Group #84446 — subgroups

Subgroups — C₃×#2582 #84446

Order 384.

484 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 248 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×C₄×Q₈	C₂	1	2	yes	maximal
2	C₃×#398	C₂	1	4	yes	maximal
3	C₃×#391	C₂	1	4	yes	maximal
4	C₃×#389	C₂	1	4	yes	maximal
5	C₃×#384	C₂	1	1	yes	maximal, characteristic
6	#2582	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₃×C₄×Q₈	—	2	8	no	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²	C₂²	1	4	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#95	C₂²	1	8	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₂×C₄×Q₈	C₂×C₃	1	2	yes	—
19	#398	C₂×C₃	1	4	yes	—
20	#391	C₂×C₃	1	4	yes	—
21	#389	C₂×C₃	1	4	yes	—
22	#384	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×Q₈	—	2	4	no	—
24	C₂×C₃×Q₈	—	2	4	no	—
25	C₂×C₃×Q₈	Q₈	1	8	yes	complemented
26	C₂²×C₃×C₄	C₂³	1	4	yes	—
27	C₂²×C₃×C₄	C₂³	1	4	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂³	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	2	yes	—
31	C₃×C₄⋊C₄	—	2	8	no	—
32	C₃×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄⋊C₄	—	2	8	no	—
34	C₃×C₄⋊C₄	—	4	2	no	—
35	C₃×C₄⋊C₄	—	4	4	no	—
36	C₃×C₄²	—	2	8	no	—
37	C₃×C₄²	D₄	1	4	yes	—
38	C₂²×Q₈	C₂²×C₃	1	2	yes	—
39	C₄×Q₈	—	2	8	no	—
40	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
41	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
42	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₄²	C₂²×C₃	1	4	yes	—
46	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
47	#95	C₂²×C₃	1	8	yes	—
48	#95	C₂²×C₃	1	4	yes	—
49	C₂³×C₃	C₂⁴	1	1	yes	characteristic
50	C₃×Q₈	—	2	16	no	—
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	—	2	1	no	—
54	C₂×C₃×C₄	—	2	1	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	8	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	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
62	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
63	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
64	C₂×C₃×C₄	#41	1	4	yes	—
65	C₂×Q₈	—	2	4	no	—
66	C₂×Q₈	—	2	4	no	—
67	C₂×Q₈	C₃×Q₈	1	8	yes	complemented
68	C₂²×C₄	C₂³×C₃	1	4	yes	—
69	C₂²×C₄	C₂³×C₃	1	4	yes	—
70	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
71	C₂²×C₄	C₂³×C₃	1	4	yes	—
72	C₂²×C₄	C₂³×C₃	1	2	yes	—
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	—	4	4	no	—
78	C₄²	—	2	8	no	—
79	C₄²	C₃×D₄	1	4	yes	—
80	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₃	C₂²×Q₈	1	2	yes	—
82	C₂²×C₃	C₂×#41	1	2	yes	—
83	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₃×C₄	—	2	8	no	—
86	C₃×C₄	—	2	2	no	—
87	C₃×C₄	—	4	4	no	—
88	C₃×C₄	—	4	4	no	—
89	C₃×C₄	C₂²⋊Q₈	1	8	yes	—
90	C₂³	C₂⁴×C₃	1	1	yes	characteristic
91	Q₈	—	2	16	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
102	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
103	C₂×C₄	C₂×C₃×Q₈	1	8	yes	—
104	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
105	C₂×C₄	C₃×#41	1	4	yes	—
106	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
107	C₂×C₃	#537	1	1	yes	characteristic
108	C₂×C₃	#557	1	2	yes	—
109	C₂×C₃	Q₈²	1	1	yes	characteristic
110	C₂×C₃	#559	1	1	yes	characteristic
111	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
112	C₂²	C₂²×C₃×Q₈	1	2	yes	—
113	C₂²	C₂×C₃×#41	1	2	yes	—
114	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
116	C₄	—	2	8	no	—
117	C₄	—	2	2	no	—
118	C₄	—	4	4	no	—
119	C₄	—	4	4	no	—
120	C₄	C₃×C₂²⋊Q₈	1	8	yes	—
121	C₃	#2582	1	1	yes	characteristic, direct product factor, Sylow
122	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
123	C₂	C₃×#537	1	1	yes	characteristic
124	C₂	C₃×#557	1	2	yes	—
125	C₂	C₃×Q₈²	1	1	yes	characteristic
126	C₂	C₃×#559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄×Q₈

C₂

yes

maximal

C₃×#398

C₂

yes

maximal

C₃×#391

C₂

yes

maximal

C₃×#389

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, characteristic

#2582

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄×Q₈

—

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

—

C₂×C₄×Q₈

C₂×C₃

yes

—

#398

C₂×C₃

yes

—

#391

C₂×C₃

yes

—

#389

C₂×C₃

yes

—

#384

C₂×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

Q₈

yes

complemented

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

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₂²×Q₈

C₂²×C₃

yes

—

C₄×Q₈

—

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

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

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×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₃×Q₈

yes

complemented

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

—

C₄⋊C₄

—

C₄²

—

C₄²

C₃×D₄

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

—

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₂³

C₂⁴×C₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₃×D₄

yes

—

102

C₂×C₄

C₂×C₃×D₄

yes

—

103

C₂×C₄

C₂×C₃×Q₈

yes

—

104

C₂×C₄

C₂×C₃×Q₈

yes

—

105

C₂×C₄

C₃×#41

yes

—

106

C₂×C₃

C₂×C₂²⋊Q₈

yes

—

107

C₂×C₃

#537

yes

characteristic

108

C₂×C₃

#557

yes

—

109

C₂×C₃

Q₈²

yes

characteristic

110

C₂×C₃

#559

yes

characteristic

111

C₂²

C₂²×C₃×D₄

yes

characteristic

112

C₂²

C₂²×C₃×Q₈

yes

—

113

C₂²

C₂×C₃×#41

yes

—

114

C₂²

C₃×H(5,𝔽₂)

yes

characteristic

115

C₂²

C₃×2₋¹⁺⁴

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₃×C₂²⋊Q₈

yes

—

121

C₃

#2582

yes

characteristic, direct product factor, Sylow

122

C₂

C₂×C₃×C₂²⋊Q₈

yes

—

123

C₂

C₃×#537

yes

characteristic

124

C₂

C₃×#557

yes

—

125

C₂

C₃×Q₈²

yes

characteristic

126

C₂

C₃×#559

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
6. #2582	C₃	direct (N × H)
25. C₂×C₃×Q₈	Q₈	semidirect
67. C₂×Q₈	C₃×Q₈	semidirect
121. C₃	#2582	direct (N × H)

H ≅ G/N

type

6. #2582

C₃

direct (N × H)

25. C₂×C₃×Q₈

Q₈

semidirect

67. C₂×Q₈

C₃×Q₈

semidirect

121. C₃

#2582

direct (N × H)

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

subgroup	index
C₂×C₃×C₄×Q₈	2
C₃×#398	2
C₃×#391	2
C₃×#389	2
C₃×#384	2
#2582	3

subgroup

index

C₂×C₃×C₄×Q₈

Subgroups — C₃×#2582 #84446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2582 #84446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2582 #84446