Subgroups — C₂²×C₃×C₄×C₈ #84733

Order 384.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₈	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	C₄	1	2	yes	—
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₄	1	16	yes	direct product factor
8	C₂²×C₃×C₈	C₂²	1	12	yes	—
9	C₂×C₃×C₄²	C₄	1	6	yes	—
10	C₂×C₃×C₄²	C₂²	1	6	yes	—
11	C₃×C₄×C₈	C₂²	1	16	yes	direct product factor
12	C₂³×C₈	C₂×C₃	1	2	yes	—
13	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂×C₃	1	12	yes	direct product factor
15	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₄	C₈	1	16	yes	direct product factor
17	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
18	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
19	C₂²×C₃×C₄	C₂×C₄	1	12	yes	—
20	C₂²×C₃×C₄	C₂³	1	3	yes	—
21	C₂×C₃×C₈	C₂×C₄	1	48	yes	direct product factor
22	C₂×C₃×C₈	C₂³	1	8	yes	—
23	C₃×C₄²	C₂×C₄	1	12	yes	—
24	C₃×C₄²	C₂³	1	4	yes	—
25	C₂³×C₄	C₃×C₄	1	2	yes	—
26	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
27	C₂²×C₈	C₃×C₄	1	16	yes	direct product factor
28	C₂²×C₈	C₂²×C₃	1	12	yes	—
29	C₂×C₄²	C₃×C₄	1	6	yes	—
30	C₂×C₄²	C₂²×C₃	1	6	yes	—
31	C₄×C₈	C₂²×C₃	1	16	yes	direct product factor
32	C₂³×C₃	C₄²	1	4	yes	—
33	C₂³×C₃	C₂×C₈	1	8	yes	—
34	C₂³×C₃	C₂²×C₄	1	3	yes	—
35	C₂×C₃×C₄	C₄²	1	12	yes	—
36	C₂×C₃×C₄	C₂×C₈	1	48	yes	direct product factor
37	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
38	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
39	C₂×C₃×C₄	C₂²×C₄	1	12	yes	—
40	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
41	C₃×C₈	C₂²×C₄	1	16	yes	direct product factor
42	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
43	C₂²×C₄	C₃×C₈	1	16	yes	direct product factor
44	C₂²×C₄	C₂×C₃×C₄	1	3	yes	—
45	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
46	C₂²×C₄	C₂×C₃×C₄	1	12	yes	—
47	C₂²×C₄	C₂³×C₃	1	3	yes	—
48	C₂×C₈	C₂×C₃×C₄	1	48	yes	direct product factor
49	C₂×C₈	C₂³×C₃	1	8	yes	—
50	C₄²	C₂×C₃×C₄	1	12	yes	—
51	C₄²	C₂³×C₃	1	4	yes	—
52	C₂²×C₃	C₄×C₈	1	16	yes	direct product factor
53	C₂²×C₃	C₂×C₄²	1	6	yes	—
54	C₂²×C₃	C₂²×C₈	1	12	yes	—
55	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
56	C₃×C₄	C₂×C₄²	1	6	yes	—
57	C₃×C₄	C₂²×C₈	1	16	yes	direct product factor
58	C₃×C₄	C₂³×C₄	1	2	yes	—
59	C₂³	C₃×C₄²	1	4	yes	—
60	C₂³	C₂×C₃×C₈	1	8	yes	—
61	C₂³	C₂²×C₃×C₄	1	3	yes	—
62	C₂×C₄	C₃×C₄²	1	12	yes	—
63	C₂×C₄	C₂×C₃×C₈	1	48	yes	direct product factor
64	C₂×C₄	C₂²×C₃×C₄	1	3	yes	—
65	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
66	C₂×C₄	C₂²×C₃×C₄	1	12	yes	—
67	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
68	C₈	C₂²×C₃×C₄	1	16	yes	direct product factor
69	C₂×C₃	C₂×C₄×C₈	1	12	yes	direct product factor
70	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
71	C₂×C₃	C₂³×C₈	1	2	yes	—
72	C₂²	C₃×C₄×C₈	1	16	yes	direct product factor
73	C₂²	C₂×C₃×C₄²	1	6	yes	—
74	C₂²	C₂²×C₃×C₈	1	12	yes	—
75	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
76	C₄	C₂×C₃×C₄²	1	6	yes	—
77	C₄	C₂²×C₃×C₈	1	16	yes	direct product factor
78	C₄	C₂³×C₃×C₄	1	2	yes	—
79	C₃	C₂²×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	C₂×C₃×C₄×C₈	1	12	yes	direct product factor
81	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
82	C₂	C₂³×C₃×C₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄×C₈	C₂	direct (N × H)
4. C₂²×C₄×C₈	C₃	direct (N × H)
7. C₂²×C₃×C₈	C₄	direct (N × H)
11. C₃×C₄×C₈	C₂²	direct (N × H)
14. C₂×C₄×C₈	C₂×C₃	direct (N × H)
16. C₂²×C₃×C₄	C₈	direct (N × H)
21. C₂×C₃×C₈	C₂×C₄	direct (N × H)
27. C₂²×C₈	C₃×C₄	direct (N × H)
31. C₄×C₈	C₂²×C₃	direct (N × H)
36. C₂×C₃×C₄	C₂×C₈	direct (N × H)
41. C₃×C₈	C₂²×C₄	direct (N × H)
43. C₂²×C₄	C₃×C₈	direct (N × H)
48. C₂×C₈	C₂×C₃×C₄	direct (N × H)
52. C₂²×C₃	C₄×C₈	direct (N × H)
57. C₃×C₄	C₂²×C₈	direct (N × H)
63. C₂×C₄	C₂×C₃×C₈	direct (N × H)
68. C₈	C₂²×C₃×C₄	direct (N × H)
69. C₂×C₃	C₂×C₄×C₈	direct (N × H)
72. C₂²	C₃×C₄×C₈	direct (N × H)
77. C₄	C₂²×C₃×C₈	direct (N × H)
79. C₃	C₂²×C₄×C₈	direct (N × H)
80. C₂	C₂×C₃×C₄×C₈	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×C₈	2
C₂²×C₃×C₄²	2
C₂×C₃×C₄×C₈	2
C₂²×C₄×C₈	3

Subgroups — C22×C3×C4×C8 #84733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×C₈ #84733