Group #82556 — subgroups

Subgroups — #82556

Order 384.

716 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 249 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×#1007	C₂	1	1	yes	maximal, characteristic
2	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	#5812	C₂	1	1	yes	maximal, characteristic, complemented
4	#5006	C₂	1	12	yes	maximal, complemented
5	#2884	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
7	#1007	—	2	4	no	—
8	#1007	—	2	4	no	—
9	C₂×#239	C₄	1	4	yes	complemented
10	C₂×D₃×Q₈	C₄	1	4	yes	complemented
11	C₂×#235	C₂²	1	6	yes	—
12	C₃×C₄×Q₈	—	2	4	no	—
13	C₃×#117	C₂²	1	6	yes	—
14	Q₈×Dic₃	—	2	4	no	—
15	#926	C₂²	1	6	yes	—
16	#806	C₂²	1	16	yes	—
17	C₂×2₋¹⁺⁴	—	3	1	no	—
18	#519	—	3	1	no	—
19	#519	D₃	1	1	yes	characteristic, complemented
20	#410	—	3	12	no	—
21	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
22	C₂×C₃×Q₈	D₄	1	4	yes	—
23	C₂×C₃×Q₈	D₄	1	4	yes	—
24	C₂²×C₃×C₄	C₂³	1	3	yes	—
25	C₂×#67	C₂³	1	4	yes	—
26	#239	—	2	16	no	—
27	D₃×Q₈	—	2	16	no	—
28	#235	—	2	12	no	—
29	#235	—	2	12	no	—
30	C₂×D₁₂	C₂×C₄	1	6	yes	—
31	C₂×C₄×D₃	C₂×C₄	1	12	yes	—
32	C₂×Dic₆	C₂×C₄	1	6	yes	—
33	C₃×C₄⋊C₄	—	2	6	no	—
34	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
35	C₃×C₄²	—	2	6	no	—
36	#217	C₂³	1	4	yes	—
37	C₄⋊Dic₃	—	2	6	no	—
38	C₄×Dic₃	—	2	6	no	—
39	2₋¹⁺⁴	—	6	4	no	—
40	2₋¹⁺⁴	—	6	4	no	—
41	C₂×#41	—	3	4	no	—
42	C₂×#41	—	3	6	no	—
43	C₂²×Q₈	—	3	4	no	—
44	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
45	C₄×Q₈	—	2	4	no	—
46	C₄×Q₈	—	6	4	no	—
47	#117	—	3	6	no	—
48	#117	C₂×D₃	1	6	yes	—
49	#99	—	3	16	no	—
50	C₂³×C₃	C₂⁴	1	1	yes	characteristic
51	C₂²×D₃	C₂²×C₄	1	4	yes	—
52	C₃×Q₈	C₂²⋊C₄	1	16	yes	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
55	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
56	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
57	#67	—	2	4	no	—
58	#67	—	2	4	no	—
59	C₂×Dic₃	—	2	4	no	—
60	C₂×Dic₃	C₂²×C₄	1	4	yes	—
61	D₁₂	—	2	12	no	—
62	C₄×D₃	—	2	24	no	—
63	Dic₆	—	2	12	no	—
64	#41	—	6	16	no	—
65	#41	—	6	12	no	—
66	#41	—	6	12	no	—
67	C₂×Q₈	—	3	6	no	—
68	C₂×Q₈	—	6	16	no	—
69	C₂×Q₈	C₄×D₃	1	4	yes	—
70	C₂×Q₈	D₁₂	1	4	yes	—
71	C₂×Q₈	#67	1	4	yes	—
72	C₂×D₄	—	3	4	no	—
73	C₂×D₄	—	3	6	no	—
74	C₂²×C₄	—	3	12	no	—
75	C₂²×C₄	C₂²×D₃	1	3	yes	—
76	C₄⋊C₄	—	2	6	no	—
77	C₄⋊C₄	—	6	6	no	—
78	C₂²⋊C₄	—	3	4	no	—
79	C₂²⋊C₄	C₂²×D₃	1	4	yes	—
80	C₄²	—	2	6	no	—
81	C₄²	—	6	6	no	—
82	C₂²×C₃	—	4	1	no	—
83	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₃	—	2	4	no	—
87	C₂×D₃	—	4	4	no	—
88	C₃×C₄	—	4	4	no	—
89	C₃×C₄	C₂×C₂²⋊C₄	1	12	yes	—
90	Dic₃	—	2	4	no	—
91	Dic₃	—	4	4	no	—
92	C₂³	—	3	4	no	—
93	C₂³	C₂³×D₃	1	1	yes	characteristic
94	Q₈	—	6	12	no	—
95	Q₈	(C₂×D₃)⋊C₄	1	16	yes	—
96	D₄	—	6	4	no	—
97	D₄	—	6	4	no	—
98	D₄	—	6	12	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	3	4	no	—
101	C₂×C₄	—	6	24	no	—
102	C₂×C₄	—	6	4	no	—
103	C₂×C₄	C₂×C₄×D₃	1	6	yes	—
104	C₂×C₄	C₂×D₁₂	1	6	yes	—
105	C₂×C₄	C₂×#67	1	6	yes	—
106	C₂×C₃	—	2	1	no	—
107	C₂×C₃	—	2	1	no	—
108	C₂×C₃	—	2	1	no	—
109	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
110	D₃	—	4	4	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	6	4	no	—
113	C₂²	—	12	4	no	—
114	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
115	C₂²	C₂²×D₁₂	1	1	yes	characteristic
116	C₂²	C₂²×#67	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	6	4	no	—
119	C₄	—	12	4	no	—
120	C₄	C₂×(C₂×D₃)⋊C₄	1	12	yes	—
121	C₃	#2884	1	1	yes	characteristic, complemented, Sylow
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	12	4	no	—
126	C₂	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1007

