Group #85429 — subgroups

Subgroups — Q₈×GL(2,𝔽₃) #85429

Order 384.

255 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 44 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	Q₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×GL(2,𝔽₃)	C₂	1	3	yes	maximal
3	#5386	C₂	1	3	yes	maximal, complemented
4	Q₈×SD₁₆	—	3	1	no	maximal, Sylow
5	C₂×D₃×Q₈	—	4	1	no	maximal
6	C₂×GL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄×SL(2,𝔽₃)	C₂²	1	3	yes	—
8	#859	C₂²	1	3	yes	—
9	Q₈²	D₃	1	1	yes	characteristic, complemented
10	D₄×Q₈	—	3	1	no	—
11	#498	—	3	3	no	—
12	#476	—	3	3	no	—
13	#475	—	3	3	no	—
14	C₈×Q₈	—	3	1	no	—
15	C₄×SD₁₆	—	3	3	no	—
16	C₂×C₃×Q₈	—	4	1	no	—
17	D₃×Q₈	—	4	8	no	—
18	C₂×C₄×D₃	—	4	3	no	—
19	C₂×Dic₆	—	4	3	no	—
20	C₂×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
21	GL(2,𝔽₃)	Q₈	1	2	yes	direct product factor
22	C₂²×Q₈	—	6	1	no	—
23	C₂×SD₁₆	—	3	1	no	—
24	C₄⋊Q₈	—	3	3	no	—
25	C₄⋊Q₈	—	3	3	no	—
26	C₂²⋊Q₈	—	6	3	no	—
27	C₄×Q₈	—	3	1	no	—
28	C₄×Q₈	C₂×D₃	1	3	yes	—
29	C₄×D₄	—	3	3	no	—
30	#106	—	3	3	no	—
31	#106	—	3	6	no	—
32	C₄⋊C₈	—	3	3	no	—
33	#103	—	3	3	no	—
34	#102	—	3	3	no	—
35	C₄×C₈	—	3	3	no	—
36	C₂²×D₃	—	4	1	no	—
37	C₃×Q₈	—	4	4	no	—
38	C₂×C₃×C₄	—	4	3	no	—
39	C₂×Dic₃	—	4	3	no	—
40	C₄×D₃	—	4	12	no	—
41	Dic₆	—	4	12	no	—
42	SL(2,𝔽₃)	C₂×Q₈	1	1	yes	characteristic, complemented
43	C₂×Q₈	—	6	1	no	—
44	C₂×Q₈	—	6	3	no	—
45	C₂×Q₈	—	12	4	no	—
46	C₂×Q₈	S₄	1	1	yes	characteristic
47	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
48	C₂×D₄	—	3	1	no	—
49	C₂²×C₄	—	6	3	no	—
50	SD₁₆	—	3	2	no	—
51	SD₁₆	—	3	2	no	—
52	C₂×C₈	—	3	1	no	—
53	C₂×C₈	—	3	3	no	—
54	C₄⋊C₄	—	3	3	no	—
55	C₄⋊C₄	—	3	3	no	—
56	C₄⋊C₄	—	3	3	no	—
57	C₄⋊C₄	—	6	3	no	—
58	C₂²⋊C₄	—	6	3	no	—
59	C₄²	—	3	3	no	—
60	C₂²×C₃	—	4	1	no	—
61	C₂×D₃	—	4	2	no	—
62	C₂×D₃	—	4	2	no	—
63	C₂×D₃	—	4	2	no	—
64	C₃×C₄	—	4	6	no	—
65	Dic₃	—	4	6	no	—
66	C₂³	—	6	1	no	—
67	Q₈	—	3	1	no	—
68	Q₈	—	12	6	no	—
69	Q₈	—	24	1	no	—
70	Q₈	GL(2,𝔽₃)	1	4	yes	direct product factor
71	Q₈	D₃×Q₈	1	1	yes	characteristic, complemented
72	D₄	—	3	2	no	—
73	D₄	—	6	1	no	—
74	C₂×C₄	—	3	1	no	—
75	C₂×C₄	—	3	3	no	—
76	C₂×C₄	—	6	3	no	—
77	C₂×C₄	—	12	6	no	—
78	C₂×C₄	C₂×S₄	1	3	yes	—
79	C₈	—	3	2	no	—
80	C₈	—	6	3	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂×C₃	—	4	1	no	—
84	D₃	—	4	4	no	—
85	C₂²	—	6	2	no	—
86	C₂²	—	12	1	no	—
87	C₂²	—	12	1	no	—
88	C₂²	C₂²×S₄	1	1	yes	characteristic
89	C₄	—	3	1	no	—
90	C₄	—	3	1	no	—
91	C₄	—	6	3	no	—
92	C₄	—	12	3	no	—
93	C₄	C₂×GL(2,𝔽₃)	1	6	yes	—
94	C₃	—	4	1	no	Sylow
95	C₂	—	12	2	no	—
96	C₂	C₂²×GL(2,𝔽₃)	1	1	yes	characteristic
97	C₂	Q₈×S₄	1	1	yes	characteristic
98	C₂	#5924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×SL(2,𝔽₃)

