Group #68504 — subgroups

Subgroups — #68504

Order 384.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#5722	C₂	1	1	yes	maximal, characteristic, complemented
2	#5085	C₂	1	1	yes	maximal, characteristic
3	#4848	C₂	1	2	yes	maximal
4	C₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	#4484	C₂	1	2	yes	maximal, complemented
6	#1623	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#944	—	2	2	no	—
9	C₂×#215	—	2	2	no	—
10	C₂×#204	—	2	2	no	—
11	#888	—	2	2	no	—
12	#887	—	2	2	no	—
13	#874	C₂²	1	1	yes	characteristic
14	C₄×Dic₆	C₂²	1	2	yes	—
15	C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	#821	—	2	2	no	—
17	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	Q₈²	—	3	1	no	—
19	#492	—	3	1	no	—
20	#460	—	3	2	no	—
21	#333	D₃	1	1	yes	characteristic, complemented
22	Q₈⋊C₈	—	3	2	no	—
23	C₂×C₃×Q₈	D₄	1	2	yes	—
24	C₂×D₁₂	—	2	2	no	—
25	C₂×D₁₂	—	2	1	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	D₄	1	2	yes	—
28	C₂×C₃×C₈	—	2	2	no	—
29	C₃×C₄⋊C₄	—	2	2	no	—
30	C₃×C₄²	C₂³	1	1	yes	characteristic
31	#215	—	4	4	no	—
32	C₄⋊Dic₃	D₄	1	2	yes	—
33	Dic₃⋊C₄	—	2	4	no	—
34	Dic₃⋊C₄	—	2	2	no	—
35	C₄×Dic₃	—	2	2	no	—
36	C₄×Dic₃	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	1	no	—
38	#204	—	4	4	no	—
39	C₂×SD₁₆	—	6	2	no	—
40	C₂×SD₁₆	—	6	2	no	—
41	C₄⋊Q₈	—	6	2	no	—
42	C₄⋊Q₈	—	6	2	no	—
43	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
44	#127	—	3	1	no	—
45	C₄×Q₈	—	3	2	no	—
46	C₄×Q₈	—	6	2	no	—
47	C₄⋊C₈	C₂×D₃	1	2	yes	—
48	#102	—	6	2	no	—
49	C₄×C₈	—	3	1	no	—
50	C₂²×D₃	—	4	1	no	—
51	C₃×Q₈	—	4	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂×Dic₃	—	2	2	no	—
56	C₂×Dic₃	—	2	2	no	—
57	D₁₂	—	4	2	no	—
58	D₁₂	—	4	2	no	—
59	D₁₂	—	4	2	no	—
60	Dic₆	—	2	2	no	—
61	Dic₆	—	8	2	no	—
62	Dic₆	SD₁₆	1	4	yes	complemented
63	C₃×C₈	—	4	2	no	—
64	C₃⋊C₈	—	2	2	no	—
65	C₂×Q₈	—	3	2	no	—
66	C₂×Q₈	—	6	2	no	—
67	C₂×Q₈	D₁₂	1	2	yes	—
68	C₂×D₄	—	6	1	no	—
69	C₂×D₄	—	6	2	no	—
70	SD₁₆	—	12	4	no	—
71	SD₁₆	—	12	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	6	1	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	3	2	no	—
76	C₄⋊C₄	—	6	4	no	—
77	C₄⋊C₄	—	6	2	no	—
78	C₄²	—	6	2	no	—
79	C₄²	—	6	2	no	—
80	C₄²	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃	#120	1	1	yes	characteristic
82	C₂×D₃	—	8	2	no	—
83	C₂×D₃	—	8	1	no	—
84	C₃×C₄	—	2	1	no	—
85	C₃×C₄	—	4	2	no	—
86	C₃×C₄	C₂×SD₁₆	1	2	yes	—
87	C₃×C₄	#137	1	2	yes	—
88	Dic₃	—	2	4	no	—
89	Dic₃	—	4	2	no	—
90	C₂³	—	12	1	no	—
91	Q₈	—	3	4	no	—
92	Q₈	—	4	2	no	—
93	Q₈	—	6	2	no	—
94	Q₈	—	24	2	no	—
95	D₄	—	12	2	no	—
96	D₄	—	12	2	no	—
97	D₄	—	12	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	6	2	no	—
100	C₂×C₄	—	6	2	no	—
101	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
102	C₂×C₄	D₃×D₄	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	6	2	no	—
105	C₂×C₃	#448	1	2	yes	—
106	C₂×C₃	#453	1	1	yes	characteristic
107	D₃	—	16	1	no	—
108	C₂²	—	24	1	no	—
109	C₂²	—	24	2	no	—
110	C₂²	#882	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	4	2	no	—
113	C₄	—	6	4	no	—
114	C₄	—	12	2	no	—
115	C₄	D₃×SD₁₆	1	2	yes	—
116	C₄	#918	1	2	yes	—
117	C₃	#1623	1	1	yes	characteristic, complemented, Sylow
118	C₂	—	48	1	no	—
119	C₂	#4747	1	1	yes	characteristic
120	C₂	#4806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5722

C₂

yes

maximal, characteristic, complemented

#5085

C₂

yes

maximal, characteristic

#4848

C₂

yes

maximal

C₃×#333

C₂

yes

maximal, characteristic, complemented

#4484

C₂

yes

maximal, complemented

#1623

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₆

C₂²

yes

—

C₃×C₄⋊C₈

C₂²

yes

—

#821

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

Q₈²

—

#492

—

#460

—

#333

D₃

yes

characteristic, complemented

Q₈⋊C₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#215

—

C₄⋊Dic₃

D₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

#204

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

—

C₄⋊C₈

C₂×D₃

yes

—

#102

—

C₄×C₈

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

Dic₆

SD₁₆

yes

complemented

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#120

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

#137

yes

—

Dic₃

—

Dic₃

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×D₁₂

yes

characteristic

102

C₂×C₄

D₃×D₄

yes

—

103

C₈

—

104

C₈

—

105

C₂×C₃

#448

yes

—

106

C₂×C₃

#453

yes

characteristic

107

D₃

—

108

C₂²

—

109

C₂²

—

110

C₂²

#882

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

D₃×SD₁₆

yes

—

116

C₄

#918

yes

—

117

C₃

#1623

yes

characteristic, complemented, Sylow

118

C₂

—

119

C₂

#4747

yes

characteristic

120

C₂

#4806

yes

—

	subgroup	G/H
center	C₂²	#882
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#333	C₂
socle	C₂²×C₃	#120
prime radical	#874	C₂²
hypercenter	#333	D₃
hypocenter	C₃	#1623
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#882

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 ◁ #333

N	H ≅ G/N	type
1. #5722	C₂	semidirect
4. C₃×#333	C₂	semidirect
5. #4484	C₂	semidirect
21. #333	D₃	semidirect
62. Dic₆	SD₁₆	semidirect
117. C₃	#1623	semidirect

H ≅ G/N

type

1. #5722

C₂

semidirect

4. C₃×#333

C₂

semidirect

5. #4484

C₂

semidirect

21. #333

D₃

semidirect

62. Dic₆

SD₁₆

semidirect

117. C₃

#1623

semidirect

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

subgroup	index
#5722	2
#5085	2
#4848	2
C₃×#333	2
#4484	2
#1623	3

subgroup

index

Subgroups — #68504

Conjugacy classes of subgroups

Complements

Maximal subgroups