
46	#95	—	2	2	no	—
47	#95	—	2	1	no	—
48	C₂×Q₈	—	2	1	no	—
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	C₄²	1	2	yes	—
51	C₂×Q₈	C₂×C₈	1	4	yes	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈	—	2	8	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₄⋊C₄	—	2	8	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	C₄²	1	2	yes	—
70	C₄⋊C₄	C₂×C₈	1	4	yes	—
71	C₄²	—	2	8	no	—
72	C₄²	—	2	4	no	—
73	C₄²	—	2	4	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	4	4	no	—
76	C₄²	—	4	2	no	—
77	C₄²	C₂×C₈	1	4	yes	—
78	C₄²	M₁₆	1	4	yes	—
79	C₄²	SD₁₆	1	2	yes	—
80	C₄²	Q₁₆	1	2	yes	—
81	C₄²	C₂²×C₄	1	2	yes	—
82	C₄²	#41	1	2	yes	—
83	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈	—	2	2	no	—
85	Q₈	—	2	4	no	—
86	Q₈	C₄×C₈	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄	#103	1	4	yes	—
100	C₂×C₄	#104	1	4	yes	—
101	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄	#117	1	1	yes	characteristic
103	C₂×C₄	C₄×D₄	1	2	yes	—
104	C₂×C₄	C₄×D₄	1	2	yes	—
105	C₂×C₄	C₂²×C₈	1	2	yes	—
106	C₂×C₄	C₂×M₁₆	1	2	yes	—
107	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄	#135	1	2	yes	—
110	C₈	—	2	8	no	—
111	C₈	—	4	4	no	—
112	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²	C₂×C₂²⋊C₈	1	2	yes	—
114	C₂²	C₂×#103	1	1	yes	characteristic
115	C₂²	#416	1	1	yes	characteristic
116	C₂²	C₂×#104	1	2	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	4	2	no	—
122	C₄	Q₈⋊C₈	1	8	yes	direct product factor
123	C₄	C₂×C₄×C₈	1	1	yes	characteristic
124	C₄	C₄×M₁₆	1	1	yes	characteristic
125	C₄	#431	1	2	yes	—
126	C₄	C₄×SD₁₆	1	2	yes	—
127	C₄	C₄×Q₁₆	1	2	yes	—
128	C₂	C₂×Q₈⋊C₈	1	2	yes	—
129	C₂	#1476	1	2	yes	—
130	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
131	C₂	C₄×#104	1	1	yes	characteristic
132	C₂	C₄×#103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×Q₈

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

#1279

C₂

yes

maximal

C₂×C₄×Q₈

C₄

yes

—

C₂×C₄×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₄

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

C₄³

—

C₄³

C₂²

yes

characteristic

#336

—

Q₈⋊C₈

C₄

yes

direct product factor

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₈

yes

complemented

C₄×Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄²

yes

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄²

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

C₄×C₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

#103

yes

—

100

C₂×C₄

#104

yes

—

101

C₂×C₄

C₂×C₄²

yes

characteristic

102

C₂×C₄

#117

yes

characteristic

103

C₂×C₄

C₄×D₄

yes

—

104

C₂×C₄

C₄×D₄

yes

—

105

C₂×C₄

C₂²×C₈

yes

—

106

C₂×C₄

C₂×M₁₆

yes

—

107

C₂×C₄

C₂×SD₁₆

yes

characteristic

108

C₂×C₄

C₂×Q₁₆

yes

characteristic

109

C₂×C₄

#135

yes

—

110

C₈

—

111

C₈

—

112

C₂²

C₄×C₂²⋊C₄

yes

characteristic

113

C₂²

C₂×C₂²⋊C₈

yes

—

114

C₂²

C₂×#103

yes

characteristic

115

C₂²

#416

yes

characteristic

116

C₂²

C₂×#104

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

Q₈⋊C₈

yes

direct product factor

123

C₄

C₂×C₄×C₈

yes

characteristic

124

C₄

C₄×M₁₆

yes

characteristic

125

C₄

#431

yes

—

126

C₄

C₄×SD₁₆

yes

—

127

C₄

C₄×Q₁₆

yes

—

128

C₂

C₂×Q₈⋊C₈

yes

—

129

C₂

#1476

yes

—

130

C₂

C₄×C₂²⋊C₈

yes

characteristic

131

C₂

C₄×#104

yes

characteristic

132

C₂

C₄×#103

yes

characteristic

	subgroup	G/H
center	C₂×C₄²	D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄²

D₄

commutator

C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
19. Q₈⋊C₈	C₄	direct (N × H)
25. C₄×Q₈	C₈	semidirect
86. Q₈	C₄×C₈	semidirect
122. C₄	Q₈⋊C₈	direct (N × H)

H ≅ G/N

type

19. Q₈⋊C₈

C₄

direct (N × H)

25. C₄×Q₈

C₈

semidirect

86. Q₈

C₄×C₈

semidirect

122. C₄

Q₈⋊C₈

direct (N × H)

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

subgroup	index
C₄²×Q₈	2
C₄×C₄⋊C₈	2
C₄²×C₈	2
C₂×Q₈⋊C₈	2
#1279	2

subgroup

index

C₄²×Q₈

C₄×C₄⋊C₈

C₄²×C₈

C₂×Q₈⋊C₈

#1279

Subgroups — C₄×Q₈⋊C₈ #8169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×Q8⋊C8 #8169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈⋊C₈ #8169