Group #68090 — subgroups

Subgroups — #68090

Order 384.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#4694	C₂	1	1	yes	maximal, characteristic
2	#4693	C₂	1	1	yes	maximal, characteristic
3	#4693	C₂	1	1	yes	maximal, characteristic
4	#4682	C₂	1	1	yes	maximal, characteristic, complemented
5	#4681	C₂	1	1	yes	maximal, characteristic, complemented
6	#4680	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	#1714	—	3	1	no	maximal, Sylow
9	C₂×#204	—	2	2	no	—
10	C₂×#204	—	2	2	no	—
11	#874	C₂²	1	1	yes	characteristic
12	#869	C₂²	1	1	yes	characteristic
13	#869	C₂²	1	1	yes	characteristic
14	#869	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	#821	—	2	2	no	—
19	#818	—	2	2	no	—
20	#817	—	2	2	no	—
21	#816	—	2	2	no	—
22	#500	—	3	1	no	—
23	#498	—	3	1	no	—
24	#492	—	3	1	no	—
25	#492	—	3	1	no	—
26	#487	—	3	1	no	—
27	#486	—	3	1	no	—
28	#322	D₃	1	1	yes	characteristic, complemented
29	C₂×D₁₂	—	2	1	no	—
30	C₂×D₁₂	—	2	1	no	—
31	C₂×D₁₂	—	2	1	no	—
32	C₂×Dic₆	—	2	1	no	—
33	C₂×Dic₆	—	2	1	no	—
34	C₂×Dic₆	—	2	1	no	—
35	C₂×C₃×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₈	D₄	1	2	yes	—
38	C₃×C₄²	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₃	—	2	1	no	—
40	C₄⋊Dic₃	—	2	1	no	—
41	C₄⋊Dic₃	—	2	1	no	—
42	C₄⋊Dic₃	—	2	1	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	#204	—	4	4	no	—
46	#204	—	4	4	no	—
47	C₂×SD₁₆	—	6	2	no	—
48	C₂×SD₁₆	—	6	2	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	—	3	1	no	—
51	C₄⋊Q₈	—	3	1	no	—
52	#127	—	3	1	no	—
53	#107	—	6	2	no	—
54	#106	—	6	2	no	—
55	#103	—	6	2	no	—
56	#102	—	6	2	no	—
57	C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×D₃	—	4	1	no	—
61	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₃	—	4	1	no	—
65	C₂×Dic₃	—	4	1	no	—
66	C₂×Dic₃	—	4	1	no	—
67	D₁₂	—	4	2	no	—
68	D₁₂	—	4	2	no	—
69	D₁₂	—	4	2	no	—
70	Dic₆	—	4	2	no	—
71	Dic₆	—	4	2	no	—
72	Dic₆	—	4	2	no	—
73	C₃×C₈	—	2	2	no	—
74	C₃×C₈	—	2	2	no	—
75	C₃×C₈	SD₁₆	1	4	yes	complemented
76	C₂×Q₈	—	6	1	no	—
77	C₂×Q₈	—	6	1	no	—
78	C₂×Q₈	—	6	1	no	—
79	C₂×D₄	—	6	1	no	—
80	C₂×D₄	—	6	1	no	—
81	C₂×D₄	—	6	1	no	—
82	SD₁₆	—	12	4	no	—
83	SD₁₆	—	12	4	no	—
84	C₂×C₈	D₁₂	1	2	yes	—
85	C₂×C₈	D₁₂	1	2	yes	—
86	C₂×C₈	D₁₂	1	2	yes	—
87	C₄⋊C₄	—	6	1	no	—
88	C₄⋊C₄	—	6	1	no	—
89	C₄⋊C₄	—	6	1	no	—
90	C₄⋊C₄	—	6	1	no	—
91	C₄⋊C₄	—	6	1	no	—
92	C₄⋊C₄	—	6	1	no	—
93	C₄²	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃	#127	1	1	yes	characteristic
95	C₂×D₃	—	8	1	no	—
96	C₂×D₃	—	8	1	no	—
97	C₂×D₃	—	8	1	no	—
98	C₃×C₄	C₂×SD₁₆	1	2	yes	—
99	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
100	C₃×C₄	#137	1	2	yes	—
101	Dic₃	—	8	1	no	—
102	Dic₃	—	8	1	no	—
103	Dic₃	—	8	1	no	—
104	C₂³	—	12	1	no	—
105	Q₈	—	12	2	no	—
106	Q₈	—	12	2	no	—
107	Q₈	—	12	2	no	—
108	D₄	—	12	2	no	—
109	D₄	—	12	2	no	—
110	D₄	—	12	2	no	—
111	C₂×C₄	—	12	1	no	—
112	C₂×C₄	—	12	1	no	—
113	C₂×C₄	—	12	1	no	—
114	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
116	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₈	#204	1	4	yes	complemented
120	C₂×C₃	#492	1	1	yes	characteristic
121	C₂×C₃	#496	1	1	yes	characteristic
122	C₂×C₃	#497	1	1	yes	characteristic
123	D₃	—	16	1	no	—
124	C₂²	—	24	1	no	—
125	C₂²	—	24	1	no	—
126	C₂²	—	24	1	no	—
127	C₂²	#874	1	1	yes	characteristic
128	C₄	—	24	1	no	—
129	C₄	—	24	1	no	—
130	C₄	—	24	1	no	—
131	C₄	C₂×#204	1	2	yes	—
132	C₄	#908	1	2	yes	—
133	C₄	#909	1	2	yes	—
134	C₃	#1714	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	48	1	no	—
136	C₂	#4693	1	1	yes	characteristic
137	C₂	#4712	1	1	yes	characteristic
138	C₂	#4715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#4694

