Subgroups — C₃×D₄×Q₈ #5879

Order 192.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
2. C₃×C₄⋊Q₈	C₂	semidirect
3. C₃×C₂²⋊Q₈	C₂	semidirect
4. C₃×C₄×Q₈	C₂	semidirect
6. D₄×Q₈	C₃	direct (N × H)
17. C₂²×Q₈	C₂×C₃	semidirect
18. C₄⋊Q₈	C₂×C₃	semidirect
19. C₂²⋊Q₈	C₂×C₃	semidirect
20. C₄×Q₈	C₂×C₃	semidirect
24. C₃×Q₈	D₄	direct (N × H)
25. C₃×D₄	Q₈	direct (N × H)
43. C₂²×C₃	C₂×Q₈	semidirect
48. C₃×C₄	C₂×Q₈	semidirect
51. Q₈	C₃×D₄	direct (N × H)
52. D₄	C₃×Q₈	direct (N × H)
64. C₂²	C₂×C₃×Q₈	semidirect
69. C₄	C₂×C₃×Q₈	semidirect
70. C₃	D₄×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₃×C₄⋊Q₈	2
C₃×C₂²⋊Q₈	2
C₃×C₄×Q₈	2
C₃×C₄×D₄	2
D₄×Q₈	3

Subgroups — C3×D4×Q8 #5879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×Q₈ #5879