Subgroups — C₂×C₃²×C₄² #64163

Order 288.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	3	yes	maximal
2	C₃²×C₄²	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄²	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₂²	1	1	yes	characteristic
5	C₂×C₃²×C₄	C₄	1	12	yes	direct product factor
6	C₂×C₃²×C₄	C₂²	1	6	yes	—
7	C₂²×C₃×C₄	C₂×C₃	1	12	yes	—
8	C₃×C₄²	C₂×C₃	1	16	yes	direct product factor
9	C₂²×C₃²	C₂×C₄	1	6	yes	—
10	C₂²×C₃²	C₂³	1	1	yes	characteristic
11	C₃²×C₄	C₂×C₄	1	12	yes	direct product factor
12	C₂×C₄²	C₃²	1	1	yes	characteristic, direct product factor, Sylow
13	C₂³×C₃	C₂²×C₃	1	4	yes	—
14	C₂×C₃×C₄	C₃×C₄	1	48	yes	direct product factor
15	C₂×C₃×C₄	C₂²×C₃	1	24	yes	—
16	C₂×C₃²	C₄²	1	4	yes	direct product factor
17	C₂×C₃²	C₂²×C₄	1	3	yes	—
18	C₂²×C₄	C₂×C₃²	1	3	yes	—
19	C₄²	C₂×C₃²	1	4	yes	direct product factor
20	C₂²×C₃	C₂×C₃×C₄	1	24	yes	—
21	C₂²×C₃	C₂³×C₃	1	4	yes	—
22	C₃×C₄	C₂×C₃×C₄	1	48	yes	direct product factor
23	C₃²	C₂×C₄²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³	C₂²×C₃²	1	1	yes	characteristic
25	C₂×C₄	C₃²×C₄	1	12	yes	direct product factor
26	C₂×C₄	C₂²×C₃²	1	6	yes	—
27	C₂×C₃	C₃×C₄²	1	16	yes	direct product factor
28	C₂×C₃	C₂²×C₃×C₄	1	12	yes	—
29	C₂²	C₂×C₃²×C₄	1	6	yes	—
30	C₂²	C₂³×C₃²	1	1	yes	characteristic
31	C₄	C₂×C₃²×C₄	1	12	yes	direct product factor
32	C₃	C₂×C₃×C₄²	1	4	yes	direct product factor
33	C₂	C₃²×C₄²	1	4	yes	direct product factor
34	C₂	C₂²×C₃²×C₄	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₃²
socle, prime radical	C₂³×C₃²	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄²	C₂	direct (N × H)
3. C₂×C₃×C₄²	C₃	direct (N × H)
5. C₂×C₃²×C₄	C₄	direct (N × H)
8. C₃×C₄²	C₂×C₃	direct (N × H)
11. C₃²×C₄	C₂×C₄	direct (N × H)
12. C₂×C₄²	C₃²	direct (N × H)
14. C₂×C₃×C₄	C₃×C₄	direct (N × H)
16. C₂×C₃²	C₄²	direct (N × H)
19. C₄²	C₂×C₃²	direct (N × H)
22. C₃×C₄	C₂×C₃×C₄	direct (N × H)
23. C₃²	C₂×C₄²	direct (N × H)
25. C₂×C₄	C₃²×C₄	direct (N × H)
27. C₂×C₃	C₃×C₄²	direct (N × H)
31. C₄	C₂×C₃²×C₄	direct (N × H)
32. C₃	C₂×C₃×C₄²	direct (N × H)
33. C₂	C₃²×C₄²	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
C₃²×C₄²	2
C₂×C₃×C₄²	3

Subgroups — C2×C32×C42 #64163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄² #64163