Subgroups — C₂×C₃²×D₈ #64181

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	2	yes	maximal, complemented
2	C₃²×D₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₈	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	—	2	2	no	—
6	C₃²×D₄	—	2	2	no	—
7	C₃²×D₄	C₂²	1	4	yes	complemented
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₃²×C₈	C₂²	1	2	yes	complemented
10	C₂×C₃×D₄	C₂×C₃	1	8	yes	complemented
11	C₃×D₈	C₂×C₃	1	16	yes	direct product factor
12	C₂×C₃×C₈	C₂×C₃	1	4	yes	complemented
13	C₂²×C₃²	—	2	4	no	—
14	C₂²×C₃²	—	4	4	no	—
15	C₂²×C₃²	D₄	1	1	yes	characteristic
16	C₃²×C₄	D₄	1	1	yes	characteristic
17	C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₂×D₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
19	C₂³×C₃	—	2	8	no	—
20	C₃×D₄	—	2	8	no	—
21	C₃×D₄	C₂²×C₃	1	16	yes	complemented
22	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
23	C₃×C₈	C₂²×C₃	1	8	yes	complemented
24	C₂×C₃²	—	4	4	no	—
25	C₂×C₃²	D₈	1	2	yes	direct product factor
26	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄	C₂×C₃²	1	2	yes	complemented
28	D₈	C₂×C₃²	1	4	yes	direct product factor
29	C₂×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
30	C₂²×C₃	—	2	16	no	—
31	C₂²×C₃	—	4	16	no	—
32	C₂²×C₃	C₃×D₄	1	4	yes	—
33	C₃×C₄	C₃×D₄	1	4	yes	—
34	C₃×C₄	C₂³×C₃	1	4	yes	—
35	C₃²	C₂×D₈	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³	—	2	2	no	—
37	D₄	—	2	2	no	—
38	D₄	C₂²×C₃²	1	4	yes	complemented
39	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
40	C₈	C₂²×C₃²	1	2	yes	complemented
41	C₂×C₃	—	4	16	no	—
42	C₂×C₃	C₃×D₈	1	8	yes	direct product factor
43	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
44	C₂²	—	2	4	no	—
45	C₂²	—	4	4	no	—
46	C₂²	C₃²×D₄	1	1	yes	characteristic
47	C₄	C₃²×D₄	1	1	yes	characteristic
48	C₄	C₂³×C₃²	1	1	yes	characteristic
49	C₃	C₂×C₃×D₈	1	4	yes	direct product factor
50	C₂	—	4	4	no	—
51	C₂	C₃²×D₈	1	2	yes	direct product factor
52	C₂	C₂×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₄	C₂	semidirect
2. C₃²×D₈	C₂	direct (N × H)
3. C₂×C₃²×C₈	C₂	semidirect
4. C₂×C₃×D₈	C₃	direct (N × H)
7. C₃²×D₄	C₂²	semidirect
9. C₃²×C₈	C₂²	semidirect
10. C₂×C₃×D₄	C₂×C₃	semidirect
11. C₃×D₈	C₂×C₃	direct (N × H)
12. C₂×C₃×C₈	C₂×C₃	semidirect
18. C₂×D₈	C₃²	direct (N × H)
21. C₃×D₄	C₂²×C₃	semidirect
23. C₃×C₈	C₂²×C₃	semidirect
25. C₂×C₃²	D₈	direct (N × H)
27. C₂×D₄	C₂×C₃²	semidirect
28. D₈	C₂×C₃²	direct (N × H)
29. C₂×C₈	C₂×C₃²	semidirect
35. C₃²	C₂×D₈	direct (N × H)
38. D₄	C₂²×C₃²	semidirect
40. C₈	C₂²×C₃²	semidirect
42. C₂×C₃	C₃×D₈	direct (N × H)
49. C₃	C₂×C₃×D₈	direct (N × H)
51. C₂	C₃²×D₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×D₄	2
C₃²×D₈	2
C₂×C₃²×C₈	2
C₂×C₃×D₈	3

Subgroups — C2×C32×D8 #64181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₈ #64181