Subgroups — C₃×#2264 #72979

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₆₄	C₂	1	2	yes	maximal, complemented
4	C₃×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#2264	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#135	—	2	1	no	—
8	C₂×C₃×Q₁₆	—	2	1	no	—
9	C₃×Q₃₂	—	2	1	no	—
10	C₃×Q₃₂	C₂²	1	2	yes	—
11	C₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₃×SD₃₂	—	2	1	no	—
13	C₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₃₂	C₂²	1	2	yes	—
16	#508	C₂×C₃	1	1	yes	characteristic
17	C₂×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
18	Q₆₄	C₂×C₃	1	2	yes	complemented
19	SD₆₄	C₂×C₃	1	2	yes	complemented
20	M₆₄	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×#41	—	4	1	no	—
22	C₂×C₃×Q₈	—	4	1	no	—
23	C₃×Q₁₆	—	2	1	no	—
24	C₃×Q₁₆	—	2	2	no	—
25	C₃×Q₁₆	—	4	1	no	—
26	C₃×SD₁₆	—	4	1	no	—
27	C₃×D₈	—	2	1	no	—
28	C₂×C₃×C₈	D₄	1	1	yes	characteristic
29	C₃×C₁₆	D₄	1	1	yes	characteristic
30	C₃×C₁₆	C₂³	1	1	yes	characteristic
31	#135	—	2	1	no	—
32	C₂×Q₁₆	—	2	1	no	—
33	Q₃₂	—	2	1	no	—
34	Q₃₂	C₂²×C₃	1	2	yes	—
35	Q₃₂	C₂²×C₃	1	1	yes	characteristic
36	SD₃₂	—	2	1	no	—
37	D₁₆	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
39	C₃₂	C₂²×C₃	1	2	yes	—
40	C₃×Q₈	—	4	1	no	—
41	C₃×Q₈	—	4	2	no	—
42	C₃×Q₈	—	8	1	no	—
43	C₃×D₄	—	4	1	no	—
44	C₃×D₄	—	8	1	no	—
45	C₂×C₃×C₄	—	8	1	no	—
46	C₂×C₃×C₄	—	8	1	no	—
47	C₂×C₃×C₄	D₈	1	1	yes	characteristic
48	C₃×C₈	D₈	1	1	yes	characteristic
49	C₃×C₈	C₂×D₄	1	1	yes	characteristic
50	#41	—	4	1	no	—
51	C₂×Q₈	—	4	1	no	—
52	Q₁₆	—	2	1	no	—
53	Q₁₆	—	2	2	no	—
54	Q₁₆	—	4	1	no	—
55	SD₁₆	—	4	1	no	—
56	D₈	—	2	1	no	—
57	C₂×C₈	C₃×D₄	1	1	yes	characteristic
58	C₁₆	C₃×D₄	1	1	yes	characteristic
59	C₁₆	C₂³×C₃	1	1	yes	characteristic
60	C₂²×C₃	—	8	1	no	—
61	C₂²×C₃	D₁₆	1	1	yes	characteristic
62	C₃×C₄	—	8	1	no	—
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	D₁₆	1	1	yes	characteristic
65	C₃×C₄	C₂×D₈	1	1	yes	characteristic
66	Q₈	—	4	1	no	—
67	Q₈	—	4	2	no	—
68	Q₈	—	8	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	8	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	C₃×D₈	1	1	yes	characteristic
74	C₈	C₃×D₈	1	1	yes	characteristic
75	C₈	C₂×C₃×D₄	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	16	1	no	—
78	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
79	C₂²	—	8	1	no	—
80	C₂²	C₃×D₁₆	1	1	yes	characteristic
81	C₄	—	8	1	no	—
82	C₄	—	8	2	no	—
83	C₄	C₃×D₁₆	1	1	yes	characteristic
84	C₄	C₂×C₃×D₈	1	1	yes	characteristic
85	C₃	#2264	1	1	yes	characteristic, direct product factor, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×C₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₃
prime radical	C₃×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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. C₂×C₃×Q₃₂	C₂	semidirect
3. C₃×Q₆₄	C₂	semidirect
4. C₃×SD₆₄	C₂	semidirect
5. C₃×M₆₄	C₂	semidirect
6. #2264	C₃	direct (N × H)
17. C₂×Q₃₂	C₂×C₃	semidirect
18. Q₆₄	C₂×C₃	semidirect
19. SD₆₄	C₂×C₃	semidirect
20. M₆₄	C₂×C₃	semidirect
85. C₃	#2264	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#508	2
C₂×C₃×Q₃₂	2
C₃×Q₆₄	2
C₃×SD₆₄	2
C₃×M₆₄	2
#2264	3

Subgroups — C3×#2264 #72979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2264 #72979