Group #3222 — subgroups

Subgroups — #3222

Order 128.

198 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 84 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	#541	C₂	1	1	yes	maximal, characteristic, complemented
2	#535	C₂	1	1	yes	maximal, characteristic
3	#497	C₂	1	1	yes	maximal, characteristic, complemented
4	#492	C₂	1	1	yes	maximal, characteristic, complemented
5	#490	C₂	1	1	yes	maximal, characteristic, complemented
6	#488	C₂	1	1	yes	maximal, characteristic, complemented
7	#464	C₂	1	2	yes	maximal, complemented
8	#460	C₂	1	2	yes	maximal, complemented
9	#452	C₂	1	2	yes	maximal, complemented
10	#450	C₂	1	2	yes	maximal, complemented
11	#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂²×D₄	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆	C₂²	1	2	yes	complemented
15	C₂×SD₁₆	C₂²	1	2	yes	—
16	C₂×SD₁₆	C₂²	1	2	yes	—
17	C₂×SD₁₆	C₂²	1	2	yes	complemented
18	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	#127	C₂²	1	1	yes	characteristic
20	#127	C₂²	1	1	yes	characteristic
21	#126	—	2	2	no	—
22	#125	—	2	2	no	—
23	#125	C₂²	1	1	yes	characteristic, complemented
24	#124	C₂²	1	1	yes	characteristic
25	#124	C₂²	1	1	yes	characteristic
26	C₂²⋊Q₈	C₂²	1	2	yes	—
27	#121	—	2	2	no	—
28	#120	—	2	2	no	—
29	C₄×Q₈	C₂²	1	2	yes	complemented
30	#117	C₂²	1	1	yes	characteristic
31	C₄⋊C₈	C₂²	1	2	yes	complemented
32	#103	C₂²	1	2	yes	complemented
33	#102	C₂²	1	2	yes	—
34	#102	C₂²	1	2	yes	—
35	#102	C₂²	1	2	yes	complemented
36	C₂²⋊C₈	C₂²	1	2	yes	—
37	#97	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴	—	2	1	no	—
40	#41	—	4	2	no	—
41	C₂×Q₈	C₂³	1	1	yes	characteristic
42	C₂×Q₈	C₂³	1	2	yes	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	2	2	no	—
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	—	4	2	no	—
48	C₂×D₄	C₂³	1	1	yes	characteristic
49	C₂×D₄	C₂³	1	1	yes	characteristic
50	C₂×D₄	C₂³	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂³	1	1	yes	characteristic
53	Q₁₆	—	4	2	no	—
54	SD₁₆	—	4	2	no	—
55	SD₁₆	—	4	2	no	—
56	SD₁₆	—	4	2	no	—
57	C₂×C₈	C₂³	1	2	yes	—
58	C₂×C₈	C₂³	1	2	yes	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	D₄	1	2	yes	—
64	C₄⋊C₄	C₂³	1	2	yes	—
65	C₂²⋊C₄	—	2	2	no	—
66	C₂²⋊C₄	—	2	2	no	—
67	C₂²⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	D₄	1	2	yes	—
69	C₄²	—	2	2	no	—
70	C₄²	C₂³	1	1	yes	characteristic
71	C₄²	C₂³	1	1	yes	characteristic
72	C₂³	—	2	1	no	—
73	C₂³	—	2	1	no	—
74	C₂³	—	2	1	no	—
75	C₂³	—	4	2	no	—
76	C₂³	—	8	2	no	—
77	C₂³	C₂×D₄	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	2	2	no	—
81	D₄	—	2	2	no	—
82	D₄	—	2	2	no	—
83	D₄	—	2	2	no	—
84	D₄	—	4	2	no	—
85	D₄	—	4	2	no	—
86	D₄	—	8	2	no	—
87	D₄	—	8	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	C₂×D₄	1	2	yes	—
94	C₂×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄	C₂⁴	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	4	1	no	—
102	C₂²	—	4	1	no	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	8	4	no	—
108	C₂²	C₂²×D₄	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	4	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	H(5,𝔽₂)	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	—	8	1	no	—
118	C₂	—	8	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#535	1	1	yes	characteristic
121	C₂	#577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#541

