Group #85156 — subgroups

Subgroups — C₃×#3292 #85156

Order 384.

476 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 184 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	C₃×#576	C₂	1	4	yes	maximal, complemented
2	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#472	C₂	1	4	yes	maximal, complemented
4	C₃×#453	C₂	1	4	yes	maximal, complemented
5	C₃×#443	C₂	1	2	yes	maximal, complemented
6	#3292	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×H(5,𝔽₂)	C₂²	1	4	yes	complemented
8	C₃×AGL(1,ℤ/8)	—	2	8	no	—
9	C₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
10	C₃×#135	—	2	4	no	—
11	C₃×#135	C₂²	1	2	yes	—
12	C₂×C₃×D₈	—	2	4	no	—
13	C₂×C₃×D₈	C₂²	1	4	yes	complemented
14	C₃×#131	C₂²	1	4	yes	complemented
15	C₃×#127	C₂²	1	2	yes	complemented
16	C₃×#108	C₂²	1	2	yes	complemented
17	C₃×#104	C₂²	1	4	yes	complemented
18	C₃×#100	C₂²	1	4	yes	complemented
19	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
20	#576	C₂×C₃	1	4	yes	complemented
21	#493	C₂×C₃	1	1	yes	characteristic, complemented
22	#472	C₂×C₃	1	4	yes	complemented
23	#453	C₂×C₃	1	4	yes	complemented
24	#443	C₂×C₃	1	2	yes	complemented
25	C₃×#41	—	2	8	no	—
26	C₃×#41	C₂³	1	4	yes	—
27	C₂×C₃×D₄	—	2	8	no	—
28	C₂×C₃×D₄	—	4	2	no	—
29	C₂×C₃×D₄	—	4	4	no	—
30	C₂×C₃×D₄	C₂³	1	4	yes	—
31	C₃×Q₁₆	D₄	1	2	yes	complemented
32	C₃×SD₁₆	—	2	8	no	—
33	C₃×SD₁₆	D₄	1	4	yes	complemented
34	C₃×D₈	—	2	8	no	—
35	C₃×D₈	—	2	4	no	—
36	C₃×D₈	—	2	4	no	—
37	C₃×D₈	D₄	1	2	yes	complemented
38	C₃×M₁₆	—	2	4	no	—
39	C₃×M₁₆	C₂³	1	4	yes	—
40	C₂×C₃×C₈	—	2	4	no	—
41	C₂×C₃×C₈	C₂³	1	2	yes	—
42	C₃×C₄²	C₂³	1	1	yes	characteristic
43	H(5,𝔽₂)	C₂²×C₃	1	4	yes	complemented
44	AGL(1,ℤ/8)	—	2	8	no	—
45	AGL(1,ℤ/8)	C₂²×C₃	1	8	yes	complemented
46	#135	—	2	4	no	—
47	#135	C₂²×C₃	1	2	yes	—
48	C₂×D₈	—	2	4	no	—
49	C₂×D₈	C₂²×C₃	1	4	yes	complemented
50	#131	C₂²×C₃	1	4	yes	complemented
51	#127	C₂²×C₃	1	2	yes	complemented
52	#108	C₂²×C₃	1	2	yes	complemented
53	#104	C₂²×C₃	1	4	yes	complemented
54	#100	C₂²×C₃	1	4	yes	complemented
55	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
56	C₂³×C₃	—	2	4	no	—
57	C₂³×C₃	—	4	4	no	—
58	C₃×Q₈	C₂×D₄	1	4	yes	—
59	C₃×D₄	—	2	4	no	—
60	C₃×D₄	—	2	8	no	—
61	C₃×D₄	—	4	8	no	—
62	C₃×D₄	—	8	4	no	—
63	C₃×D₄	C₂×D₄	1	4	yes	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	—	4	4	no	—
67	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
68	C₃×C₈	—	2	4	no	—
69	C₃×C₈	C₂×D₄	1	2	yes	—
70	C₃×C₈	C₂×D₄	1	2	yes	—
71	#41	—	2	8	no	—
72	#41	C₂³×C₃	1	4	yes	—
73	C₂×D₄	—	2	8	no	—
74	C₂×D₄	—	4	2	no	—
75	C₂×D₄	—	4	4	no	—
76	C₂×D₄	C₂³×C₃	1	4	yes	—
77	Q₁₆	C₃×D₄	1	2	yes	complemented
78	SD₁₆	—	2	8	no	—
79	SD₁₆	C₃×D₄	1	4	yes	complemented
80	D₈	—	2	8	no	—
81	D₈	—	2	4	no	—
82	D₈	—	2	4	no	—
83	D₈	C₃×D₄	1	2	yes	complemented
84	M₁₆	—	2	4	no	—
85	M₁₆	C₂³×C₃	1	4	yes	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	C₂³×C₃	1	2	yes	—
88	C₄²	C₂³×C₃	1	1	yes	characteristic
89	C₂²×C₃	—	2	4	no	—
90	C₂²×C₃	—	4	4	no	—
91	C₂²×C₃	—	8	4	no	—
92	C₂²×C₃	—	16	4	no	—
93	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₃×C₄	—	2	2	no	—
95	C₃×C₄	—	2	4	no	—
96	C₃×C₄	C₂²×D₄	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	4	no	—
99	Q₈	C₂×C₃×D₄	1	4	yes	—
100	D₄	—	2	4	no	—
101	D₄	—	2	8	no	—
102	D₄	—	4	8	no	—
103	D₄	—	8	4	no	—
104	D₄	C₂×C₃×D₄	1	4	yes	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	C₂×C₃×D₄	1	2	yes	—
111	C₈	C₂×C₃×D₄	1	2	yes	—
112	C₂×C₃	—	2	1	no	—
113	C₂×C₃	—	4	4	no	—
114	C₂×C₃	—	8	4	no	—
115	C₂×C₃	D₄²	1	1	yes	characteristic
116	C₂²	—	2	4	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	16	4	no	—
120	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	C₂²×C₃×D₄	1	2	yes	—
124	C₃	#3292	1	1	yes	characteristic, direct product factor, Sylow
125	C₂	—	2	1	no	—
126	C₂	—	4	4	no	—
127	C₂	—	8	4	no	—
128	C₂	C₃×D₄²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#576

