Group #84683 — subgroups

Subgroups — C₃×#2819 #84683

Order 384.

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

normal subgroups only

This group has 580 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	C₂×C₃×#124	C₂	1	2	yes	maximal, complemented
2	C₃×#394	C₂	1	4	yes	maximal, complemented
3	C₃×#393	C₂	1	4	yes	maximal, complemented
4	C₃×#392	C₂	1	4	yes	maximal, complemented
5	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	#2819	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₂²×C₃×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×D₄	C₂²	1	4	yes	—
10	C₃×#124	—	2	8	no	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#95	C₂²	1	4	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₂×#124	C₂×C₃	1	2	yes	complemented
19	#394	C₂×C₃	1	4	yes	complemented
20	#393	C₂×C₃	1	4	yes	complemented
21	#392	C₂×C₃	1	4	yes	complemented
22	#382	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂⁴×C₃	C₂³	1	4	yes	—
24	C₂×C₃×Q₈	—	2	4	no	—
25	C₂×C₃×Q₈	—	4	4	no	—
26	C₂×C₃×D₄	—	2	4	no	—
27	C₂×C₃×D₄	—	4	4	no	—
28	C₂×C₃×D₄	—	4	8	no	—
29	C₂×C₃×D₄	—	4	4	no	—
30	C₂²×C₃×C₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	4	yes	—
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₃×C₄⋊C₄	—	4	2	no	—
35	C₃×C₂²⋊C₄	—	2	8	no	—
36	C₃×C₂²⋊C₄	—	2	8	no	—
37	C₃×C₂²⋊C₄	—	4	8	no	—
38	C₃×C₄²	D₄	1	4	yes	complemented
39	C₂²×Q₈	C₂²×C₃	1	2	yes	—
40	C₂²×D₄	C₂²×C₃	1	2	yes	—
41	C₂²×D₄	C₂²×C₃	1	4	yes	—
42	#124	—	2	8	no	—
43	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
47	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
48	#95	C₂²×C₃	1	4	yes	—
49	#95	C₂²×C₃	1	4	yes	—
50	C₂³×C₃	—	2	4	no	—
51	C₂³×C₃	—	2	8	no	—
52	C₂³×C₃	—	2	8	no	—
53	C₂³×C₃	—	2	4	no	—
54	C₂³×C₃	—	2	4	no	—
55	C₂³×C₃	C₂⁴	1	1	yes	characteristic
56	C₃×Q₈	—	4	8	no	—
57	C₃×D₄	—	4	8	no	—
58	C₃×D₄	—	4	16	no	—
59	C₂×C₃×C₄	—	2	1	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	1	no	—
62	C₂×C₃×C₄	—	2	4	no	—
63	C₂×C₃×C₄	—	2	8	no	—
64	C₂×C₃×C₄	—	2	8	no	—
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
68	C₂⁴	C₂³×C₃	1	4	yes	—
69	C₂×Q₈	—	2	4	no	—
70	C₂×Q₈	—	4	4	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	4	4	no	—
73	C₂×D₄	—	4	8	no	—
74	C₂×D₄	—	4	4	no	—
75	C₂²×C₄	C₂³×C₃	1	4	yes	—
76	C₂²×C₄	C₂³×C₃	1	4	yes	—
77	C₂²×C₄	C₂³×C₃	1	2	yes	—
78	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₄⋊C₄	—	4	2	no	—
80	C₂²⋊C₄	—	2	8	no	—
81	C₂²⋊C₄	—	2	8	no	—
82	C₂²⋊C₄	—	4	8	no	—
83	C₄²	C₃×D₄	1	4	yes	complemented
84	C₂²×C₃	—	4	4	no	—
85	C₂²×C₃	—	4	4	no	—
86	C₂²×C₃	—	4	4	no	—
87	C₂²×C₃	—	4	8	no	—
88	C₂²×C₃	—	4	8	no	—
89	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
91	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
93	C₃×C₄	—	2	4	no	—
94	C₃×C₄	—	2	2	no	—
95	C₃×C₄	—	4	4	no	—
96	C₃×C₄	—	4	4	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	8	no	—
101	C₂³	—	2	4	no	—
102	C₂³	C₂⁴×C₃	1	1	yes	characteristic
103	Q₈	—	4	8	no	—
104	D₄	—	4	8	no	—
105	D₄	—	4	16	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
114	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
115	C₂×C₃	—	8	4	no	—
116	C₂×C₃	#535	1	2	yes	—
117	C₂×C₃	#537	1	1	yes	characteristic
118	C₂×C₃	H(3,𝔽₄)	1	2	yes	—
119	C₂×C₃	#562	1	2	yes	—
120	C₂²	—	4	8	no	—
121	C₂²	—	4	8	no	—
122	C₂²	—	4	4	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
126	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
128	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	4	no	—
132	C₄	—	4	4	no	—
133	C₃	#2819	1	1	yes	characteristic, direct product factor, Sylow
134	C₂	—	8	4	no	—
135	C₂	C₃×#535	1	2	yes	—
136	C₂	C₃×#537	1	1	yes	characteristic
137	C₂	C₃×H(3,𝔽₄)	1	2	yes	—
138	C₂	C₃×#562	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#124

