Subgroups — C₂²×C₃×D₈ #5899

Order 192.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×D₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	—	2	2	no	—
6	C₂×C₃×D₄	—	2	6	no	—
7	C₂×C₃×D₄	C₂²	1	12	yes	complemented
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×D₈	C₂²	1	16	yes	direct product factor
10	C₂×C₃×C₈	C₂²	1	6	yes	complemented
11	C₂²×D₄	C₂×C₃	1	2	yes	complemented
12	C₂×D₈	C₂×C₃	1	12	yes	direct product factor
13	C₂²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂³×C₃	—	2	12	no	—
15	C₂³×C₃	—	4	8	no	—
16	C₂³×C₃	D₄	1	1	yes	characteristic
17	C₃×D₄	—	2	12	no	—
18	C₃×D₄	C₂³	1	8	yes	complemented
19	C₂×C₃×C₄	D₄	1	3	yes	—
20	C₂×C₃×C₄	C₂³	1	3	yes	—
21	C₃×C₈	C₂³	1	4	yes	complemented
22	C₂⁴	—	2	2	no	—
23	C₂×D₄	—	2	6	no	—
24	C₂×D₄	C₂²×C₃	1	12	yes	complemented
25	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
26	D₈	C₂²×C₃	1	16	yes	direct product factor
27	C₂×C₈	C₂²×C₃	1	6	yes	complemented
28	C₂²×C₃	—	2	8	no	—
29	C₂²×C₃	—	4	24	no	—
30	C₂²×C₃	D₈	1	4	yes	direct product factor
31	C₂²×C₃	C₂×D₄	1	3	yes	—
32	C₃×C₄	C₂×D₄	1	3	yes	—
33	C₃×C₄	C₂⁴	1	1	yes	characteristic
34	C₂³	—	2	12	no	—
35	C₂³	—	4	8	no	—
36	C₂³	C₃×D₄	1	1	yes	characteristic
37	D₄	—	2	12	no	—
38	D₄	C₂³×C₃	1	8	yes	complemented
39	C₂×C₄	C₃×D₄	1	3	yes	—
40	C₂×C₄	C₂³×C₃	1	3	yes	—
41	C₈	C₂³×C₃	1	4	yes	complemented
42	C₂×C₃	—	4	8	no	—
43	C₂×C₃	C₂×D₈	1	6	yes	direct product factor
44	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
45	C₂²	—	2	8	no	—
46	C₂²	—	4	24	no	—
47	C₂²	C₃×D₈	1	4	yes	direct product factor
48	C₂²	C₂×C₃×D₄	1	3	yes	—
49	C₄	C₂×C₃×D₄	1	3	yes	—
50	C₄	C₂⁴×C₃	1	1	yes	characteristic
51	C₃	C₂²×D₈	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	—	4	8	no	—
53	C₂	C₂×C₃×D₈	1	6	yes	direct product factor
54	C₂	C₂²×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₄	C₂	semidirect
2. C₂×C₃×D₈	C₂	direct (N × H)
3. C₂²×C₃×C₈	C₂	semidirect
4. C₂²×D₈	C₃	direct (N × H)
7. C₂×C₃×D₄	C₂²	semidirect
9. C₃×D₈	C₂²	direct (N × H)
10. C₂×C₃×C₈	C₂²	semidirect
11. C₂²×D₄	C₂×C₃	semidirect
12. C₂×D₈	C₂×C₃	direct (N × H)
13. C₂²×C₈	C₂×C₃	semidirect
18. C₃×D₄	C₂³	semidirect
21. C₃×C₈	C₂³	semidirect
24. C₂×D₄	C₂²×C₃	semidirect
26. D₈	C₂²×C₃	direct (N × H)
27. C₂×C₈	C₂²×C₃	semidirect
30. C₂²×C₃	D₈	direct (N × H)
38. D₄	C₂³×C₃	semidirect
41. C₈	C₂³×C₃	semidirect
43. C₂×C₃	C₂×D₈	direct (N × H)
47. C₂²	C₃×D₈	direct (N × H)
51. C₃	C₂²×D₈	direct (N × H)
53. C₂	C₂×C₃×D₈	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×D₄	2
C₂×C₃×D₈	2
C₂²×C₃×C₈	2
C₂²×D₈	3

Subgroups — C22×C3×D8 #5899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₈ #5899