Group #68091 — subgroups

Subgroups — #68091

Order 384.

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

normal subgroups only

This group has 220 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
2	#4694	C₂	1	2	yes	maximal, complemented
3	#4693	C₂	1	2	yes	maximal, complemented
4	#4680	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#1715	—	3	1	no	maximal, Sylow
7	C₂×D₂₄	—	2	2	no	—
8	C₂×#204	—	2	4	no	—
9	#874	C₂²	1	1	yes	characteristic
10	#874	C₂²	1	1	yes	characteristic
11	#869	C₂²	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₈	C₂²	1	2	yes	—
14	#821	—	2	4	no	—
15	#817	—	2	2	no	—
16	#498	—	3	1	no	—
17	#493	—	3	1	no	—
18	#492	—	3	2	no	—
19	#486	—	3	2	no	—
20	#322	D₃	1	1	yes	characteristic, complemented
21	C₂×D₁₂	—	2	2	no	—
22	C₂×D₁₂	—	2	2	no	—
23	C₂×D₁₂	—	2	1	no	—
24	C₂×D₁₂	—	2	1	no	—
25	C₂×Dic₆	—	2	2	no	—
26	C₂×C₃×C₈	D₄	1	4	yes	—
27	C₂×C₃×C₈	D₄	1	2	yes	—
28	C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	D₂₄	—	4	4	no	—
32	#204	—	4	8	no	—
33	C₂×SD₁₆	—	6	4	no	—
34	C₂×D₈	—	6	2	no	—
35	C₄⋊Q₈	—	3	2	no	—
36	#127	—	3	1	no	—
37	#127	—	3	1	no	—
38	#106	—	6	2	no	—
39	#102	—	6	4	no	—
40	C₄×C₈	C₂×D₃	1	2	yes	—
41	C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×D₃	—	4	1	no	—
43	C₂²×D₃	—	4	1	no	—
44	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
46	C₂×Dic₃	—	4	2	no	—
47	D₁₂	—	4	2	no	—
48	D₁₂	—	4	2	no	—
49	D₁₂	—	4	4	no	—
50	D₁₂	—	4	4	no	—
51	Dic₆	—	4	4	no	—
52	C₃×C₈	—	2	4	no	—
53	C₃×C₈	SD₁₆	1	4	yes	complemented
54	C₂×Q₈	—	6	2	no	—
55	C₂×D₄	—	6	2	no	—
56	C₂×D₄	—	6	1	no	—
57	C₂×D₄	—	6	1	no	—
58	C₂×D₄	—	6	2	no	—
59	SD₁₆	—	12	8	no	—
60	D₈	—	12	4	no	—
61	C₂×C₈	D₁₂	1	4	yes	—
62	C₂×C₈	D₁₂	1	2	yes	—
63	C₄⋊C₄	—	6	2	no	—
64	C₄⋊C₄	—	6	2	no	—
65	C₄²	C₂²×D₃	1	1	yes	characteristic
66	C₂²×C₃	#127	1	1	yes	characteristic
67	C₂×D₃	—	8	1	no	—
68	C₂×D₃	—	8	1	no	—
69	C₂×D₃	—	8	2	no	—
70	C₂×D₃	—	8	2	no	—
71	C₃×C₄	C₂×SD₁₆	1	2	yes	—
72	C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
73	Dic₃	—	8	2	no	—
74	C₂³	—	12	1	no	—
75	C₂³	—	12	1	no	—
76	Q₈	—	12	4	no	—
77	D₄	—	12	2	no	—
78	D₄	—	12	2	no	—
79	D₄	—	12	4	no	—
80	D₄	—	12	4	no	—
81	C₂×C₄	—	12	2	no	—
82	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
83	C₂×C₄	C₂×D₁₂	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#204	1	4	yes	complemented
86	C₂×C₃	#492	1	1	yes	characteristic
87	C₂×C₃	#496	1	2	yes	—
88	D₃	—	16	1	no	—
89	D₃	—	16	1	no	—
90	C₂²	—	24	1	no	—
91	C₂²	—	24	1	no	—
92	C₂²	—	24	2	no	—
93	C₂²	—	24	2	no	—
94	C₂²	#874	1	1	yes	characteristic
95	C₄	—	24	2	no	—
96	C₄	C₂×#204	1	2	yes	—
97	C₄	#908	1	4	yes	—
98	C₃	#1715	1	1	yes	characteristic, complemented, Sylow
99	C₂	—	48	1	no	—
100	C₂	—	48	1	no	—
101	C₂	#4693	1	1	yes	characteristic
102	C₂	#4712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#4695

C₂

yes

maximal, characteristic

#4694

C₂

yes

maximal, complemented

#4693

C₂

yes

maximal, complemented

#4680

C₂

yes

maximal, characteristic, complemented

C₃×#322

C₂

yes

maximal, characteristic, complemented

#1715

—

maximal, Sylow

C₂×D₂₄

—

C₂×#204

—

#874

C₂²

yes

characteristic

#874

C₂²

yes

characteristic

#869

C₂²

yes

—

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

—

—

—

—

—

—

—

D₃

yes

characteristic, complemented

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

—

C₄⋊Dic₃

—

D₂₄

—

#204

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

—

—

—

—

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

—

Dic₆

—

C₃×C₈

—

C₃×C₈

SD₁₆

yes

complemented

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

Dic₃

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

—

C₈

—

C₈

yes

C₂×C₃

yes

C₂×C₃

yes

—

D₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#874

yes

characteristic

C₄

—

C₄

C₂×#204

yes

—

C₄

#908

yes

—

C₃

#1715

yes

characteristic, complemented, Sylow

C₂

—

100

C₂

—

101

C₂

#4693

yes

characteristic

102

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
prime radical	#4695	C₂
hypercenter	#322	D₃
hypocenter	C₃	#1715
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

C₂

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
2. #4694	C₂	semidirect
3. #4693	C₂	semidirect
4. #4680	C₂	semidirect
5. C₃×#322	C₂	semidirect
20. #322	D₃	semidirect
53. C₃×C₈	SD₁₆	semidirect
85. C₈	#204	semidirect
98. C₃	#1715	semidirect

H ≅ G/N

type

2. #4694

C₂

semidirect

3. #4693

C₂

semidirect

4. #4680

C₂

semidirect

5. C₃×#322

C₂

semidirect

20. #322

D₃

semidirect

53. C₃×C₈

SD₁₆

semidirect

85. C₈

#204

semidirect

98. C₃

#1715

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
#4680	2
C₃×#322	2
#1715	3

subgroup

index

Subgroups — #68091

Conjugacy classes of subgroups

Complements

Maximal subgroups