Group #68094 — subgroups

Subgroups — #68094

Order 384.

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

normal subgroups only

This group has 242 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	#4695	C₂	1	1	yes	maximal, characteristic, complemented
2	#4695	C₂	1	2	yes	maximal, complemented
3	#4694	C₂	1	2	yes	maximal, complemented
4	#4693	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#1721	—	3	1	no	maximal, Sylow
7	C₂×D₂₄	—	2	4	no	—
8	C₂×D₂₄	—	2	2	no	—
9	C₂×#204	—	2	2	no	—
10	#874	C₂²	1	1	yes	characteristic
11	#874	C₂²	1	2	yes	—
12	#869	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₈	C₂²	1	2	yes	—
15	#821	—	2	4	no	—
16	#493	—	3	2	no	—
17	#493	—	3	1	no	—
18	#492	—	3	1	no	—
19	#486	—	3	2	no	—
20	#322	D₃	1	1	yes	characteristic, complemented
21	C₂×D₁₂	—	2	2	no	—
22	C₂×D₁₂	—	2	1	no	—
23	C₂×D₁₂	—	2	2	no	—
24	C₂×D₁₂	—	2	2	no	—
25	C₂×D₁₂	—	2	2	no	—
26	C₂×Dic₆	—	2	1	no	—
27	C₂×C₃×C₈	D₄	1	4	yes	—
28	C₂×C₃×C₈	D₄	1	2	yes	—
29	C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₃	—	2	2	no	—
31	D₂₄	—	4	8	no	—
32	D₂₄	—	4	4	no	—
33	#204	—	4	4	no	—
34	C₂×SD₁₆	—	6	2	no	—
35	C₂×D₈	—	6	4	no	—
36	C₂×D₈	—	6	2	no	—
37	C₄⋊Q₈	—	3	1	no	—
38	#127	—	3	2	no	—
39	#127	—	3	1	no	—
40	#102	—	6	4	no	—
41	C₄×C₈	C₂×D₃	1	2	yes	—
42	C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂²×D₃	—	4	2	no	—
44	C₂²×D₃	—	4	1	no	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
47	C₂×Dic₃	—	4	1	no	—
48	D₁₂	—	4	2	no	—
49	D₁₂	—	4	4	no	—
50	D₁₂	—	4	4	no	—
51	D₁₂	—	4	4	no	—
52	D₁₂	—	4	4	no	—
53	Dic₆	—	4	2	no	—
54	C₃×C₈	—	2	4	no	—
55	C₃×C₈	D₈	1	4	yes	complemented
56	C₂×Q₈	—	6	1	no	—
57	C₂×D₄	—	6	2	no	—
58	C₂×D₄	—	6	2	no	—
59	C₂×D₄	—	6	2	no	—
60	C₂×D₄	—	6	2	no	—
61	C₂×D₄	—	6	1	no	—
62	SD₁₆	—	12	4	no	—
63	D₈	—	12	8	no	—
64	D₈	—	12	4	no	—
65	C₂×C₈	D₁₂	1	4	yes	—
66	C₂×C₈	D₁₂	1	2	yes	—
67	C₄⋊C₄	—	6	2	no	—
68	C₄²	C₂²×D₃	1	1	yes	characteristic
69	C₂²×C₃	#127	1	1	yes	characteristic
70	C₂×D₃	—	8	1	no	—
71	C₂×D₃	—	8	2	no	—
72	C₂×D₃	—	8	2	no	—
73	C₂×D₃	—	8	2	no	—
74	C₂×D₃	—	8	2	no	—
75	C₃×C₄	C₂×D₈	1	2	yes	—
76	C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
77	Dic₃	—	8	1	no	—
78	C₂³	—	12	2	no	—
79	C₂³	—	12	1	no	—
80	Q₈	—	12	2	no	—
81	D₄	—	12	4	no	—
82	D₄	—	12	2	no	—
83	D₄	—	12	4	no	—
84	D₄	—	12	4	no	—
85	D₄	—	12	4	no	—
86	C₂×C₄	—	12	1	no	—
87	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₂	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	D₂₄	1	4	yes	complemented
91	C₂×C₃	#493	1	1	yes	characteristic
92	C₂×C₃	#496	1	2	yes	—
93	D₃	—	16	1	no	—
94	D₃	—	16	2	no	—
95	C₂²	—	24	2	no	—
96	C₂²	—	24	1	no	—
97	C₂²	—	24	2	no	—
98	C₂²	—	24	2	no	—
99	C₂²	—	24	2	no	—
100	C₂²	#874	1	1	yes	characteristic
101	C₄	—	24	1	no	—
102	C₄	C₂×D₂₄	1	2	yes	—
103	C₄	#908	1	4	yes	—
104	C₃	#1721	1	1	yes	characteristic, complemented, Sylow
105	C₂	—	48	2	no	—
106	C₂	—	48	1	no	—
107	C₂	#4695	1	1	yes	characteristic
108	C₂	#4712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#4695

C₂

yes

maximal, characteristic, complemented

#4695

C₂

yes

maximal, complemented

#4694

C₂

yes

maximal, complemented

#4693

C₂

yes

maximal, characteristic, complemented

C₃×#322

C₂

yes

maximal, characteristic, complemented

#1721

—

maximal, Sylow

C₂×D₂₄

—

C₂×D₂₄

—

C₂×#204

—

#874

C₂²

yes

characteristic

#874

C₂²

yes

—

#869

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×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₄⋊Dic₃

—

D₂₄

—

D₂₄

—

#204

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

C₄×C₈

C₂×D₃

yes

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

C₃×C₈

—

C₃×C₈

D₈

yes

complemented

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₂×D₈

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

Dic₃

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

—

C₈

—

C₈

D₂₄

yes

C₂×C₃

yes

C₂×C₃

yes

—

D₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

#874

yes

characteristic

101

C₄

—

102

C₄

C₂×D₂₄

yes

—

103

C₄

#908

yes

—

104

C₃

#1721

yes

characteristic, complemented, Sylow

105

C₂

—

106

C₂

—

107

C₂

#4695

yes

characteristic

108

C₂

#4712

yes

—

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle	C₂²×C₃	#127
hypercenter	#322	D₃
hypocenter	C₃	#1721
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
1. #4695	C₂	semidirect
2. #4695	C₂	semidirect
3. #4694	C₂	semidirect
4. #4693	C₂	semidirect
5. C₃×#322	C₂	semidirect
20. #322	D₃	semidirect
55. C₃×C₈	D₈	semidirect
90. C₈	D₂₄	semidirect
104. C₃	#1721	semidirect

H ≅ G/N

type

1. #4695

C₂

semidirect

2. #4695

C₂

semidirect

3. #4694

C₂

semidirect

4. #4693

C₂

semidirect

5. C₃×#322

C₂

semidirect

20. #322

D₃

semidirect

55. C₃×C₈

D₈

semidirect

90. C₈

D₂₄

semidirect

104. C₃

#1721

semidirect

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

subgroup	index
#4695	2
#4694	2
#4693	2
C₃×#322	2
#1721	3

subgroup

index

Subgroups — #68094

Conjugacy classes of subgroups

Complements

Maximal subgroups