C₂

yes

maximal, characteristic

C₃×#519

C₂

yes

maximal, characteristic, complemented

#5812

C₂

yes

maximal, characteristic, complemented

#5006

C₂

yes

maximal, complemented

#2884

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

—

C₄

yes

complemented

C₂×D₃×Q₈

C₄

yes

complemented

C₂×#235

C₂²

yes

—

C₃×C₄×Q₈

—

C₃×#117

C₂²

yes

—

Q₈×Dic₃

—

#926

C₂²

yes

—

#806

C₂²

yes

—

C₂×2₋¹⁺⁴

—

#519

—

#519

D₃

yes

characteristic, complemented

#410

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂×C₃×Q₈

D₄

yes

—

C₂×C₃×Q₈

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂×#67

C₂³

yes

—

#239

—

D₃×Q₈

—

#235

—

#235

—

C₂×D₁₂

C₂×C₄

yes

—

C₂×C₄×D₃

C₂×C₄

yes

—

C₂×Dic₆

C₂×C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

C₂³

yes

—

C₃×C₄²

—

#217

C₂³

yes

—

C₄⋊Dic₃

—

C₄×Dic₃

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃

yes

—

#99

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂²×D₃

C₂²×C₄

yes

—

C₃×Q₈

C₂²⋊C₄

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

#67

—

#67

—

C₂×Dic₃

—

C₂×Dic₃

C₂²×C₄

yes

—

D₁₂

—

C₄×D₃

—

Dic₆

—

#41

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₃

yes

—

C₂×Q₈

D₁₂

yes

—

C₂×Q₈

#67

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₃

yes

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

C₂×C₂²⋊C₄

yes

—

Dic₃

—

Dic₃

—

C₂³

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

Q₈

(C₂×D₃)⋊C₄

yes

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄×D₃

yes

—

104

C₂×C₄

C₂×D₁₂

yes

—

105

C₂×C₄

C₂×#67

yes

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₂²×C₂²⋊C₄

yes

characteristic

110

D₃

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×C₄×D₃

yes

characteristic

115

C₂²

C₂²×D₁₂

yes

characteristic

116

C₂²

C₂²×#67

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×(C₂×D₃)⋊C₄

yes

—

121

C₃

#2884

yes

characteristic, complemented, Sylow

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂²×(C₂×D₃)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂³	C₂³×D₃
Fitting	C₃×#519	C₂
socle	C₂×C₃	C₂²×C₂²⋊C₄
prime radical	C₂×#1007	C₂
cosocle	C₂³×C₃	C₂⁴
hypercenter	#519	D₃
hypocenter	C₃	#2884
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×(C₂×D₃)⋊C₄

commutator

C₂²×C₃

C₂³×C₄

Frattini

C₂³

C₂³×D₃

Fitting

C₃×#519

C₂

socle

C₂×C₃

C₂²×C₂²⋊C₄

prime radical

C₂×#1007

C₂

cosocle

C₂³×C₃

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. C₃×#519	C₂	semidirect
3. #5812	C₂	semidirect
4. #5006	C₂	semidirect
9. C₂×#239	C₄	semidirect
10. C₂×D₃×Q₈	C₄	semidirect
19. #519	D₃	semidirect
121. C₃	#2884	semidirect

H ≅ G/N

type

2. C₃×#519

C₂

semidirect

3. #5812

C₂

semidirect

4. #5006

C₂

semidirect

9. C₂×#239

C₄

semidirect

10. C₂×D₃×Q₈

C₄

semidirect

19. #519

D₃

semidirect

121. C₃

#2884

semidirect

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

subgroup	index
C₂×#1007	2
C₃×#519	2
#5812	2
#5006	2
#2884	3

subgroup

index

Subgroups — #82556

Conjugacy classes of subgroups

Complements

Maximal subgroups