Subgroups — C₂²×C₃²×D₄ #64371

Order 288.

708 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×D₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×D₄	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	—	2	8	no	—
6	C₂³×C₃²	C₂²	1	1	yes	characteristic
7	C₂³×C₃²	C₂²	1	12	yes	complemented
8	C₃²×D₄	C₂²	1	16	yes	direct product factor
9	C₂×C₃²×C₄	C₂²	1	6	yes	complemented
10	C₂⁴×C₃	C₂×C₃	1	8	yes	complemented
11	C₂×C₃×D₄	C₂×C₃	1	48	yes	direct product factor
12	C₂²×C₃×C₄	C₂×C₃	1	4	yes	complemented
13	C₂²×C₃²	—	2	24	no	—
14	C₂²×C₃²	D₄	1	4	yes	direct product factor
15	C₂²×C₃²	C₂³	1	3	yes	—
16	C₂²×C₃²	C₂³	1	8	yes	complemented
17	C₃²×C₄	C₂³	1	4	yes	complemented
18	C₂²×D₄	C₃²	1	1	yes	characteristic, direct product factor, Sylow
19	C₂³×C₃	—	2	32	no	—
20	C₂³×C₃	C₂²×C₃	1	4	yes	—
21	C₂³×C₃	C₂²×C₃	1	48	yes	complemented
22	C₃×D₄	C₂²×C₃	1	64	yes	direct product factor
23	C₂×C₃×C₄	C₂²×C₃	1	24	yes	complemented
24	C₂×C₃²	—	2	8	no	—
25	C₂×C₃²	C₂×D₄	1	6	yes	direct product factor
26	C₂×C₃²	C₂⁴	1	1	yes	characteristic
27	C₂⁴	C₂×C₃²	1	2	yes	complemented
28	C₂×D₄	C₂×C₃²	1	12	yes	direct product factor
29	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
30	C₂²×C₃	—	2	96	no	—
31	C₂²×C₃	C₃×D₄	1	16	yes	direct product factor
32	C₂²×C₃	C₂³×C₃	1	12	yes	—
33	C₂²×C₃	C₂³×C₃	1	32	yes	complemented
34	C₃×C₄	C₂³×C₃	1	16	yes	complemented
35	C₃²	C₂²×D₄	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³	—	2	8	no	—
37	C₂³	C₂²×C₃²	1	1	yes	characteristic
38	C₂³	C₂²×C₃²	1	12	yes	complemented
39	D₄	C₂²×C₃²	1	16	yes	direct product factor
40	C₂×C₄	C₂²×C₃²	1	6	yes	complemented
41	C₂×C₃	—	2	32	no	—
42	C₂×C₃	C₂×C₃×D₄	1	24	yes	direct product factor
43	C₂×C₃	C₂⁴×C₃	1	4	yes	—
44	C₂²	—	2	24	no	—
45	C₂²	C₃²×D₄	1	4	yes	direct product factor
46	C₂²	C₂³×C₃²	1	3	yes	—
47	C₂²	C₂³×C₃²	1	8	yes	complemented
48	C₄	C₂³×C₃²	1	4	yes	complemented
49	C₃	C₂²×C₃×D₄	1	4	yes	direct product factor
50	C₂	—	2	8	no	—
51	C₂	C₂×C₃²×D₄	1	6	yes	direct product factor
52	C₂	C₂⁴×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²	C₂	semidirect
2. C₂×C₃²×D₄	C₂	direct (N × H)
3. C₂²×C₃²×C₄	C₂	semidirect
4. C₂²×C₃×D₄	C₃	direct (N × H)
7. C₂³×C₃²	C₂²	semidirect
8. C₃²×D₄	C₂²	direct (N × H)
9. C₂×C₃²×C₄	C₂²	semidirect
10. C₂⁴×C₃	C₂×C₃	semidirect
11. C₂×C₃×D₄	C₂×C₃	direct (N × H)
12. C₂²×C₃×C₄	C₂×C₃	semidirect
14. C₂²×C₃²	D₄	direct (N × H)
16. C₂²×C₃²	C₂³	semidirect
17. C₃²×C₄	C₂³	semidirect
18. C₂²×D₄	C₃²	direct (N × H)
21. C₂³×C₃	C₂²×C₃	semidirect
22. C₃×D₄	C₂²×C₃	direct (N × H)
23. C₂×C₃×C₄	C₂²×C₃	semidirect
25. C₂×C₃²	C₂×D₄	direct (N × H)
27. C₂⁴	C₂×C₃²	semidirect
28. C₂×D₄	C₂×C₃²	direct (N × H)
29. C₂²×C₄	C₂×C₃²	semidirect
31. C₂²×C₃	C₃×D₄	direct (N × H)
33. C₂²×C₃	C₂³×C₃	semidirect
34. C₃×C₄	C₂³×C₃	semidirect
35. C₃²	C₂²×D₄	direct (N × H)
38. C₂³	C₂²×C₃²	semidirect
39. D₄	C₂²×C₃²	direct (N × H)
40. C₂×C₄	C₂²×C₃²	semidirect
42. C₂×C₃	C₂×C₃×D₄	direct (N × H)
45. C₂²	C₃²×D₄	direct (N × H)
47. C₂²	C₂³×C₃²	semidirect
48. C₄	C₂³×C₃²	semidirect
49. C₃	C₂²×C₃×D₄	direct (N × H)
51. C₂	C₂×C₃²×D₄	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×C₃²	2
C₂×C₃²×D₄	2
C₂²×C₃²×C₄	2
C₂²×C₃×D₄	3

Subgroups — C22×C32×D4 #64371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₄ #64371