C₂

yes

maximal, characteristic

#4693

C₂

yes

maximal, characteristic

#4693

C₂

yes

maximal, characteristic

#4682

C₂

yes

maximal, characteristic, complemented

#4681

C₂

yes

maximal, characteristic, complemented

#4680

C₂

yes

maximal, characteristic, complemented

C₃×#322

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

—

—

—

—

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#204

—

#204

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

SD₁₆

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

100

C₃×C₄

#137

yes

—

101

Dic₃

—

102

Dic₃

—

103

Dic₃

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₁₂

yes

characteristic

115

C₂×C₄

C₂×D₁₂

yes

characteristic

116

C₂×C₄

C₂×D₁₂

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

#204

yes

complemented

120

C₂×C₃

#492

yes

characteristic

121

C₂×C₃

#496

yes

characteristic

122

C₂×C₃

#497

yes

characteristic

123

D₃

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#874

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×#204

yes

—

132

C₄

#908

yes

—

133

C₄

#909

yes

—

134

C₃

#1714

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

#4693

yes

characteristic

137

C₂

#4712

yes

characteristic

138

C₂

#4715

yes

characteristic

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	#322	D₃
hypocenter	C₃	#1714
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#874

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #4682	C₂	semidirect
5. #4681	C₂	semidirect
6. #4680	C₂	semidirect
7. C₃×#322	C₂	semidirect
28. #322	D₃	semidirect
75. C₃×C₈	SD₁₆	semidirect
119. C₈	#204	semidirect
134. C₃	#1714	semidirect

H ≅ G/N

type

4. #4682

C₂

semidirect

5. #4681

C₂

semidirect

6. #4680

C₂

semidirect

7. C₃×#322

C₂

semidirect

28. #322

D₃

semidirect

75. C₃×C₈

SD₁₆

semidirect

119. C₈

#204

semidirect

134. C₃

#1714

semidirect

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

subgroup	index
#4694	2
#4693	2
#4682	2
#4681	2
#4680	2
C₃×#322	2
#1714	3

subgroup

index

Subgroups — #68090

Conjugacy classes of subgroups

Complements

Maximal subgroups