Group #68096 — subgroups

Subgroups — #68096

Order 384.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#4682	C₂	1	1	yes	maximal, characteristic
2	#4681	C₂	1	1	yes	maximal, characteristic
3	#4681	C₂	1	1	yes	maximal, characteristic
4	#4681	C₂	1	1	yes	maximal, characteristic
5	#4681	C₂	1	1	yes	maximal, characteristic
6	#4680	C₂	1	1	yes	maximal, characteristic
7	C₃×#322	C₂	1	1	yes	maximal, characteristic
8	#1718	—	3	1	no	maximal, Sylow
9	#869	C₂²	1	1	yes	characteristic
10	#869	C₂²	1	1	yes	characteristic
11	#869	C₂²	1	1	yes	characteristic
12	#869	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#818	—	2	2	no	—
17	#817	—	2	2	no	—
18	#816	—	2	2	no	—
19	#816	—	2	2	no	—
20	#816	—	2	2	no	—
21	#816	—	2	2	no	—
22	#500	—	3	1	no	—
23	#498	—	3	1	no	—
24	#487	—	3	1	no	—
25	#487	—	3	1	no	—
26	#487	—	3	1	no	—
27	#487	—	3	1	no	—
28	#322	D₃	1	1	yes	characteristic
29	C₂×Dic₆	—	2	1	no	—
30	C₂×Dic₆	—	2	1	no	—
31	C₂×Dic₆	—	2	1	no	—
32	C₂×Dic₆	—	2	1	no	—
33	C₂×C₃×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₈	D₄	1	2	yes	—
36	C₃×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₃	—	2	1	no	—
38	C₄⋊Dic₃	—	2	1	no	—
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	C₄⋊Q₈	—	3	1	no	—
46	C₄⋊Q₈	—	3	1	no	—
47	C₄⋊Q₈	—	3	1	no	—
48	C₄⋊Q₈	—	3	1	no	—
49	#107	—	6	2	no	—
50	#106	—	6	2	no	—
51	#103	—	6	2	no	—
52	#103	—	6	2	no	—
53	#103	—	6	2	no	—
54	#103	—	6	2	no	—
55	C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₃	—	4	1	no	—
62	C₂×Dic₃	—	4	1	no	—
63	C₂×Dic₃	—	4	1	no	—
64	C₂×Dic₃	—	4	1	no	—
65	Dic₆	—	4	2	no	—
66	Dic₆	—	4	2	no	—
67	Dic₆	—	4	2	no	—
68	Dic₆	—	4	2	no	—
69	C₃×C₈	—	2	2	no	—
70	C₃×C₈	—	2	2	no	—
71	C₃×C₈	—	2	2	no	—
72	C₂×Q₈	—	6	1	no	—
73	C₂×Q₈	—	6	1	no	—
74	C₂×Q₈	—	6	1	no	—
75	C₂×Q₈	—	6	1	no	—
76	C₂×C₈	D₁₂	1	2	yes	—
77	C₂×C₈	D₁₂	1	2	yes	—
78	C₂×C₈	D₁₂	1	2	yes	—
79	C₄⋊C₄	—	6	1	no	—
80	C₄⋊C₄	—	6	1	no	—
81	C₄⋊C₄	—	6	1	no	—
82	C₄⋊C₄	—	6	1	no	—
83	C₄⋊C₄	—	6	1	no	—
84	C₄⋊C₄	—	6	1	no	—
85	C₄⋊C₄	—	6	1	no	—
86	C₄⋊C₄	—	6	1	no	—
87	C₄²	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₃	#127	1	1	yes	characteristic
89	C₃×C₄	#135	1	2	yes	—
90	C₃×C₄	#137	1	2	yes	—
91	C₃×C₄	#137	1	2	yes	—
92	Dic₃	—	8	1	no	—
93	Dic₃	—	8	1	no	—
94	Dic₃	—	8	1	no	—
95	Dic₃	—	8	1	no	—
96	Q₈	—	12	2	no	—
97	Q₈	—	12	2	no	—
98	Q₈	—	12	2	no	—
99	Q₈	—	12	2	no	—
100	C₂×C₄	—	12	1	no	—
101	C₂×C₄	—	12	1	no	—
102	C₂×C₄	—	12	1	no	—
103	C₂×C₄	—	12	1	no	—
104	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₂×C₃	#495	1	1	yes	characteristic
111	C₂×C₃	#497	1	1	yes	characteristic
112	C₂×C₃	#497	1	1	yes	characteristic
113	C₂²	#874	1	1	yes	characteristic
114	C₄	—	24	1	no	—
115	C₄	—	24	1	no	—
116	C₄	—	24	1	no	—
117	C₄	—	24	1	no	—
118	C₄	#904	1	2	yes	—
119	C₄	#909	1	2	yes	—
120	C₄	#909	1	2	yes	—
121	C₃	#1718	1	1	yes	characteristic, complemented, Sylow
122	C₂	#4696	1	1	yes	characteristic
123	C₂	#4715	1	1	yes	characteristic
124	C₂	#4715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#4682

C₂

yes

maximal, characteristic

#4681

C₂

yes

maximal, characteristic

#4681

C₂

yes

maximal, characteristic

#4681

C₂

yes

maximal, characteristic

#4681

C₂

yes

maximal, characteristic

#4680

C₂

yes

maximal, characteristic

C₃×#322

C₂

yes

maximal, characteristic

#1718

—

maximal, Sylow

#869

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

C₂×Dic₆

—

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

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

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

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

C₂²×C₃

yes

C₃×C₄

yes

—

C₃×C₄

#137

yes

—

C₃×C₄

#137

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₁₂

yes

characteristic

105

C₂×C₄

C₂×D₁₂

yes

characteristic

106

C₂×C₄

C₂×D₁₂

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂×C₃

#495

yes

characteristic

111

C₂×C₃

#497

yes

characteristic

112

C₂×C₃

#497

yes

characteristic

113

C₂²

#874

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#904

yes

—

119

C₄

#909

yes

—

120

C₄

#909

yes

—

121

C₃

#1718

yes

characteristic, complemented, Sylow

122

C₂

#4696

yes

characteristic

123

C₂

#4715

yes

characteristic

124

C₂

#4715

yes

characteristic

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#322	D₃
hypocenter	C₃	#1718
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

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
121. C₃	#1718	semidirect

H ≅ G/N

type

121. C₃

#1718

semidirect

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

subgroup	index
#4682	2
#4681	2
#4680	2
C₃×#322	2
#1718	3

subgroup

index

Subgroups — #68096

Conjugacy classes of subgroups

Complements

Maximal subgroups