Subgroups — C₂³×C₃×D₈ #87458

Order 384.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×D₈	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁵×C₃	—	2	2	no	—
6	C₂²×C₃×D₄	—	2	14	no	—
7	C₂²×C₃×D₄	C₂²	1	28	yes	complemented
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₈	C₂²	1	112	yes	direct product factor
10	C₂²×C₃×C₈	C₂²	1	14	yes	complemented
11	C₂³×D₄	C₂×C₃	1	2	yes	complemented
12	C₂²×D₈	C₂×C₃	1	28	yes	direct product factor
13	C₂³×C₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂⁴×C₃	—	2	28	no	—
15	C₂⁴×C₃	—	4	16	no	—
16	C₂⁴×C₃	D₄	1	1	yes	characteristic
17	C₂×C₃×D₄	—	2	84	no	—
18	C₂×C₃×D₄	C₂³	1	56	yes	complemented
19	C₂²×C₃×C₄	D₄	1	7	yes	—
20	C₂²×C₃×C₄	C₂³	1	7	yes	—
21	C₃×D₈	C₂³	1	64	yes	direct product factor
22	C₂×C₃×C₈	C₂³	1	28	yes	complemented
23	C₂⁵	—	2	2	no	—
24	C₂²×D₄	—	2	14	no	—
25	C₂²×D₄	C₂²×C₃	1	28	yes	complemented
26	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
27	C₂×D₈	C₂²×C₃	1	112	yes	direct product factor
28	C₂²×C₈	C₂²×C₃	1	14	yes	complemented
29	C₂³×C₃	—	2	56	no	—
30	C₂³×C₃	—	4	112	no	—
31	C₂³×C₃	D₈	1	8	yes	direct product factor
32	C₂³×C₃	C₂×D₄	1	7	yes	—
33	C₃×D₄	—	2	56	no	—
34	C₃×D₄	C₂⁴	1	16	yes	complemented
35	C₂×C₃×C₄	C₂×D₄	1	21	yes	—
36	C₂×C₃×C₄	C₂⁴	1	7	yes	—
37	C₃×C₈	C₂⁴	1	8	yes	complemented
38	C₂⁴	—	2	28	no	—
39	C₂⁴	—	4	16	no	—
40	C₂⁴	C₃×D₄	1	1	yes	characteristic
41	C₂×D₄	—	2	84	no	—
42	C₂×D₄	C₂³×C₃	1	56	yes	complemented
43	C₂²×C₄	C₃×D₄	1	7	yes	—
44	C₂²×C₄	C₂³×C₃	1	7	yes	—
45	D₈	C₂³×C₃	1	64	yes	direct product factor
46	C₂×C₈	C₂³×C₃	1	28	yes	complemented
47	C₂²×C₃	—	2	16	no	—
48	C₂²×C₃	—	4	112	no	—
49	C₂²×C₃	C₂×D₈	1	28	yes	direct product factor
50	C₂²×C₃	C₂²×D₄	1	7	yes	—
51	C₃×C₄	C₂²×D₄	1	7	yes	—
52	C₃×C₄	C₂⁵	1	1	yes	characteristic
53	C₂³	—	2	56	no	—
54	C₂³	—	4	112	no	—
55	C₂³	C₃×D₈	1	8	yes	direct product factor
56	C₂³	C₂×C₃×D₄	1	7	yes	—
57	D₄	—	2	56	no	—
58	D₄	C₂⁴×C₃	1	16	yes	complemented
59	C₂×C₄	C₂×C₃×D₄	1	21	yes	—
60	C₂×C₄	C₂⁴×C₃	1	7	yes	—
61	C₈	C₂⁴×C₃	1	8	yes	complemented
62	C₂×C₃	—	4	16	no	—
63	C₂×C₃	C₂²×D₈	1	14	yes	direct product factor
64	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
65	C₂²	—	2	16	no	—
66	C₂²	—	4	112	no	—
67	C₂²	C₂×C₃×D₈	1	28	yes	direct product factor
68	C₂²	C₂²×C₃×D₄	1	7	yes	—
69	C₄	C₂²×C₃×D₄	1	7	yes	—
70	C₄	C₂⁵×C₃	1	1	yes	characteristic
71	C₃	C₂³×D₈	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	4	16	no	—
73	C₂	C₂²×C₃×D₈	1	14	yes	direct product factor
74	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₄	C₂	semidirect
2. C₂²×C₃×D₈	C₂	direct (N × H)
3. C₂³×C₃×C₈	C₂	semidirect
4. C₂³×D₈	C₃	direct (N × H)
7. C₂²×C₃×D₄	C₂²	semidirect
9. C₂×C₃×D₈	C₂²	direct (N × H)
10. C₂²×C₃×C₈	C₂²	semidirect
11. C₂³×D₄	C₂×C₃	semidirect
12. C₂²×D₈	C₂×C₃	direct (N × H)
13. C₂³×C₈	C₂×C₃	semidirect
18. C₂×C₃×D₄	C₂³	semidirect
21. C₃×D₈	C₂³	direct (N × H)
22. C₂×C₃×C₈	C₂³	semidirect
25. C₂²×D₄	C₂²×C₃	semidirect
27. C₂×D₈	C₂²×C₃	direct (N × H)
28. C₂²×C₈	C₂²×C₃	semidirect
31. C₂³×C₃	D₈	direct (N × H)
34. C₃×D₄	C₂⁴	semidirect
37. C₃×C₈	C₂⁴	semidirect
42. C₂×D₄	C₂³×C₃	semidirect
45. D₈	C₂³×C₃	direct (N × H)
46. C₂×C₈	C₂³×C₃	semidirect
49. C₂²×C₃	C₂×D₈	direct (N × H)
55. C₂³	C₃×D₈	direct (N × H)
58. D₄	C₂⁴×C₃	semidirect
61. C₈	C₂⁴×C₃	semidirect
63. C₂×C₃	C₂²×D₈	direct (N × H)
67. C₂²	C₂×C₃×D₈	direct (N × H)
71. C₃	C₂³×D₈	direct (N × H)
73. C₂	C₂²×C₃×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
C₂³×C₃×D₄	2
C₂²×C₃×D₈	2
C₂³×C₃×C₈	2
C₂³×D₈	3

Subgroups — C23×C3×D8 #87458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×D₈ #87458