Subgroups — C₂³×C₃×C₄² #87302

Order 384.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	3	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁵×C₃	C₂²	1	1	yes	characteristic
5	C₂³×C₃×C₄	C₄	1	48	yes	direct product factor
6	C₂³×C₃×C₄	C₂²	1	42	yes	—
7	C₂×C₃×C₄²	C₂²	1	112	yes	direct product factor
8	C₂⁴×C₄	C₂×C₃	1	3	yes	—
9	C₂²×C₄²	C₂×C₃	1	28	yes	direct product factor
10	C₂⁴×C₃	C₂×C₄	1	24	yes	—
11	C₂⁴×C₃	C₂³	1	7	yes	—
12	C₂²×C₃×C₄	C₂×C₄	1	336	yes	direct product factor
13	C₂²×C₃×C₄	C₂³	1	84	yes	—
14	C₃×C₄²	C₂³	1	64	yes	direct product factor
15	C₂⁵	C₂²×C₃	1	1	yes	characteristic
16	C₂³×C₄	C₃×C₄	1	48	yes	direct product factor
17	C₂³×C₄	C₂²×C₃	1	42	yes	—
18	C₂×C₄²	C₂²×C₃	1	112	yes	direct product factor
19	C₂³×C₃	C₄²	1	64	yes	direct product factor
20	C₂³×C₃	C₂²×C₄	1	84	yes	—
21	C₂³×C₃	C₂⁴	1	7	yes	—
22	C₂×C₃×C₄	C₂²×C₄	1	336	yes	direct product factor
23	C₂×C₃×C₄	C₂⁴	1	24	yes	—
24	C₂⁴	C₂×C₃×C₄	1	24	yes	—
25	C₂⁴	C₂³×C₃	1	7	yes	—
26	C₂²×C₄	C₂×C₃×C₄	1	336	yes	direct product factor
27	C₂²×C₄	C₂³×C₃	1	84	yes	—
28	C₄²	C₂³×C₃	1	64	yes	direct product factor
29	C₂²×C₃	C₂×C₄²	1	112	yes	direct product factor
30	C₂²×C₃	C₂³×C₄	1	42	yes	—
31	C₂²×C₃	C₂⁵	1	1	yes	characteristic
32	C₃×C₄	C₂³×C₄	1	48	yes	direct product factor
33	C₂³	C₃×C₄²	1	64	yes	direct product factor
34	C₂³	C₂²×C₃×C₄	1	84	yes	—
35	C₂³	C₂⁴×C₃	1	7	yes	—
36	C₂×C₄	C₂²×C₃×C₄	1	336	yes	direct product factor
37	C₂×C₄	C₂⁴×C₃	1	24	yes	—
38	C₂×C₃	C₂²×C₄²	1	28	yes	direct product factor
39	C₂×C₃	C₂⁴×C₄	1	3	yes	—
40	C₂²	C₂×C₃×C₄²	1	112	yes	direct product factor
41	C₂²	C₂³×C₃×C₄	1	42	yes	—
42	C₂²	C₂⁵×C₃	1	1	yes	characteristic
43	C₄	C₂³×C₃×C₄	1	48	yes	direct product factor
44	C₃	C₂³×C₄²	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	C₂²×C₃×C₄²	1	28	yes	direct product factor
46	C₂	C₂⁴×C₃×C₄	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₄²	C₂	direct (N × H)
3. C₂³×C₄²	C₃	direct (N × H)
5. C₂³×C₃×C₄	C₄	direct (N × H)
7. C₂×C₃×C₄²	C₂²	direct (N × H)
9. C₂²×C₄²	C₂×C₃	direct (N × H)
12. C₂²×C₃×C₄	C₂×C₄	direct (N × H)
14. C₃×C₄²	C₂³	direct (N × H)
16. C₂³×C₄	C₃×C₄	direct (N × H)
18. C₂×C₄²	C₂²×C₃	direct (N × H)
19. C₂³×C₃	C₄²	direct (N × H)
22. C₂×C₃×C₄	C₂²×C₄	direct (N × H)
26. C₂²×C₄	C₂×C₃×C₄	direct (N × H)
28. C₄²	C₂³×C₃	direct (N × H)
29. C₂²×C₃	C₂×C₄²	direct (N × H)
32. C₃×C₄	C₂³×C₄	direct (N × H)
33. C₂³	C₃×C₄²	direct (N × H)
36. C₂×C₄	C₂²×C₃×C₄	direct (N × H)
38. C₂×C₃	C₂²×C₄²	direct (N × H)
40. C₂²	C₂×C₃×C₄²	direct (N × H)
43. C₄	C₂³×C₃×C₄	direct (N × H)
44. C₃	C₂³×C₄²	direct (N × H)
45. C₂	C₂²×C₃×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
C₂⁴×C₃×C₄	2
C₂²×C₃×C₄²	2
C₂³×C₄²	3

Subgroups — C23×C3×C42 #87302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₄² #87302