Group #22720 — subgroups

Subgroups — Q₈×C₄⋊C₈ #22720

Order 256.

448 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 301 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	2	yes	maximal
2	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	#1939	C₂	1	6	yes	maximal
4	#1842	C₂	1	3	yes	maximal
5	C₄×C₄⋊C₈	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
9	C₈×Q₈	—	2	8	no	—
10	C₂×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₄⋊C₈	C₂²	1	6	yes	—
12	C₂×C₄⋊C₈	C₂²	1	6	yes	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	6	yes	—
15	C₄×C₄⋊C₄	C₄	1	3	yes	—
16	C₄×C₄⋊C₄	C₄	1	3	yes	—
17	C₄×C₄⋊C₄	C₂²	1	3	yes	—
18	C₄³	C₂²	1	3	yes	—
19	#336	C₂²	1	6	yes	—
20	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	6	yes	—
22	C₂²×C₈	C₂³	1	2	yes	—
23	C₄×Q₈	—	2	4	no	—
24	C₄×Q₈	C₈	1	8	yes	complemented
25	C₄×Q₈	D₄	1	4	yes	—
26	C₄×Q₈	Q₈	1	4	yes	—
27	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
28	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₄²	C₂×C₄	1	3	yes	—
31	C₂×C₄²	C₂×C₄	1	3	yes	—
32	C₂×C₄²	C₂×C₄	1	3	yes	—
33	C₂×C₄²	C₂×C₄	1	3	yes	—
34	C₂×C₄²	C₂³	1	3	yes	—
35	C₂×C₄²	C₂³	1	3	yes	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	12	no	—
38	C₄⋊C₈	—	2	6	no	—
39	C₄⋊C₈	—	4	6	no	—
40	C₄⋊C₈	Q₈	1	4	yes	direct product factor
41	C₄×C₈	—	2	12	no	—
42	#95	C₂×C₄	1	6	yes	—
43	C₂×Q₈	C₄⋊C₄	1	4	yes	—
44	C₂×Q₈	C₂×C₈	1	4	yes	—
45	C₂×Q₈	M₁₆	1	4	yes	—
46	C₂²×C₄	C₂²×C₄	1	3	yes	—
47	C₂²×C₄	C₂²×C₄	1	3	yes	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂²×C₄	1	3	yes	—
50	C₂²×C₄	C₂²×C₄	1	3	yes	—
51	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄	C₂⁴	1	1	yes	characteristic
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	6	no	—
56	C₂×C₈	—	2	6	no	—
57	C₂×C₈	—	2	6	no	—
58	C₂×C₈	C₂×Q₈	1	4	yes	—
59	C₄⋊C₄	—	2	6	no	—
60	C₄⋊C₄	C₄⋊C₄	1	12	yes	—
61	C₄⋊C₄	C₂×C₈	1	12	yes	—
62	C₄²	—	2	6	no	—
63	C₄²	—	2	6	no	—
64	C₄²	—	2	3	no	—
65	C₄²	—	2	3	no	—
66	C₄²	—	4	3	no	—
67	C₄²	C₂×C₈	1	12	yes	—
68	C₄²	C₂×D₄	1	6	yes	—
69	C₄²	C₂×Q₈	1	6	yes	—
70	C₄²	C₂×Q₈	1	2	yes	—
71	C₄²	#41	1	2	yes	—
72	C₂³	C₂³×C₄	1	1	yes	characteristic
73	Q₈	C₄⋊C₈	1	16	yes	direct product factor
74	C₂×C₄	—	2	3	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄	C₄×Q₈	1	2	yes	—
86	C₂×C₄	C₄×Q₈	1	2	yes	—
87	C₂×C₄	C₂²×C₈	1	6	yes	—
88	C₂×C₄	C₂²×C₈	1	6	yes	—
89	C₂×C₄	C₂²×C₈	1	2	yes	—
90	C₂×C₄	C₂×M₁₆	1	6	yes	—
91	C₂×C₄	#131	1	2	yes	—
92	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄	C₂×#41	1	1	yes	characteristic
96	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
98	C₈	—	2	4	no	—
99	C₈	—	4	6	no	—
100	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂²	#520	1	1	yes	characteristic
103	C₂²	C₂³×C₈	1	1	yes	characteristic
104	C₂²	C₂²×M₁₆	1	1	yes	characteristic
105	C₂²	C₂×#131	1	1	yes	characteristic
106	C₂²	#568	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	2	2	no	—
109	C₄	—	2	6	no	—
110	C₄	—	4	3	no	—
111	C₄	C₂×C₄⋊C₈	1	12	yes	—
112	C₄	C₈×Q₈	1	4	yes	complemented
113	C₄	D₄×Q₈	1	1	yes	characteristic
114	C₄	#550	1	1	yes	characteristic
115	C₄	#557	1	1	yes	characteristic
116	C₄	Q₈²	1	1	yes	characteristic
117	C₂	Q₈×C₄⋊C₄	1	1	yes	characteristic
118	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic
119	C₂	#2906	1	1	yes	characteristic
120	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
121	C₂	Q₈×M₁₆	1	1	yes	characteristic
122	C₂	#2988	1	1	yes	characteristic
123	C₂	#2991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal

