Subgroups — C₃×#12521 #10631128

Order 768.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2252	C₂	1	4	yes	maximal
2	C₃×#1713	C₂	1	2	yes	maximal
3	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#12521	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×#500	—	2	2	no	—
6	C₃×#500	C₂²	1	4	yes	—
7	C₃×#494	—	2	2	no	—
8	C₃×#366	—	2	8	no	—
9	C₃×C₄×C₁₆	C₂²	1	2	yes	—
10	C₃×C₈²	C₂²	1	1	yes	characteristic
11	#2252	C₂×C₃	1	4	yes	—
12	#1713	C₂×C₃	1	2	yes	—
13	C₈×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×Q₁₆	—	4	4	no	—
15	C₃×C₄⋊Q₈	—	2	4	no	—
16	C₂×C₃×C₁₆	D₄	1	4	yes	—
17	C₃×#107	—	2	8	no	—
18	C₃×#107	—	4	4	no	—
19	C₃×C₄×C₈	D₄	1	1	yes	characteristic
20	C₃×C₄×C₈	D₄	1	1	yes	characteristic
21	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
22	#500	—	2	2	no	—
23	#500	C₂²×C₃	1	4	yes	—
24	#494	—	2	2	no	—
25	#366	—	2	8	no	—
26	C₄×C₁₆	C₂²×C₃	1	2	yes	—
27	C₈²	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₃×Q₈	—	4	4	no	—
29	C₃×Q₁₆	—	8	8	no	—
30	C₂×C₃×C₈	D₈	1	2	yes	—
31	C₂×C₃×C₈	D₈	1	2	yes	—
32	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
33	C₃×C₄⋊C₄	—	4	4	no	—
34	C₃×C₄⋊C₄	—	4	4	no	—
35	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
36	C₃×C₁₆	Q₁₆	1	8	yes	complemented
37	C₂×Q₁₆	—	4	4	no	—
38	C₄⋊Q₈	—	2	4	no	—
39	C₂×C₁₆	C₃×D₄	1	4	yes	—
40	#107	—	2	8	no	—
41	#107	—	4	4	no	—
42	C₄×C₈	C₃×D₄	1	1	yes	characteristic
43	C₄×C₈	C₃×D₄	1	1	yes	characteristic
44	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
45	C₃×Q₈	—	8	8	no	—
46	C₂×C₃×C₄	—	8	4	no	—
47	C₂×C₃×C₄	#127	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
50	C₃×C₈	D₁₆	1	4	yes	—
51	C₃×C₈	Q₃₂	1	4	yes	—
52	C₃×C₈	C₂×Q₁₆	1	4	yes	—
53	C₂×Q₈	—	4	4	no	—
54	Q₁₆	—	8	8	no	—
55	C₂×C₈	C₃×D₈	1	2	yes	—
56	C₂×C₈	C₃×D₈	1	2	yes	—
57	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
58	C₄⋊C₄	—	4	4	no	—
59	C₄⋊C₄	—	4	4	no	—
60	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
61	C₁₆	C₃×Q₁₆	1	8	yes	complemented
62	C₂²×C₃	#493	1	1	yes	characteristic
63	C₃×C₄	—	16	4	no	—
64	C₃×C₄	#494	1	2	yes	—
65	C₃×C₄	C₂×D₁₆	1	2	yes	—
66	C₃×C₄	C₂×Q₃₂	1	2	yes	—
67	Q₈	—	8	8	no	—
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	C₃×#127	1	1	yes	characteristic
70	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
71	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
72	C₈	C₃×D₁₆	1	4	yes	—
73	C₈	C₃×Q₃₂	1	4	yes	—
74	C₈	C₂×C₃×Q₁₆	1	4	yes	—
75	C₂×C₃	#1713	1	1	yes	characteristic
76	C₂×C₃	#2246	1	1	yes	characteristic
77	C₂×C₃	#2247	1	1	yes	characteristic
78	C₂²	C₃×#493	1	1	yes	characteristic
79	C₄	—	16	4	no	—
80	C₄	C₃×#494	1	2	yes	—
81	C₄	C₂×C₃×D₁₆	1	2	yes	—
82	C₄	C₂×C₃×Q₃₂	1	2	yes	—
83	C₃	#12521	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	C₃×#1713	1	1	yes	characteristic
85	C₂	C₃×#2246	1	1	yes	characteristic
86	C₂	C₃×#2247	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
4. #12521	C₃	direct (N × H)
36. C₃×C₁₆	Q₁₆	semidirect
61. C₁₆	C₃×Q₁₆	semidirect
83. C₃	#12521	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×#2252	2
C₃×#1713	2
C₃×C₈×C₁₆	2
#12521	3

Subgroups — C3×#12521 #10631128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12521 #10631128