Group #33434 — subgroups

Subgroups — #33434

Order 256.

345 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 203 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×#504	C₂	1	2	yes	maximal, complemented
2	#2931	C₂	1	1	yes	maximal, characteristic
3	#2169	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆	C₂	1	4	yes	maximal, complemented
5	#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	#2113	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	C₂²	1	2	yes	—
9	#532	C₄	1	1	yes	characteristic
10	#504	—	2	8	no	—
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆	C₂²	1	4	yes	—
15	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	#436	C₄	1	4	yes	—
17	C₈×D₄	C₂²	1	4	yes	—
18	#423	C₄	1	1	yes	characteristic
19	#408	C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₂²⋊C₁₆	C₂²	1	8	yes	—
23	C₄×C₁₆	C₂²	1	2	yes	—
24	C₄×C₁₆	C₂²	1	2	yes	—
25	C₂×#41	C₂×C₄	1	2	yes	—
26	#131	—	2	4	no	—
27	#131	—	2	4	no	—
28	C₂×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×M₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	4	yes	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈	C₈	1	1	yes	characteristic
34	#127	C₈	1	1	yes	characteristic
35	#124	C₈	1	2	yes	—
36	#121	C₈	1	4	yes	—
37	C₄×D₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
39	M₃₂	—	2	4	no	—
40	M₃₂	—	2	8	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	—	2	8	no	—
43	C₂×C₁₆	D₄	1	4	yes	complemented
44	C₂×C₁₆	C₂³	1	2	yes	—
45	C₂×C₁₆	C₂³	1	4	yes	—
46	C₂×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₆	C₂³	1	1	yes	characteristic
48	C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈	C₂×C₄	1	8	yes	—
50	C₄×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	#41	—	2	4	no	—
53	#41	—	2	4	no	—
54	C₂×Q₈	C₂×C₈	1	2	yes	—
55	C₂×D₄	C₂×C₈	1	4	yes	—
56	C₂×D₄	C₂×C₈	1	4	yes	—
57	C₂×D₄	C₂×C₈	1	2	yes	—
58	C₂²×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₄	C₂²×C₄	1	4	yes	—
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	M₁₆	—	2	4	no	—
62	M₁₆	—	2	8	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₈	#41	1	2	yes	—
71	C₂×C₈	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄	C₂×C₈	1	8	yes	—
74	C₄²	C₂×C₈	1	1	yes	characteristic
75	C₄²	C₂×C₈	1	1	yes	characteristic
76	C₄²	C₂²×C₄	1	2	yes	—
77	C₁₆	—	2	4	no	—
78	C₁₆	—	2	2	no	—
79	C₁₆	C₂×D₄	1	2	yes	—
80	C₁₆	C₂×D₄	1	2	yes	—
81	C₂³	C₂²×C₈	1	4	yes	—
82	C₂³	C₂²×C₈	1	1	yes	characteristic
83	Q₈	—	2	4	no	—
84	D₄	—	2	8	no	—
85	D₄	—	2	4	no	—
86	D₄	—	2	8	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×D₄	1	2	yes	—
91	C₂×C₄	C₂²×C₈	1	2	yes	—
92	C₂×C₄	C₂²×C₈	1	4	yes	—
93	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄	C₂²×C₈	1	2	yes	—
95	C₂×C₄	#131	1	2	yes	—
96	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	2	2	no	—
99	C₈	C₄×D₄	1	2	yes	—
100	C₈	C₂²×D₄	1	1	yes	characteristic
101	C₈	C₂×#41	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	—	2	8	no	—
104	C₂²	—	2	4	no	—
105	C₂²	C₈×D₄	1	2	yes	—
106	C₂²	C₂³×C₈	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	2	4	no	—
109	C₄	C₈×D₄	1	2	yes	—
110	C₄	#504	1	4	yes	complemented
111	C₄	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄	C₂×#131	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	4	4	no	—
115	C₂	C₂×C₈×D₄	1	1	yes	characteristic
116	C₂	C₂×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504

C₂

yes

maximal, complemented

#2931

C₂

yes

maximal, characteristic

#2169

C₂

yes

maximal, complemented

D₄×C₁₆

C₂

yes

maximal, complemented

#2150

C₂

yes

maximal, characteristic, complemented

#2113

C₂

yes

maximal, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂²

yes

—

#532

C₄

yes

characteristic

#504

—

C₂×M₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

#436

C₄

yes

—

C₈×D₄

C₂²

yes

—

#423

C₄

yes

characteristic

#408

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₂×#41

C₂×C₄

yes

—

#131

—

#131

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₈

yes

characteristic

#127

C₈

yes

characteristic

#124

C₈

yes

—

#121

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

complemented

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

—

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

—

C₈

—

C₈

C₄×D₄

yes

—

100

C₈

C₂²×D₄

yes

characteristic

101

C₈

C₂×#41

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₈×D₄

yes

—

106

C₂²

C₂³×C₈

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₈×D₄

yes

—

110

C₄

#504

yes

complemented

111

C₄

C₂×C₄×D₄

yes

characteristic

112

C₄

C₂×#131

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₈×D₄

yes

characteristic

116

C₂

C₂×#504

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂³

commutator, socle

C₂²

C₂³×C₈

Frattini, cosocle

C₂×C₈

C₂⁴

C₄

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₁₆ ◁ G

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
3. #2169	C₂	semidirect
4. D₄×C₁₆	C₂	semidirect
5. #2150	C₂	semidirect
6. #2113	C₂	semidirect
7. C₂×C₄×C₁₆	C₂	semidirect
43. C₂×C₁₆	D₄	semidirect
110. C₄	#504	semidirect

H ≅ G/N

type

1. C₂×#504

C₂

semidirect

3. #2169

C₂

semidirect

4. D₄×C₁₆

C₂

semidirect

5. #2150

C₂

semidirect

6. #2113

C₂

semidirect

7. C₂×C₄×C₁₆

C₂

semidirect

43. C₂×C₁₆

D₄

semidirect

110. C₄

#504

semidirect

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

subgroup	index
C₂×#504	2
#2931	2
#2169	2
D₄×C₁₆	2
#2150	2
#2113	2
C₂×C₄×C₁₆	2

subgroup

index

D₄×C₁₆

C₂×C₄×C₁₆

Subgroups — #33434

Conjugacy classes of subgroups

Complements

Maximal subgroups