C₂

yes

maximal, complemented

C₃×#493

C₂

yes

maximal, characteristic, complemented

C₃×#472

C₂

yes

maximal, complemented

C₃×#453

C₂

yes

maximal, complemented

C₃×#443

C₂

yes

maximal, complemented

#3292

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₃×H(5,𝔽₂)

C₂²

yes

complemented

C₃×AGL(1,ℤ/8)

—

C₃×AGL(1,ℤ/8)

C₂²

yes

complemented

C₃×#135

—

C₃×#135

C₂²

yes

—

C₂×C₃×D₈

—

C₂×C₃×D₈

C₂²

yes

complemented

C₃×#131

C₂²

yes

complemented

C₃×#127

C₂²

yes

complemented

C₃×#108

C₂²

yes

complemented

C₃×#104

C₂²

yes

complemented

C₃×#100

C₂²

yes

complemented

C₃×C₄×C₈

C₂²

yes

characteristic, complemented

#576

C₂×C₃

yes

complemented

#493

C₂×C₃

yes

characteristic, complemented

#472

C₂×C₃

yes

complemented

#453

C₂×C₃

yes

complemented

#443

C₂×C₃

yes

complemented

C₃×#41

—

C₃×#41

C₂³

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂³

yes

—

C₃×Q₁₆

D₄

yes

complemented

C₃×SD₁₆

—

C₃×SD₁₆

D₄

yes

complemented

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

D₄

yes

complemented

C₃×M₁₆

—

C₃×M₁₆

C₂³

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂³

yes

—

C₃×C₄²

C₂³

yes

characteristic

H(5,𝔽₂)

C₂²×C₃

yes

complemented

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

C₂²×C₃

yes

complemented

#135

—

#135

C₂²×C₃

yes

—

C₂×D₈

—

C₂×D₈

C₂²×C₃

yes

complemented

#131

C₂²×C₃

yes

complemented

#127

C₂²×C₃

yes

complemented

#108

C₂²×C₃

yes

complemented

#104

C₂²×C₃

yes

complemented

#100

C₂²×C₃

yes

complemented

C₄×C₈

C₂²×C₃

yes

characteristic, complemented

C₂³×C₃

—

C₂³×C₃

—

C₃×Q₈

C₂×D₄

yes

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₂×D₄

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₃×C₈

—

C₃×C₈

C₂×D₄

yes

—

C₃×C₈

C₂×D₄

yes

—

#41

—

#41

C₂³×C₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³×C₃

yes

—

Q₁₆

C₃×D₄

yes

complemented

SD₁₆

—

SD₁₆

C₃×D₄

yes

complemented

D₈

—

D₈

—

D₈

—

D₈

C₃×D₄

yes

complemented

M₁₆

—

M₁₆

C₂³×C₃

yes

—

C₂×C₈

—

C₂×C₈

C₂³×C₃

yes

—

C₄²

C₂³×C₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²×D₄

yes

—

C₂³

—

C₂³

—

Q₈

C₂×C₃×D₄

yes

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

C₂×C₃×D₄

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂⁴×C₃

yes

characteristic

109

C₈

—

110

C₈

C₂×C₃×D₄

yes

—

111

C₈

C₂×C₃×D₄

yes

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

D₄²

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₃×H(5,𝔽₂)

