Group #84246 — subgroups

Subgroups — C₃×#2382 #84246

Order 384.

408 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 264 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#389	C₂	1	4	yes	maximal
3	C₃×#387	C₂	1	4	yes	maximal
4	C₃×#387	C₂	1	4	yes	maximal
5	C₃×#382	C₂	1	2	yes	maximal
6	#2382	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#125	C₄	1	8	yes	complemented
8	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄²	C₂²	1	4	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₃×#95	C₂²	1	4	yes	—
16	C₃×#95	C₂²	1	4	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₃×#95	C₂²	1	4	yes	—
19	C₂×#125	C₂×C₃	1	1	yes	characteristic
20	#389	C₂×C₃	1	4	yes	—
21	#387	C₂×C₃	1	4	yes	—
22	#387	C₂×C₃	1	4	yes	—
23	#382	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₄	C₂³	1	4	yes	—
25	C₂²×C₃×C₄	C₂³	1	4	yes	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂³	1	4	yes	—
28	C₂²×C₃×C₄	C₂³	1	2	yes	—
29	C₃×C₄⋊C₄	—	4	4	no	—
30	C₃×C₄⋊C₄	—	4	2	no	—
31	C₃×C₄⋊C₄	—	4	2	no	—
32	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
34	C₃×C₄²	—	4	4	no	—
35	C₃×C₄²	C₂×C₄	1	4	yes	—
36	#125	C₃×C₄	1	8	yes	complemented
37	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
38	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
40	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
41	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
42	C₂×C₄²	C₂²×C₃	1	4	yes	—
43	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
44	#95	C₂²×C₃	1	4	yes	—
45	#95	C₂²×C₃	1	4	yes	—
46	#95	C₂²×C₃	1	4	yes	—
47	#95	C₂²×C₃	1	4	yes	—
48	C₂³×C₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₃×C₄	—	2	4	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	2	1	no	—
52	C₂×C₃×C₄	—	2	1	no	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	4	no	—
57	C₂×C₃×C₄	—	2	4	no	—
58	C₂×C₃×C₄	—	2	4	no	—
59	C₂×C₃×C₄	—	2	4	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
62	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
63	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
64	C₂²×C₄	C₂³×C₃	1	4	yes	—
65	C₂²×C₄	C₂³×C₃	1	4	yes	—
66	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
67	C₂²×C₄	C₂³×C₃	1	4	yes	—
68	C₂²×C₄	C₂³×C₃	1	2	yes	—
69	C₄⋊C₄	—	4	4	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
73	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
74	C₄²	—	4	4	no	—
75	C₄²	C₂×C₃×C₄	1	4	yes	—
76	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
77	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
80	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
81	C₃×C₄	—	2	2	no	—
82	C₃×C₄	—	2	8	no	—
83	C₃×C₄	—	2	4	no	—
84	C₃×C₄	—	4	4	no	—
85	C₃×C₄	—	4	4	no	—
86	C₂³	C₂⁴×C₃	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
100	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
101	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
102	C₂×C₃	#519	1	1	yes	characteristic
103	C₂×C₃	#520	1	2	yes	—
104	C₂×C₃	#562	1	1	yes	characteristic
105	C₂×C₃	#563	1	2	yes	—
106	C₂×C₃	#564	1	1	yes	characteristic
107	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
108	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
109	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
110	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
111	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	—	2	8	no	—
114	C₄	—	2	4	no	—
115	C₄	—	4	4	no	—
116	C₄	—	4	4	no	—
117	C₃	#2382	1	1	yes	characteristic, direct product factor, Sylow
118	C₂	C₃×#519	1	1	yes	characteristic
119	C₂	C₃×#520	1	2	yes	—
120	C₂	C₃×#562	1	1	yes	characteristic
121	C₂	C₃×#563	1	2	yes	—
122	C₂	C₃×#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic

C₃×#389

C₂

yes

maximal

C₃×#387

C₂

yes

maximal

C₃×#387

C₂

yes

maximal

C₃×#382

C₂

yes

maximal

#2382

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₃×#125

C₄

yes

complemented

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂×#125

C₂×C₃

yes

characteristic

#389

C₂×C₃

yes

—

#387

C₂×C₃

yes

—

#387

C₂×C₃

yes

—

#382

C₂×C₃

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

#125

C₃×C₄

yes

complemented

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

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×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃×C₄

yes

—

C₄⋊C₄

C₂×C₃×C₄

yes

—

C₄²

—

C₄²

C₂×C₃×C₄

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

C₂⁴×C₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₃×C₄

yes

—

100

C₂×C₄

C₂²×C₃×C₄

yes

—

101

C₂×C₄

C₂²×C₃×C₄

yes

—

102

C₂×C₃

#519

yes

characteristic

103

C₂×C₃

#520

yes

—

104

C₂×C₃

#562

yes

characteristic

105

C₂×C₃

#563

yes

—

106

C₂×C₃

#564

yes

characteristic

107

C₂²

C₂³×C₃×C₄

yes

characteristic

108

C₂²

C₃×H(5,𝔽₂)

yes

—

109

C₂²

C₃×2₋¹⁺⁴

yes

characteristic

110

C₂²

C₃×2₋¹⁺⁴

yes

characteristic

111

C₂²

C₃×2₋¹⁺⁴

yes

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₃

#2382

yes

characteristic, direct product factor, Sylow

118

C₂

C₃×#519

yes

characteristic

119

C₂

C₃×#520

yes

—

120

C₂

C₃×#562

yes

characteristic

121

C₂

C₃×#563

yes

—

122

C₂

C₃×#564

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂⁴

commutator

C₂²

C₂³×C₃×C₄

Frattini, cosocle

C₂³

C₂⁴×C₃

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
6. #2382	C₃	direct (N × H)
7. C₃×#125	C₄	semidirect
36. #125	C₃×C₄	semidirect
117. C₃	#2382	direct (N × H)

H ≅ G/N

type

6. #2382

C₃

direct (N × H)

7. C₃×#125

C₄

semidirect

36. #125

C₃×C₄

semidirect

117. C₃

#2382

direct (N × H)

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

subgroup	index
C₂×C₃×#125	2
C₃×#389	2
C₃×#387	2
C₃×#382	2
#2382	3

subgroup

index

C₂×C₃×#125

Subgroups — C₃×#2382 #84246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2382 #84246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2382 #84246