Subgroups — C₂²×C₃×D₁₆ #85272

Order 384.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×D₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×D₄	—	2	2	no	—
6	C₂×C₃×D₈	—	2	6	no	—
7	C₂×C₃×D₈	C₂²	1	12	yes	complemented
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₃×D₁₆	C₂²	1	16	yes	direct product factor
10	C₂×C₃×C₁₆	C₂²	1	6	yes	complemented
11	C₂²×D₈	C₂×C₃	1	2	yes	complemented
12	C₂×D₁₆	C₂×C₃	1	12	yes	direct product factor
13	C₂²×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂⁴×C₃	—	4	2	no	—
15	C₂×C₃×D₄	—	2	12	no	—
16	C₂×C₃×D₄	—	4	6	no	—
17	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
18	C₃×D₈	—	2	12	no	—
19	C₃×D₈	C₂³	1	8	yes	complemented
20	C₂×C₃×C₈	D₄	1	3	yes	—
21	C₂×C₃×C₈	C₂³	1	3	yes	—
22	C₃×C₁₆	C₂³	1	4	yes	complemented
23	C₂²×D₄	—	2	2	no	—
24	C₂×D₈	—	2	6	no	—
25	C₂×D₈	C₂²×C₃	1	12	yes	complemented
26	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
27	D₁₆	C₂²×C₃	1	16	yes	direct product factor
28	C₂×C₁₆	C₂²×C₃	1	6	yes	complemented
29	C₂³×C₃	—	4	12	no	—
30	C₂³×C₃	—	8	8	no	—
31	C₂³×C₃	D₈	1	1	yes	characteristic
32	C₃×D₄	—	2	8	no	—
33	C₃×D₄	—	4	12	no	—
34	C₂×C₃×C₄	D₈	1	3	yes	—
35	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
36	C₃×C₈	C₂×D₄	1	3	yes	—
37	C₃×C₈	C₂⁴	1	1	yes	characteristic
38	C₂⁴	—	4	2	no	—
39	C₂×D₄	—	2	12	no	—
40	C₂×D₄	—	4	6	no	—
41	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
42	D₈	—	2	12	no	—
43	D₈	C₂³×C₃	1	8	yes	complemented
44	C₂×C₈	C₃×D₄	1	3	yes	—
45	C₂×C₈	C₂³×C₃	1	3	yes	—
46	C₁₆	C₂³×C₃	1	4	yes	complemented
47	C₂²×C₃	—	4	8	no	—
48	C₂²×C₃	—	8	24	no	—
49	C₂²×C₃	D₁₆	1	4	yes	direct product factor
50	C₂²×C₃	C₂×D₈	1	3	yes	—
51	C₃×C₄	C₂×D₈	1	3	yes	—
52	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
53	C₂³	—	4	12	no	—
54	C₂³	—	8	8	no	—
55	C₂³	C₃×D₈	1	1	yes	characteristic
56	D₄	—	2	8	no	—
57	D₄	—	4	12	no	—
58	C₂×C₄	C₃×D₈	1	3	yes	—
59	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
60	C₈	C₂×C₃×D₄	1	3	yes	—
61	C₈	C₂⁴×C₃	1	1	yes	characteristic
62	C₂×C₃	—	8	8	no	—
63	C₂×C₃	C₂×D₁₆	1	6	yes	direct product factor
64	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
65	C₂²	—	4	8	no	—
66	C₂²	—	8	24	no	—
67	C₂²	C₃×D₁₆	1	4	yes	direct product factor
68	C₂²	C₂×C₃×D₈	1	3	yes	—
69	C₄	C₂×C₃×D₈	1	3	yes	—
70	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
71	C₃	C₂²×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	8	8	no	—
73	C₂	C₂×C₃×D₁₆	1	6	yes	direct product factor
74	C₂	C₂²×C₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₈	C₂	semidirect
2. C₂×C₃×D₁₆	C₂	direct (N × H)
3. C₂²×C₃×C₁₆	C₂	semidirect
4. C₂²×D₁₆	C₃	direct (N × H)
7. C₂×C₃×D₈	C₂²	semidirect
9. C₃×D₁₆	C₂²	direct (N × H)
10. C₂×C₃×C₁₆	C₂²	semidirect
11. C₂²×D₈	C₂×C₃	semidirect
12. C₂×D₁₆	C₂×C₃	direct (N × H)
13. C₂²×C₁₆	C₂×C₃	semidirect
19. C₃×D₈	C₂³	semidirect
22. C₃×C₁₆	C₂³	semidirect
25. C₂×D₈	C₂²×C₃	semidirect
27. D₁₆	C₂²×C₃	direct (N × H)
28. C₂×C₁₆	C₂²×C₃	semidirect
43. D₈	C₂³×C₃	semidirect
46. C₁₆	C₂³×C₃	semidirect
49. C₂²×C₃	D₁₆	direct (N × H)
63. C₂×C₃	C₂×D₁₆	direct (N × H)
67. C₂²	C₃×D₁₆	direct (N × H)
71. C₃	C₂²×D₁₆	direct (N × H)
73. C₂	C₂×C₃×D₁₆	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×D₈	2
C₂×C₃×D₁₆	2
C₂²×C₃×C₁₆	2
C₂²×D₁₆	3

Subgroups — C22×C3×D16 #85272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₁₆ #85272