Subgroups — Q₈×Dic₂₁ #10621428

Order 672.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₂₁	C₂	1	3	yes	maximal
3	C₄×Dic₂₁	C₂	1	3	yes	maximal
4	Q₈×Dic₇	—	3	1	no	maximal, Hall
5	C₃×C₇×Q₈	C₄	1	4	yes	complemented
6	C₂×C₃×C₄×C₇	C₂²	1	3	yes	—
7	C₂×Dic₂₁	C₂²	1	3	yes	—
8	C₂×Dic₂₁	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₈	D₃	1	1	yes	characteristic
10	C₄⋊Dic₇	—	3	3	no	—
11	C₄×Dic₇	—	3	3	no	—
12	Q₈×Dic₃	—	7	1	no	maximal, Hall
13	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
14	C₃×C₄×C₇	C₂×C₄	1	6	yes	—
15	Dic₂₁	—	2	3	no	—
16	Dic₂₁	Q₈	1	2	yes	direct product factor
17	C₇×Q₈	Dic₃	1	4	yes	complemented
18	C₂×C₄×C₇	C₂×D₃	1	3	yes	—
19	C₂×Dic₇	—	3	1	no	—
20	C₂×Dic₇	—	3	3	no	—
21	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
22	C₄⋊Dic₃	—	7	3	no	—
23	C₄×Dic₃	—	7	3	no	—
24	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₃×C₇	#41	1	1	yes	characteristic
27	C₄×Q₈	—	21	1	no	Sylow
28	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
29	C₄×C₇	C₂×Dic₃	1	6	yes	—
30	Dic₇	—	3	2	no	—
31	Dic₇	—	6	3	no	—
32	C₃×Q₈	Dic₇	1	4	yes	complemented
33	C₂×C₃×C₄	C₂×D₇	1	3	yes	—
34	C₂×Dic₃	—	7	1	no	—
35	C₂×Dic₃	—	7	3	no	—
36	C₃×C₇	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
37	C₂×Q₈	D₂₁	1	1	yes	characteristic
38	C₄⋊C₄	—	21	3	no	—
39	C₄²	—	21	3	no	—
40	C₂×C₇	D₃×Q₈	1	1	yes	characteristic
41	C₂×C₇	#239	1	1	yes	characteristic
42	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
43	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
44	C₃×C₄	C₂×Dic₇	1	6	yes	—
45	Dic₃	—	7	2	no	—
46	Dic₃	—	14	3	no	—
47	Q₈	Dic₂₁	1	4	yes	direct product factor
48	C₂×C₄	—	21	1	no	—
49	C₂×C₄	—	21	3	no	—
50	C₂×C₄	C₂×D₂₁	1	3	yes	—
51	C₇	Q₈×Dic₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	Q₈×D₇	1	1	yes	characteristic
53	C₂×C₃	#1161	1	1	yes	characteristic
54	C₂×C₃	C₂²×Dic₇	1	1	yes	characteristic
55	C₂²	C₂²×D₂₁	1	1	yes	characteristic
56	C₄	—	21	2	no	—
57	C₄	—	42	3	no	—
58	C₄	C₂×Dic₂₁	1	6	yes	—
59	C₃	Q₈×Dic₇	1	1	yes	characteristic, complemented, Sylow
60	C₂	Q₈×D₂₁	1	1	yes	characteristic
61	C₂	#66732	1	1	yes	characteristic
62	C₂	C₂²×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₁
commutator	C₂×C₃×C₇	C₂²×C₄
Fitting	C₂×C₃×C₇×Q₈	C₂
socle, prime radical, cosocle	C₂²×C₃×C₇	C₂³
hypercenter	C₂×Q₈	D₂₁
hypocenter	C₃×C₇	C₄×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. C₃×C₇×Q₈	C₄	semidirect
16. Dic₂₁	Q₈	direct (N × H)
17. C₇×Q₈	Dic₃	semidirect
32. C₃×Q₈	Dic₇	semidirect
36. C₃×C₇	C₄×Q₈	semidirect
47. Q₈	Dic₂₁	direct (N × H)
51. C₇	Q₈×Dic₃	semidirect
59. C₃	Q₈×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×Q₈	2
C₄⋊Dic₂₁	2
C₄×Dic₂₁	2
Q₈×Dic₇	3
Q₈×Dic₃	7

Subgroups — Q8×Dic21 #10621428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₂₁ #10621428