Subgroups — C₂²×C₃²×C₈ #64178

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₈	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₄	C₄	1	3	yes	—
6	C₂×C₃²×C₄	C₂²	1	3	yes	—
7	C₃²×C₈	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₈	C₂×C₃	1	24	yes	direct product factor
10	C₂²×C₃²	C₈	1	4	yes	direct product factor
11	C₂²×C₃²	C₂×C₄	1	3	yes	—
12	C₃²×C₄	C₂×C₄	1	3	yes	—
13	C₃²×C₄	C₂³	1	1	yes	characteristic
14	C₂²×C₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
15	C₂³×C₃	C₃×C₄	1	4	yes	—
16	C₂×C₃×C₄	C₃×C₄	1	12	yes	—
17	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
18	C₃×C₈	C₂²×C₃	1	16	yes	direct product factor
19	C₂×C₃²	C₂×C₈	1	6	yes	direct product factor
20	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic
22	C₂×C₈	C₂×C₃²	1	6	yes	direct product factor
23	C₂²×C₃	C₃×C₈	1	16	yes	direct product factor
24	C₂²×C₃	C₂×C₃×C₄	1	12	yes	—
25	C₃×C₄	C₂×C₃×C₄	1	12	yes	—
26	C₃×C₄	C₂³×C₃	1	4	yes	—
27	C₃²	C₂²×C₈	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³	C₃²×C₄	1	1	yes	characteristic
29	C₂×C₄	C₃²×C₄	1	3	yes	—
30	C₂×C₄	C₂²×C₃²	1	3	yes	—
31	C₈	C₂²×C₃²	1	4	yes	direct product factor
32	C₂×C₃	C₂×C₃×C₈	1	24	yes	direct product factor
33	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
34	C₂²	C₃²×C₈	1	4	yes	direct product factor
35	C₂²	C₂×C₃²×C₄	1	3	yes	—
36	C₄	C₂×C₃²×C₄	1	3	yes	—
37	C₄	C₂³×C₃²	1	1	yes	characteristic
38	C₃	C₂²×C₃×C₈	1	4	yes	direct product factor
39	C₂	C₂×C₃²×C₈	1	6	yes	direct product factor
40	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₃²
socle, prime radical	C₂³×C₃²	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₈	C₂	direct (N × H)
3. C₂²×C₃×C₈	C₃	direct (N × H)
7. C₃²×C₈	C₂²	direct (N × H)
9. C₂×C₃×C₈	C₂×C₃	direct (N × H)
10. C₂²×C₃²	C₈	direct (N × H)
14. C₂²×C₈	C₃²	direct (N × H)
18. C₃×C₈	C₂²×C₃	direct (N × H)
19. C₂×C₃²	C₂×C₈	direct (N × H)
22. C₂×C₈	C₂×C₃²	direct (N × H)
23. C₂²×C₃	C₃×C₈	direct (N × H)
27. C₃²	C₂²×C₈	direct (N × H)
31. C₈	C₂²×C₃²	direct (N × H)
32. C₂×C₃	C₂×C₃×C₈	direct (N × H)
34. C₂²	C₃²×C₈	direct (N × H)
38. C₃	C₂²×C₃×C₈	direct (N × H)
39. C₂	C₂×C₃²×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×C₃²×C₄	2
C₂×C₃²×C₈	2
C₂²×C₃×C₈	3

Subgroups — C22×C32×C8 #64178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₈ #64178