Subgroups — C₂×C₃×M₆₄ #72972

Order 384.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×M₆₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₃₂	C₂	1	2	yes	maximal, complemented
4	C₂×M₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×C₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₃₂	C₂²	1	4	yes	complemented
11	C₂²×C₁₆	C₂×C₃	1	1	yes	characteristic
12	M₆₄	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₃₂	C₂×C₃	1	2	yes	complemented
14	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
15	C₂×C₃×C₈	C₈	1	1	yes	characteristic
16	C₂×C₃×C₈	C₈	1	2	yes	—
17	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
19	C₃×C₁₆	C₂×C₄	1	2	yes	—
20	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₃×C₁₆	C₂³	1	1	yes	characteristic
22	C₂²×C₈	C₃×C₄	1	1	yes	characteristic
23	C₂×C₁₆	C₃×C₄	1	1	yes	characteristic
24	C₂×C₁₆	C₃×C₄	1	2	yes	—
25	C₂×C₁₆	C₂²×C₃	1	2	yes	—
26	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
27	C₃₂	C₂²×C₃	1	4	yes	complemented
28	C₂³×C₃	C₁₆	1	1	yes	characteristic
29	C₂×C₃×C₄	C₁₆	1	2	yes	—
30	C₂×C₃×C₄	C₁₆	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
32	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
33	C₃×C₈	C₂×C₈	1	2	yes	—
34	C₃×C₈	C₂×C₈	1	1	yes	characteristic
35	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄	C₃×C₈	1	1	yes	characteristic
37	C₂×C₈	C₃×C₈	1	2	yes	—
38	C₂×C₈	C₃×C₈	1	1	yes	characteristic
39	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
40	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
41	C₁₆	C₂×C₃×C₄	1	2	yes	—
42	C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
43	C₁₆	C₂³×C₃	1	1	yes	characteristic
44	C₂²×C₃	—	2	2	no	—
45	C₂²×C₃	C₂×C₁₆	1	2	yes	—
46	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
47	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
48	C₃×C₄	C₂×C₁₆	1	2	yes	—
49	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₂³	C₃×C₁₆	1	1	yes	characteristic
51	C₂×C₄	C₃×C₁₆	1	2	yes	—
52	C₂×C₄	C₃×C₁₆	1	1	yes	characteristic
53	C₂×C₄	C₂×C₃×C₈	1	2	yes	—
54	C₂×C₄	C₂×C₃×C₈	1	1	yes	characteristic
55	C₈	C₂×C₃×C₈	1	2	yes	—
56	C₈	C₂×C₃×C₈	1	1	yes	characteristic
57	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
58	C₂×C₃	—	2	2	no	—
59	C₂×C₃	M₆₄	1	2	yes	direct product factor
60	C₂×C₃	C₂²×C₁₆	1	1	yes	characteristic
61	C₂²	—	2	2	no	—
62	C₂²	C₂×C₃×C₁₆	1	2	yes	—
63	C₂²	C₂×C₃×C₁₆	1	1	yes	characteristic
64	C₄	C₂×C₃×C₁₆	1	1	yes	characteristic
65	C₄	C₂×C₃×C₁₆	1	2	yes	—
66	C₄	C₂²×C₃×C₈	1	1	yes	characteristic
67	C₃	C₂×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	—	2	2	no	—
69	C₂	C₃×M₆₄	1	2	yes	direct product factor
70	C₂	C₂²×C₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₁₆	C₂²
commutator	C₂	C₂²×C₃×C₁₆
Frattini, cosocle	C₁₆	C₂³×C₃
socle	C₂²×C₃	C₂×C₁₆
prime radical	C₂³×C₃	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×M₆₄	C₂	direct (N × H)
3. C₂×C₃×C₃₂	C₂	semidirect
4. C₂×M₆₄	C₃	direct (N × H)
10. C₃×C₃₂	C₂²	semidirect
12. M₆₄	C₂×C₃	direct (N × H)
13. C₂×C₃₂	C₂×C₃	semidirect
27. C₃₂	C₂²×C₃	semidirect
59. C₂×C₃	M₆₄	direct (N × H)
67. C₃	C₂×M₆₄	direct (N × H)
69. C₂	C₃×M₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₁₆	2
C₃×M₆₄	2
C₂×C₃×C₃₂	2
C₂×M₆₄	3

Subgroups — C2×C3×M64 #72972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×M₆₄ #72972