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

C₂²×C₃×D₄

yes

—

124

C₃

#3292

yes

characteristic, direct product factor, Sylow

125

C₂

—

126

C₂

—

127

C₂

—

128

C₂

C₃×D₄²

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

subgroup

G/H

center, socle

C₂×C₃

D₄²

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

N	H ≅ G/N	type
1. C₃×#576	C₂	semidirect
2. C₃×#493	C₂	semidirect
3. C₃×#472	C₂	semidirect
4. C₃×#453	C₂	semidirect
5. C₃×#443	C₂	semidirect
6. #3292	C₃	direct (N × H)
7. C₃×H(5,𝔽₂)	C₂²	semidirect
9. C₃×AGL(1,ℤ/8)	C₂²	semidirect
13. C₂×C₃×D₈	C₂²	semidirect
14. C₃×#131	C₂²	semidirect
15. C₃×#127	C₂²	semidirect
16. C₃×#108	C₂²	semidirect
17. C₃×#104	C₂²	semidirect
18. C₃×#100	C₂²	semidirect
19. C₃×C₄×C₈	C₂²	semidirect
20. #576	C₂×C₃	semidirect
21. #493	C₂×C₃	semidirect
22. #472	C₂×C₃	semidirect
23. #453	C₂×C₃	semidirect
24. #443	C₂×C₃	semidirect
31. C₃×Q₁₆	D₄	semidirect
33. C₃×SD₁₆	D₄	semidirect
37. C₃×D₈	D₄	semidirect
43. H(5,𝔽₂)	C₂²×C₃	semidirect
45. AGL(1,ℤ/8)	C₂²×C₃	semidirect
49. C₂×D₈	C₂²×C₃	semidirect
50. #131	C₂²×C₃	semidirect
51. #127	C₂²×C₃	semidirect
52. #108	C₂²×C₃	semidirect
53. #104	C₂²×C₃	semidirect
54. #100	C₂²×C₃	semidirect
55. C₄×C₈	C₂²×C₃	semidirect
77. Q₁₆	C₃×D₄	semidirect
79. SD₁₆	C₃×D₄	semidirect
83. D₈	C₃×D₄	semidirect
124. C₃	#3292	direct (N × H)

H ≅ G/N

type

1. C₃×#576

C₂

semidirect

2. C₃×#493

C₂

semidirect

3. C₃×#472

C₂

semidirect

4. C₃×#453

C₂

semidirect

5. C₃×#443

C₂

semidirect

6. #3292

C₃

direct (N × H)

7. C₃×H(5,𝔽₂)

C₂²

semidirect

9. C₃×AGL(1,ℤ/8)

C₂²

semidirect

13. C₂×C₃×D₈

C₂²

semidirect

14. C₃×#131

C₂²

semidirect

15. C₃×#127

C₂²

semidirect

16. C₃×#108

C₂²

semidirect

17. C₃×#104

C₂²

semidirect

18. C₃×#100

C₂²

semidirect

19. C₃×C₄×C₈

C₂²

semidirect

20. #576

C₂×C₃

semidirect

21. #493

C₂×C₃

semidirect

22. #472

C₂×C₃

semidirect

23. #453

C₂×C₃

semidirect

24. #443

C₂×C₃

semidirect

31. C₃×Q₁₆

D₄

semidirect

33. C₃×SD₁₆

D₄

semidirect

37. C₃×D₈

D₄

semidirect

43. H(5,𝔽₂)

C₂²×C₃

semidirect

45. AGL(1,ℤ/8)

C₂²×C₃

semidirect

49. C₂×D₈

C₂²×C₃

semidirect

50. #131

C₂²×C₃

semidirect

51. #127

C₂²×C₃

semidirect

52. #108

C₂²×C₃

semidirect

53. #104

C₂²×C₃

semidirect

54. #100

C₂²×C₃

semidirect

55. C₄×C₈

C₂²×C₃

semidirect

77. Q₁₆

C₃×D₄

semidirect

79. SD₁₆

C₃×D₄

semidirect

83. D₈

C₃×D₄

semidirect

124. C₃

#3292

direct (N × H)

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

subgroup	index
C₃×#576	2
C₃×#493	2
C₃×#472	2
C₃×#453	2
C₃×#443	2
#3292	3

subgroup

index

Subgroups — C₃×#3292 #85156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#3292 #85156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3292 #85156