C₂

yes

maximal, characteristic, complemented

#535

C₂

yes

maximal, characteristic

#497

C₂

yes

maximal, characteristic, complemented

#492

C₂

yes

maximal, characteristic, complemented

#490

C₂

yes

maximal, characteristic, complemented

#488

C₂

yes

maximal, characteristic, complemented

#464

C₂

yes

maximal, complemented

#460

C₂

yes

maximal, complemented

#452

C₂

yes

maximal, complemented

#450

C₂

yes

maximal, complemented

#433

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

characteristic

C₂²×D₄

C₂²

yes

characteristic

C₂×Q₁₆

C₂²

yes

complemented

C₂×SD₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

complemented

C₄⋊Q₈

C₂²

yes

characteristic, complemented

#127

C₂²

yes

characteristic

#127

C₂²

yes

—

—

C₂²

yes

characteristic, complemented

#124

C₂²

yes

characteristic

#124

C₂²

yes

characteristic

C₂²⋊Q₈

C₂²

yes

—

#121

—

#120

—

C₄×Q₈

C₂²

yes

complemented

#117

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

complemented

#103

C₂²

yes

complemented

#102

C₂²

yes

—

#102

C₂²

yes

—

#102

C₂²

yes

complemented

C₂²⋊C₈

C₂²

yes

—

#97

C₂²

yes

characteristic, complemented

C₄×C₈

C₂²

yes

characteristic, complemented

C₂⁴

—

#41

—

C₂×Q₈

C₂³

yes

characteristic

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

characteristic

C₂×D₄

C₂³

yes

characteristic

C₂×D₄

C₂³

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄

yes

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×D₄

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

H(5,𝔽₂)

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#535

yes

characteristic

121

C₂

#577

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ #117 ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #541	C₂	semidirect
3. #497	C₂	semidirect
4. #492	C₂	semidirect
5. #490	C₂	semidirect
6. #488	C₂	semidirect
7. #464	C₂	semidirect
8. #460	C₂	semidirect
9. #452	C₂	semidirect
10. #450	C₂	semidirect
11. #433	C₂	semidirect
14. C₂×Q₁₆	C₂²	semidirect
17. C₂×SD₁₆	C₂²	semidirect
18. C₄⋊Q₈	C₂²	semidirect
23. #125	C₂²	semidirect
29. C₄×Q₈	C₂²	semidirect
31. C₄⋊C₈	C₂²	semidirect
32. #103	C₂²	semidirect
35. #102	C₂²	semidirect
37. #97	C₂²	semidirect
38. C₄×C₈	C₂²	semidirect

H ≅ G/N

type

1. #541

C₂

semidirect

3. #497

C₂

semidirect

4. #492

C₂

semidirect

5. #490

C₂

semidirect

6. #488

C₂

semidirect

7. #464

C₂

semidirect

8. #460

C₂

semidirect

9. #452

C₂

semidirect

10. #450

C₂

semidirect

11. #433

C₂

semidirect

14. C₂×Q₁₆

C₂²

semidirect

17. C₂×SD₁₆

C₂²

semidirect

18. C₄⋊Q₈

C₂²

semidirect

23. #125

C₂²

semidirect

29. C₄×Q₈

C₂²

semidirect

31. C₄⋊C₈

C₂²

semidirect

32. #103

C₂²

semidirect

35. #102

C₂²

semidirect

37. #97

C₂²

semidirect

38. C₄×C₈

C₂²

semidirect

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

subgroup	index
#541	2
#535	2
#497	2
#492	2
#490	2
#488	2
#464	2
#460	2
#452	2
#450	2
#433	2

subgroup

index

Subgroups — #3222

Conjugacy classes of subgroups

Complements

Maximal subgroups