Group #12598 — subgroups

Subgroups — C₄×C₂²⋊C₁₆ #12598

Order 256.

307 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆	C₂	1	2	yes	maximal, complemented
4	#1374	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²	C₄	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	4	yes	—
9	C₂×C₄×C₈	—	2	4	no	—
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆	C₄	1	8	yes	direct product factor
15	C₄×C₁₆	—	2	4	no	—
16	C₂³×C₄	C₈	1	2	yes	—
17	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	C₈	1	4	yes	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²	—	2	2	no	—
29	C₂×C₄²	—	2	2	no	—
30	C₂×C₄²	C₈	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆	—	2	4	no	—
34	C₂×C₁₆	—	2	4	no	—
35	C₂×C₁₆	C₂×C₄	1	8	yes	complemented
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	—	2	2	no	—
39	C₄×C₈	D₄	1	4	yes	—
40	C₂⁴	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄	—	2	4	no	—
42	C₂²×C₄	—	2	4	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₁₆	1	8	yes	complemented
48	C₂²×C₄	C₄²	1	2	yes	—
49	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄	C₂×C₈	1	2	yes	—
51	C₂²×C₄	C₂×C₈	1	2	yes	—
52	C₂²×C₄	C₂×C₈	1	2	yes	—
53	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	8	no	—
59	C₂×C₈	—	2	8	no	—
60	C₂×C₈	C₄²	1	2	yes	—
61	C₂×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈	C₂×C₈	1	2	yes	—
63	C₂×C₈	C₂×C₈	1	2	yes	—
64	C₂×C₈	M₁₆	1	2	yes	—
65	C₂×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₈	#41	1	2	yes	—
68	C₄²	—	2	4	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	C₂²⋊C₄	1	2	yes	—
72	C₄²	M₁₆	1	2	yes	—
73	C₁₆	—	2	8	no	—
74	C₂³	—	2	1	no	—
75	C₂³	—	2	1	no	—
76	C₂³	—	2	2	no	—
77	C₂³	C₄×C₈	1	2	yes	—
78	C₂³	C₂×C₁₆	1	4	yes	—
79	C₂³	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	16	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₄×C₈	1	2	yes	—
90	C₂×C₄	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄	C₂×C₁₆	1	4	yes	—
92	C₂×C₄	C₂×C₁₆	1	4	yes	—
93	C₂×C₄	M₃₂	1	4	yes	—
94	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄	#117	1	1	yes	characteristic
97	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	C₂²⋊C₈	1	4	yes	—
103	C₈	C₄×D₄	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	C₄×C₁₆	1	4	yes	complemented
109	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
110	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²	#431	1	1	yes	characteristic
112	C₂²	C₂²×C₁₆	1	2	yes	—
113	C₂²	C₂×M₃₂	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	2	2	no	—
116	C₄	C₂²⋊C₁₆	1	8	yes	direct product factor
117	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄	C₄×M₁₆	1	1	yes	characteristic
119	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄	#431	1	1	yes	characteristic
121	C₂	—	2	4	no	—
122	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
123	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
124	C₂	C₄×M₃₂	1	1	yes	characteristic
125	C₂	C₂×C₂²⋊C₁₆	1	2	yes	—
126	C₂	#2162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₁₆

C₂

yes

maximal

C₂×C₄×C₁₆

C₂

yes

maximal, complemented

#1374

C₂

yes

maximal, complemented

C₂³×C₈

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂²×C₄²

C₄

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂²⋊C₁₆

C₄

yes

direct product factor

C₄×C₁₆

—

C₂³×C₄

C₈

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₈

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₁₆

yes

complemented

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄×C₈

yes

—

C₂³

C₂×C₁₆

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

C₂²⋊C₈

yes

—

103

C₈

C₄×D₄

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₄×C₁₆

yes

complemented

109

C₂²

C₂×C₄×C₈

yes

characteristic

110

C₂²

C₂×C₂²⋊C₈

yes

characteristic

111

C₂²

#431

yes

characteristic

112

C₂²

C₂²×C₁₆

yes

—

113

C₂²

C₂×M₃₂

yes

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊C₁₆

yes

direct product factor

117

C₄

C₄×C₂²⋊C₄

yes

characteristic

118

C₄

C₄×M₁₆

yes

characteristic

119

C₄

C₂×C₂²⋊C₈

yes

characteristic

120

C₄

#431

yes

characteristic

121

C₂

—

122

C₂

C₄×C₂²⋊C₈

yes

characteristic

123

C₂

C₂×C₄×C₁₆

yes

characteristic

124

C₂

C₄×M₃₂

yes

characteristic

125

C₂

C₂×C₂²⋊C₁₆

yes

—

126

C₂

#2162

yes

—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄×C₈

C₂²

commutator

C₂

C₂×C₄×C₁₆

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂³

C₂²×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. C₂×C₄×C₁₆	C₂	semidirect
4. #1374	C₂	semidirect
14. C₂²⋊C₁₆	C₄	direct (N × H)
35. C₂×C₁₆	C₂×C₄	semidirect
47. C₂²×C₄	C₁₆	semidirect
108. C₂²	C₄×C₁₆	semidirect
116. C₄	C₂²⋊C₁₆	direct (N × H)

H ≅ G/N

type

3. C₂×C₄×C₁₆

C₂

semidirect

4. #1374

C₂

semidirect

14. C₂²⋊C₁₆

C₄

direct (N × H)

35. C₂×C₁₆

C₂×C₄

semidirect

47. C₂²×C₄

C₁₆

semidirect

108. C₂²

C₄×C₁₆

semidirect

116. C₄

C₂²⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×C₄×C₈	2
C₂×C₂²⋊C₁₆	2
C₂×C₄×C₁₆	2
#1374	2

subgroup

index

C₂²×C₄×C₈

C₂×C₂²⋊C₁₆

C₂×C₄×C₁₆

#1374

Subgroups — C₄×C₂²⋊C₁₆ #12598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C22⋊C16 #12598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊C₁₆ #12598