Subgroups — Q₈×Dic₆ #5566

Order 192.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃	C₂	1	2	yes	maximal
3	#888	C₂	1	6	yes	maximal
4	#869	C₂	1	3	yes	maximal
5	C₄×Dic₆	C₂	1	3	yes	maximal
6	Q₈²	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×Dic₆	C₂²	1	6	yes	—
9	C₂×Dic₆	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₃×C₄²	C₂²	1	3	yes	—
12	C₄⋊Dic₃	C₂²	1	3	yes	—
13	C₄⋊Dic₃	C₂²	1	6	yes	—
14	Dic₃⋊C₄	C₂²	1	6	yes	—
15	C₄×Dic₃	C₂²	1	6	yes	—
16	C₄⋊Q₈	—	3	6	no	—
17	C₄⋊Q₈	—	3	3	no	—
18	C₄×Q₈	—	3	2	no	—
19	C₄×Q₈	—	3	3	no	—
20	C₄×Q₈	D₃	1	1	yes	characteristic
21	C₃×Q₈	Q₈	1	4	yes	complemented
22	C₂×C₃×C₄	C₂³	1	3	yes	—
23	C₂×C₃×C₄	C₂³	1	3	yes	—
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	C₂³	1	6	yes	—
26	C₂×Dic₃	C₂³	1	2	yes	—
27	Dic₆	—	4	6	no	—
28	Dic₆	Q₈	1	4	yes	direct product factor
29	C₂×Q₈	—	3	1	no	—
30	C₂×Q₈	—	3	6	no	—
31	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
32	C₄⋊C₄	—	3	6	no	—
33	C₄⋊C₄	—	3	6	no	—
34	C₄⋊C₄	—	3	3	no	—
35	C₄⋊C₄	C₂×D₃	1	3	yes	—
36	C₄²	—	3	6	no	—
37	C₄²	C₂×D₃	1	3	yes	—
38	C₂²×C₃	C₂⁴	1	1	yes	characteristic
39	C₃×C₄	—	2	3	no	—
40	C₃×C₄	C₂×Q₈	1	6	yes	—
41	C₃×C₄	C₂×Q₈	1	2	yes	—
42	Dic₃	—	2	6	no	—
43	Dic₃	C₂×Q₈	1	4	yes	—
44	Q₈	—	3	4	no	—
45	Q₈	—	12	6	no	—
46	Q₈	Dic₆	1	4	yes	direct product factor
47	C₂×C₄	—	3	2	no	—
48	C₂×C₄	—	3	6	no	—
49	C₂×C₄	C₂²×D₃	1	3	yes	—
50	C₂×C₄	C₂²×D₃	1	3	yes	—
51	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂²	C₂³×D₃	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	—	3	4	no	—
58	C₄	—	6	6	no	—
59	C₄	C₂×Dic₆	1	6	yes	—
60	C₄	D₃×Q₈	1	2	yes	—
61	C₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
62	C₂	C₂²×Dic₆	1	1	yes	characteristic
63	C₂	C₂×D₃×Q₈	1	1	yes	characteristic
64	C₂	#1009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄×Q₈	C₂
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	Q₈²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₃×Q₈	Q₈	semidirect
28. Dic₆	Q₈	direct (N × H)
46. Q₈	Dic₆	direct (N × H)
61. C₃	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈	2
Q₈×Dic₃	2
#888	2
#869	2
C₄×Dic₆	2
Q₈²	3

Subgroups — Q8×Dic6 #5566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₆ #5566