C₄²×Q₈

C₂

yes

maximal, characteristic

#1939

C₂

yes

maximal

#1842

C₂

yes

maximal

C₄×C₄⋊C₈

C₂

yes

maximal

C₂×C₄×Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₄×C₄⋊C₄

C₄

yes

—

C₄×C₄⋊C₄

C₄

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

—

#336

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄×Q₈

—

C₄×Q₈

C₈

yes

complemented

C₄×Q₈

D₄

yes

—

C₄×Q₈

Q₈

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Q₈

yes

direct product factor

C₄×C₈

—

#95

C₂×C₄

yes

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂×Q₈

C₂×C₈

yes

—

C₂×Q₈

M₁₆

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄⋊C₄

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂³×C₄

yes

characteristic

Q₈

C₄⋊C₈

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×Q₈

yes

characteristic

C₂×C₄

C₂²×Q₈

yes

characteristic

C₂×C₄

C₂×#41

yes

characteristic

C₂×C₄

H(5,𝔽₂)

yes

characteristic

C₂×C₄

2₋¹⁺⁴

yes

characteristic

C₈

—

C₈

—

100

C₂²

C₂²×C₄⋊C₄

yes

characteristic

101

C₂²

C₂×C₄×Q₈

yes

characteristic

102

C₂²

#520

yes

characteristic

103

C₂²

C₂³×C₈

yes

characteristic

104

C₂²

C₂²×M₁₆

yes

characteristic

105

C₂²

C₂×#131

yes

characteristic

106

C₂²

#568

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

C₂×C₄⋊C₈

yes

—

112

C₄

C₈×Q₈

yes

complemented

113

C₄

D₄×Q₈

yes

characteristic

114

C₄

#550

yes

characteristic

115

C₄

#557

yes

characteristic

116

C₄

Q₈²

yes

characteristic

117

C₂

Q₈×C₄⋊C₄

yes

characteristic

118

C₂

C₂²×C₄⋊C₈

yes

characteristic

119

C₂

#2906

yes

characteristic

120

C₂

C₂×C₈×Q₈

yes

characteristic

121

C₂

Q₈×M₁₆

yes

characteristic

122

C₂

#2988

yes

characteristic

123

C₂

#2991

yes

characteristic

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₄

C₂⁴

commutator

C₂²

C₂³×C₈

socle, prime radical

C₂³

C₂³×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
24. C₄×Q₈	C₈	semidirect
40. C₄⋊C₈	Q₈	direct (N × H)
73. Q₈	C₄⋊C₈	direct (N × H)
112. C₄	C₈×Q₈	semidirect

H ≅ G/N

type

24. C₄×Q₈

C₈

semidirect

40. C₄⋊C₈

Q₈

direct (N × H)

73. Q₈

C₄⋊C₈

direct (N × H)

112. C₄

C₈×Q₈

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
C₄²×Q₈	2
#1939	2
#1842	2
C₄×C₄⋊C₈	2

subgroup

index

C₂×C₈×Q₈

C₄²×Q₈

#1939

#1842

C₄×C₄⋊C₈

Subgroups — Q₈×C₄⋊C₈ #22720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C4⋊C8 #22720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄⋊C₈ #22720