Subgroups — C₂×C₃×C₄×C₈ #5276

Order 192.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₄	C₄	1	2	yes	—
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₈	C₄	1	8	yes	direct product factor
8	C₂×C₃×C₈	C₂²	1	4	yes	—
9	C₃×C₄²	C₄	1	2	yes	—
10	C₃×C₄²	C₂²	1	2	yes	—
11	C₂²×C₈	C₂×C₃	1	2	yes	—
12	C₂×C₄²	C₂×C₃	1	1	yes	characteristic
13	C₄×C₈	C₂×C₃	1	4	yes	direct product factor
14	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄	C₈	1	8	yes	direct product factor
16	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
18	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
19	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₃×C₈	C₂×C₄	1	8	yes	direct product factor
21	C₂²×C₄	C₃×C₄	1	2	yes	—
22	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₈	C₃×C₄	1	8	yes	direct product factor
24	C₂×C₈	C₂²×C₃	1	4	yes	—
25	C₄²	C₃×C₄	1	2	yes	—
26	C₄²	C₂²×C₃	1	2	yes	—
27	C₂²×C₃	C₄²	1	2	yes	—
28	C₂²×C₃	C₂×C₈	1	4	yes	—
29	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
30	C₃×C₄	C₄²	1	2	yes	—
31	C₃×C₄	C₂×C₈	1	8	yes	direct product factor
32	C₃×C₄	C₂²×C₄	1	2	yes	—
33	C₂³	C₂×C₃×C₄	1	1	yes	characteristic
34	C₂×C₄	C₃×C₈	1	8	yes	direct product factor
35	C₂×C₄	C₂×C₃×C₄	1	1	yes	characteristic
36	C₂×C₄	C₂×C₃×C₄	1	4	yes	—
37	C₂×C₄	C₂×C₃×C₄	1	4	yes	—
38	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
39	C₈	C₂×C₃×C₄	1	8	yes	direct product factor
40	C₂×C₃	C₄×C₈	1	4	yes	direct product factor
41	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₃	C₂²×C₈	1	2	yes	—
43	C₂²	C₃×C₄²	1	2	yes	—
44	C₂²	C₂×C₃×C₈	1	4	yes	—
45	C₂²	C₂²×C₃×C₄	1	1	yes	characteristic
46	C₄	C₃×C₄²	1	2	yes	—
47	C₄	C₂×C₃×C₈	1	8	yes	direct product factor
48	C₄	C₂²×C₃×C₄	1	2	yes	—
49	C₃	C₂×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
50	C₂	C₃×C₄×C₈	1	4	yes	direct product factor
51	C₂	C₂×C₃×C₄²	1	1	yes	characteristic
52	C₂	C₂²×C₃×C₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₈	C₂	direct (N × H)
4. C₂×C₄×C₈	C₃	direct (N × H)
7. C₂×C₃×C₈	C₄	direct (N × H)
13. C₄×C₈	C₂×C₃	direct (N × H)
15. C₂×C₃×C₄	C₈	direct (N × H)
20. C₃×C₈	C₂×C₄	direct (N × H)
23. C₂×C₈	C₃×C₄	direct (N × H)
31. C₃×C₄	C₂×C₈	direct (N × H)
34. C₂×C₄	C₃×C₈	direct (N × H)
39. C₈	C₂×C₃×C₄	direct (N × H)
40. C₂×C₃	C₄×C₈	direct (N × H)
47. C₄	C₂×C₃×C₈	direct (N × H)
49. C₃	C₂×C₄×C₈	direct (N × H)
50. C₂	C₃×C₄×C₈	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×C₈	2
C₂×C₃×C₄²	2
C₃×C₄×C₈	2
C₂×C₄×C₈	3

Subgroups — C2×C3×C4×C8 #5276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₈ #5276