C₂

yes

maximal, complemented

C₃×#394

C₂

yes

maximal, complemented

C₃×#393

C₂

yes

maximal, complemented

C₃×#392

C₂

yes

maximal, complemented

C₃×#382

C₂

yes

maximal, characteristic, complemented

#2819

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₂²×C₃×D₄

C₂²

yes

—

C₂²×C₃×D₄

C₂²

yes

—

C₃×#124

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂×#124

C₂×C₃

yes

complemented

#394

C₂×C₃

yes

complemented

#393

C₂×C₃

yes

complemented

#392

C₂×C₃

yes

complemented

#382

C₂×C₃

yes

characteristic, complemented

C₂⁴×C₃

C₂³

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

D₄

yes

complemented

C₂²×Q₈

C₂²×C₃

yes

—

C₂²×D₄

C₂²×C₃

yes

—

C₂²×D₄

C₂²×C₃

yes

—

#124

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂⁴

C₂³×C₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₃×D₄

yes

complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂⁴×C₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₃×D₄

yes

—

114

C₂×C₄

C₂×C₃×D₄

yes

—

115

C₂×C₃

—

116

C₂×C₃

#535

yes

—

117

C₂×C₃

#537

yes

characteristic

118

C₂×C₃

H(3,𝔽₄)

yes

—

119

C₂×C₃

#562

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×C₃×D₄

yes

characteristic

126

C₂²

C₃×H(5,𝔽₂)

yes

characteristic

127

C₂²

C₃×H(5,𝔽₂)

yes

—

128

C₂²

C₃×2₋¹⁺⁴

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₃

#2819

yes

characteristic, direct product factor, Sylow

134

C₂

—

135

C₂

C₃×#535

yes

—

136

C₂

C₃×#537

yes

characteristic

137

C₂

C₃×H(3,𝔽₄)

yes

—

138

C₂

C₃×#562

yes

—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
1. C₂×C₃×#124	C₂	semidirect
2. C₃×#394	C₂	semidirect
3. C₃×#393	C₂	semidirect
4. C₃×#392	C₂	semidirect
5. C₃×#382	C₂	semidirect
6. #2819	C₃	direct (N × H)
18. C₂×#124	C₂×C₃	semidirect
19. #394	C₂×C₃	semidirect
20. #393	C₂×C₃	semidirect
21. #392	C₂×C₃	semidirect
22. #382	C₂×C₃	semidirect
38. C₃×C₄²	D₄	semidirect
83. C₄²	C₃×D₄	semidirect
133. C₃	#2819	direct (N × H)

H ≅ G/N

type

1. C₂×C₃×#124

C₂

semidirect

2. C₃×#394

C₂

semidirect

3. C₃×#393

C₂

semidirect

4. C₃×#392

C₂

semidirect

5. C₃×#382

C₂

semidirect

6. #2819

C₃

direct (N × H)

18. C₂×#124

C₂×C₃

semidirect

19. #394

C₂×C₃

semidirect

20. #393

C₂×C₃

semidirect

21. #392

C₂×C₃

semidirect

22. #382

C₂×C₃

semidirect

38. C₃×C₄²

D₄

semidirect

83. C₄²

C₃×D₄

semidirect

133. C₃

#2819

direct (N × H)

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

subgroup	index
C₂×C₃×#124	2
C₃×#394	2
C₃×#393	2
C₃×#392	2
C₃×#382	2
#2819	3

subgroup

index

C₂×C₃×#124

Subgroups — C₃×#2819 #84683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2819 #84683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2819 #84683