Subgroups — C₂×C₃×Q₆₄ #72976

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	2	yes	maximal
2	C₃×Q₆₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×Q₁₆	—	2	2	no	—
6	C₃×Q₃₂	—	2	2	no	—
7	C₃×Q₃₂	C₂²	1	4	yes	—
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×C₃₂	C₂²	1	2	yes	—
10	C₂×Q₃₂	C₂×C₃	1	2	yes	—
11	Q₆₄	C₂×C₃	1	4	yes	direct product factor
12	C₂×C₃₂	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×Q₈	—	4	2	no	—
14	C₃×Q₁₆	—	2	4	no	—
15	C₃×Q₁₆	—	4	2	no	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₃×C₁₆	D₄	1	1	yes	characteristic
18	C₃×C₁₆	C₂³	1	1	yes	characteristic
19	C₂×Q₁₆	—	2	2	no	—
20	Q₃₂	—	2	2	no	—
21	Q₃₂	C₂²×C₃	1	4	yes	—
22	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
23	C₃₂	C₂²×C₃	1	2	yes	—
24	C₃×Q₈	—	4	4	no	—
25	C₃×Q₈	—	8	2	no	—
26	C₂×C₃×C₄	—	8	2	no	—
27	C₂×C₃×C₄	D₈	1	1	yes	characteristic
28	C₃×C₈	D₈	1	1	yes	characteristic
29	C₃×C₈	C₂×D₄	1	1	yes	characteristic
30	C₂×Q₈	—	4	2	no	—
31	Q₁₆	—	2	4	no	—
32	Q₁₆	—	4	2	no	—
33	C₂×C₈	C₃×D₄	1	1	yes	characteristic
34	C₁₆	C₃×D₄	1	1	yes	characteristic
35	C₁₆	C₂³×C₃	1	1	yes	characteristic
36	C₂²×C₃	D₁₆	1	1	yes	characteristic
37	C₃×C₄	—	8	4	no	—
38	C₃×C₄	D₁₆	1	1	yes	characteristic
39	C₃×C₄	C₂×D₈	1	1	yes	characteristic
40	Q₈	—	4	4	no	—
41	Q₈	—	8	2	no	—
42	C₂×C₄	—	8	2	no	—
43	C₂×C₄	C₃×D₈	1	1	yes	characteristic
44	C₈	C₃×D₈	1	1	yes	characteristic
45	C₈	C₂×C₃×D₄	1	1	yes	characteristic
46	C₂×C₃	Q₆₄	1	2	yes	direct product factor
47	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
48	C₂²	C₃×D₁₆	1	1	yes	characteristic
49	C₄	—	8	4	no	—
50	C₄	C₃×D₁₆	1	1	yes	characteristic
51	C₄	C₂×C₃×D₈	1	1	yes	characteristic
52	C₃	C₂×Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
53	C₂	C₃×Q₆₄	1	2	yes	direct product factor
54	C₂	C₂×C₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×Q₆₄	C₂	direct (N × H)
4. C₂×Q₆₄	C₃	direct (N × H)
11. Q₆₄	C₂×C₃	direct (N × H)
46. C₂×C₃	Q₆₄	direct (N × H)
52. C₃	C₂×Q₆₄	direct (N × H)
53. C₂	C₃×Q₆₄	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×C₃×Q₃₂	2
C₃×Q₆₄	2
C₂×C₃×C₃₂	2
C₂×Q₆₄	3

Subgroups — C2×C3×Q64 #72976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Q₆₄ #72976