Subgroups — C₂×C₃×SD₆₄ #72975

Order 384.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×SD₆₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×SD₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×Q₁₆	—	2	1	no	—
7	C₂×C₃×D₈	—	2	1	no	—
8	C₃×Q₃₂	—	2	1	no	—
9	C₃×Q₃₂	C₂²	1	2	yes	complemented
10	C₃×D₁₆	—	2	1	no	—
11	C₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₃×C₃₂	C₂²	1	2	yes	complemented
14	C₂×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂×D₁₆	C₂×C₃	1	1	yes	characteristic
16	SD₆₄	C₂×C₃	1	4	yes	direct product factor
17	C₂×C₃₂	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×Q₈	—	4	1	no	—
19	C₂×C₃×D₄	—	4	1	no	—
20	C₃×Q₁₆	—	2	2	no	—
21	C₃×Q₁₆	—	4	1	no	—
22	C₃×D₈	—	2	2	no	—
23	C₃×D₈	—	4	1	no	—
24	C₂×C₃×C₈	D₄	1	1	yes	characteristic
25	C₃×C₁₆	D₄	1	1	yes	characteristic
26	C₃×C₁₆	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×D₈	—	2	1	no	—
29	Q₃₂	—	2	1	no	—
30	Q₃₂	C₂²×C₃	1	2	yes	complemented
31	D₁₆	—	2	1	no	—
32	D₁₆	C₂²×C₃	1	2	yes	—
33	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
34	C₃₂	C₂²×C₃	1	2	yes	complemented
35	C₂³×C₃	—	8	1	no	—
36	C₃×Q₈	—	4	2	no	—
37	C₃×Q₈	—	8	1	no	—
38	C₃×D₄	—	4	2	no	—
39	C₃×D₄	—	8	1	no	—
40	C₂×C₃×C₄	—	8	1	no	—
41	C₂×C₃×C₄	D₈	1	1	yes	characteristic
42	C₃×C₈	D₈	1	1	yes	characteristic
43	C₃×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×Q₈	—	4	1	no	—
45	C₂×D₄	—	4	1	no	—
46	Q₁₆	—	2	2	no	—
47	Q₁₆	—	4	1	no	—
48	D₈	—	2	2	no	—
49	D₈	—	4	1	no	—
50	C₂×C₈	C₃×D₄	1	1	yes	characteristic
51	C₁₆	C₃×D₄	1	1	yes	characteristic
52	C₁₆	C₂³×C₃	1	1	yes	characteristic
53	C₂²×C₃	—	8	2	no	—
54	C₂²×C₃	—	16	2	no	—
55	C₂²×C₃	D₁₆	1	1	yes	characteristic
56	C₃×C₄	—	8	2	no	—
57	C₃×C₄	D₁₆	1	1	yes	characteristic
58	C₃×C₄	C₂×D₈	1	1	yes	characteristic
59	C₂³	—	8	1	no	—
60	Q₈	—	4	2	no	—
61	Q₈	—	8	1	no	—
62	D₄	—	4	2	no	—
63	D₄	—	8	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	C₃×D₈	1	1	yes	characteristic
66	C₈	C₃×D₈	1	1	yes	characteristic
67	C₈	C₂×C₃×D₄	1	1	yes	characteristic
68	C₂×C₃	—	16	2	no	—
69	C₂×C₃	SD₆₄	1	2	yes	direct product factor
70	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
71	C₂²	—	8	2	no	—
72	C₂²	—	16	2	no	—
73	C₂²	C₃×D₁₆	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	C₃×D₁₆	1	1	yes	characteristic
76	C₄	C₂×C₃×D₈	1	1	yes	characteristic
77	C₃	C₂×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	—	16	2	no	—
79	C₂	C₃×SD₆₄	1	2	yes	direct product factor
80	C₂	C₂×C₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₃
prime radical	C₂×C₃×D₁₆	C₂
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
1. C₂×C₃×Q₃₂	C₂	semidirect
3. C₃×SD₆₄	C₂	direct (N × H)
4. C₂×C₃×C₃₂	C₂	semidirect
5. C₂×SD₆₄	C₃	direct (N × H)
9. C₃×Q₃₂	C₂²	semidirect
13. C₃×C₃₂	C₂²	semidirect
14. C₂×Q₃₂	C₂×C₃	semidirect
16. SD₆₄	C₂×C₃	direct (N × H)
17. C₂×C₃₂	C₂×C₃	semidirect
30. Q₃₂	C₂²×C₃	semidirect
34. C₃₂	C₂²×C₃	semidirect
69. C₂×C₃	SD₆₄	direct (N × H)
77. C₃	C₂×SD₆₄	direct (N × H)
79. C₂	C₃×SD₆₄	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
C₂×C₃×D₁₆	2
C₃×SD₆₄	2
C₂×C₃×C₃₂	2
C₂×SD₆₄	3

Subgroups — C2×C3×SD64 #72975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×SD₆₄ #72975