C₂

yes

maximal, characteristic, complemented

C₄×GL(2,𝔽₃)

C₂

yes

maximal

#5386

C₂

yes

maximal, complemented

Q₈×SD₁₆

—

maximal, Sylow

C₂×D₃×Q₈

—

maximal

C₂×GL(2,𝔽₃)

C₂²

yes

characteristic

C₄×SL(2,𝔽₃)

C₂²

yes

—

#859

C₂²

yes

—

Q₈²

D₃

yes

characteristic, complemented

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂×C₃×Q₈

—

D₃×Q₈

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×SL(2,𝔽₃)

C₂³

yes

characteristic

GL(2,𝔽₃)

Q₈

yes

direct product factor

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃

yes

—

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

Dic₆

—

SL(2,𝔽₃)

C₂×Q₈

yes

characteristic, complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

S₄

yes

characteristic

C₂×Q₈

C₂²×D₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

Dic₃

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

GL(2,𝔽₃)

yes

direct product factor

Q₈

D₃×Q₈

yes

characteristic, complemented

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×S₄

yes

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×S₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×GL(2,𝔽₃)

yes

—

C₃

—

Sylow

C₂

—

C₂

C₂²×GL(2,𝔽₃)

yes

characteristic

C₂

Q₈×S₄

yes

characteristic

C₂

#5924

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂²	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³
Fitting	Q₈²	D₃
prime radical	C₂×GL(2,𝔽₃)	C₂²
hypercenter	C₂×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini, socle

C₂²

C₂²×S₄

commutator, cosocle

C₂×SL(2,𝔽₃)

C₂³

Fitting

Q₈²

D₃

prime radical

C₂×GL(2,𝔽₃)

C₂²

hypercenter

C₂×Q₈

S₄

hypocenter

SL(2,𝔽₃)

C₂×Q₈

LCS

G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
1. Q₈×SL(2,𝔽₃)	C₂	semidirect
3. #5386	C₂	semidirect
9. Q₈²	D₃	semidirect
21. GL(2,𝔽₃)	Q₈	direct (N × H)
42. SL(2,𝔽₃)	C₂×Q₈	semidirect
70. Q₈	GL(2,𝔽₃)	direct (N × H)
71. Q₈	D₃×Q₈	semidirect

H ≅ G/N

type

1. Q₈×SL(2,𝔽₃)

C₂

semidirect

3. #5386

C₂

semidirect

9. Q₈²

D₃

semidirect

21. GL(2,𝔽₃)

Q₈

direct (N × H)

42. SL(2,𝔽₃)

C₂×Q₈

semidirect

70. Q₈

GL(2,𝔽₃)

direct (N × H)

71. Q₈

D₃×Q₈

semidirect

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

subgroup	index
Q₈×SL(2,𝔽₃)	2
C₄×GL(2,𝔽₃)	2
#5386	2
Q₈×SD₁₆	3
C₂×D₃×Q₈	4

subgroup

index

Q₈×SL(2,𝔽₃)

C₄×GL(2,𝔽₃)

#5386

Q₈×SD₁₆

C₂×D₃×Q₈

Subgroups — Q₈×GL(2,𝔽₃) #85429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×GL(2,𝔽3) #85429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×GL(2,𝔽₃) #85429