Subgroups — Q₈×D₁₂ #5575

Order 192.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#897	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₂	C₂	1	3	yes	maximal
5	#869	C₂	1	3	yes	maximal, complemented
6	D₄×Q₈	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	D₃×Q₈	—	2	8	no	—
9	C₂×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃	C₂²	1	6	yes	—
11	C₂×Dic₆	C₂²	1	6	yes	—
12	C₃×C₄⋊C₄	C₂²	1	3	yes	—
13	C₃×C₄²	C₂²	1	3	yes	—
14	(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
15	C₄⋊Dic₃	C₂²	1	3	yes	—
16	C₄⋊Dic₃	C₂²	1	6	yes	—
17	C₂²×Q₈	—	3	2	no	—
18	C₄⋊Q₈	—	3	3	no	—
19	C₂²⋊Q₈	—	3	6	no	—
20	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
21	C₄×D₄	—	3	3	no	—
22	C₂²×D₃	C₂³	1	2	yes	—
23	C₃×Q₈	D₄	1	4	yes	complemented
24	C₂×C₃×C₄	C₂³	1	3	yes	—
25	C₂×C₃×C₄	C₂³	1	3	yes	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	C₂³	1	6	yes	—
28	D₁₂	Q₈	1	4	yes	direct product factor
29	C₄×D₃	—	2	12	no	—
30	Dic₆	—	2	12	no	—
31	C₂×Q₈	—	3	6	no	—
32	C₂×Q₈	—	6	8	no	—
33	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
34	C₂×D₄	—	3	1	no	—
35	C₂²×C₄	—	3	6	no	—
36	C₄⋊C₄	—	3	3	no	—
37	C₄⋊C₄	—	3	6	no	—
38	C₄⋊C₄	C₂×D₃	1	3	yes	—
39	C₂²⋊C₄	—	3	6	no	—
40	C₄²	C₂×D₃	1	3	yes	—
41	C₂²×C₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₃	—	2	2	no	—
43	C₂×D₃	—	2	2	no	—
44	C₂×D₃	C₂×Q₈	1	4	yes	complemented
45	C₃×C₄	—	2	3	no	—
46	C₃×C₄	C₂×D₄	1	6	yes	—
47	C₃×C₄	C₂×Q₈	1	2	yes	complemented
48	Dic₃	—	2	6	no	—
49	C₂³	—	3	2	no	—
50	Q₈	—	6	12	no	—
51	Q₈	D₁₂	1	4	yes	direct product factor
52	D₄	—	3	4	no	—
53	C₂×C₄	—	3	6	no	—
54	C₂×C₄	—	6	12	no	—
55	C₂×C₄	C₂²×D₃	1	3	yes	—
56	C₂×C₄	C₂²×D₃	1	3	yes	—
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
61	D₃	—	2	4	no	—
62	C₂²	—	3	4	no	—
63	C₂²	—	6	2	no	—
64	C₂²	—	6	2	no	—
65	C₂²	C₂³×D₃	1	1	yes	characteristic
66	C₄	—	2	3	no	—
67	C₄	—	6	6	no	—
68	C₄	C₂×D₁₂	1	6	yes	—
69	C₄	D₃×Q₈	1	2	yes	complemented
70	C₃	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	6	4	no	—
72	C₂	C₂²×D₁₂	1	1	yes	characteristic
73	C₂	C₂×D₃×Q₈	1	1	yes	characteristic
74	C₂	#1010	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄×Q₈	C₂
prime radical	C₂×D₁₂	C₂²
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	D₄×Q₈
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
1. C₂×D₃×Q₈	C₂	semidirect
2. C₃×C₄×Q₈	C₂	semidirect
3. #897	C₂	semidirect
5. #869	C₂	semidirect
20. C₄×Q₈	D₃	semidirect
23. C₃×Q₈	D₄	semidirect
28. D₁₂	Q₈	direct (N × H)
44. C₂×D₃	C₂×Q₈	semidirect
47. C₃×C₄	C₂×Q₈	semidirect
51. Q₈	D₁₂	direct (N × H)
69. C₄	D₃×Q₈	semidirect
70. C₃	D₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₈	2
C₃×C₄×Q₈	2
#897	2
C₄×D₁₂	2
#869	2
D₄×Q₈	3

Subgroups — Q8×D12 #5575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₂ #5575