Subgroups — C₃×#12523 #10631130

Order 768.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2254	C₂	1	4	yes	maximal
2	C₃×#1713	C₂	1	2	yes	maximal
3	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#12523	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×#367	—	2	8	no	—
9	C₃×C₄×C₁₆	C₂²	1	2	yes	—
10	C₃×C₈²	C₂²	1	1	yes	characteristic
11	#2254	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	2	yes	—
20	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
21	#500	—	2	2	no	—
22	#500	C₂²×C₃	1	4	yes	—
23	#494	—	2	2	no	—
24	#367	—	2	8	no	—
25	C₄×C₁₆	C₂²×C₃	1	2	yes	—
26	C₈²	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₃×Q₈	—	4	4	no	—
28	C₃×Q₁₆	—	8	8	no	—
29	C₂×C₃×C₈	D₈	1	4	yes	—
30	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
31	C₃×C₄⋊C₄	—	4	4	no	—
32	C₃×C₄⋊C₄	—	4	4	no	—
33	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
34	C₃×C₁₆	Q₁₆	1	8	yes	complemented
35	C₂×Q₁₆	—	4	4	no	—
36	C₄⋊Q₈	—	2	4	no	—
37	C₂×C₁₆	C₃×D₄	1	4	yes	—
38	#107	—	2	8	no	—
39	#107	—	4	4	no	—
40	C₄×C₈	C₃×D₄	1	2	yes	—
41	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
42	C₃×Q₈	—	8	8	no	—
43	C₂×C₃×C₄	—	8	4	no	—
44	C₂×C₃×C₄	#127	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
46	C₃×C₈	SD₃₂	1	8	yes	—
47	C₃×C₈	C₂×Q₁₆	1	4	yes	—
48	C₂×Q₈	—	4	4	no	—
49	Q₁₆	—	8	8	no	—
50	C₂×C₈	C₃×D₈	1	4	yes	—
51	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
52	C₄⋊C₄	—	4	4	no	—
53	C₄⋊C₄	—	4	4	no	—
54	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
55	C₁₆	C₃×Q₁₆	1	8	yes	complemented
56	C₂²×C₃	#493	1	1	yes	characteristic
57	C₃×C₄	—	16	4	no	—
58	C₃×C₄	#494	1	2	yes	—
59	C₃×C₄	C₂×SD₃₂	1	4	yes	—
60	Q₈	—	8	8	no	—
61	C₂×C₄	—	8	4	no	—
62	C₂×C₄	C₃×#127	1	1	yes	characteristic
63	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
64	C₈	C₃×SD₃₂	1	8	yes	—
65	C₈	C₂×C₃×Q₁₆	1	4	yes	—
66	C₂×C₃	#1713	1	1	yes	characteristic
67	C₂×C₃	#2248	1	2	yes	—
68	C₂²	C₃×#493	1	1	yes	characteristic
69	C₄	—	16	4	no	—
70	C₄	C₃×#494	1	2	yes	—
71	C₄	C₂×C₃×SD₃₂	1	4	yes	—
72	C₃	#12523	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₃×#1713	1	1	yes	characteristic
74	C₂	C₃×#2248	1	2	yes	—

	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. #12523	C₃	direct (N × H)
34. C₃×C₁₆	Q₁₆	semidirect
55. C₁₆	C₃×Q₁₆	semidirect
72. C₃	#12523	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×#2254	2
C₃×#1713	2
C₃×C₈×C₁₆	2
#12523	3

Subgroups — C3×#12523 #10631130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12523 #10631130