Subgroups — C₃×#12522 #10631129

Order 768.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2254	C₂	1	2	yes	maximal
2	C₃×#2252	C₂	1	2	yes	maximal
3	C₃×#1710	C₂	1	2	yes	maximal
4	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#12522	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#500	C₂²	1	4	yes	—
7	C₃×#487	—	2	4	no	—
8	C₃×#367	—	2	4	no	—
9	C₃×#366	—	2	4	no	—
10	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₈²	C₂²	1	1	yes	characteristic
13	#2254	C₂×C₃	1	2	yes	—
14	#2252	C₂×C₃	1	2	yes	—
15	#1710	C₂×C₃	1	2	yes	—
16	C₈×C₁₆	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	—	2	4	no	—
18	C₂×C₃×C₁₆	D₄	1	2	yes	—
19	C₂×C₃×C₁₆	D₄	1	2	yes	—
20	C₃×#107	—	2	4	no	—
21	C₃×#107	—	2	4	no	—
22	C₃×#103	—	4	8	no	—
23	C₃×C₄×C₈	D₄	1	2	yes	—
24	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
25	#500	C₂²×C₃	1	4	yes	—
26	#487	—	2	4	no	—
27	#367	—	2	4	no	—
28	#366	—	2	4	no	—
29	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
30	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
31	C₈²	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₃×Q₈	—	4	4	no	—
33	C₂×C₃×C₈	D₈	1	4	yes	—
34	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
36	C₃×C₄⋊C₄	—	4	4	no	—
37	C₃×C₄⋊C₄	—	4	4	no	—
38	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
39	C₃×C₁₆	Q₁₆	1	4	yes	—
40	C₃×C₁₆	Q₁₆	1	4	yes	—
41	C₄⋊Q₈	—	2	4	no	—
42	C₂×C₁₆	C₃×D₄	1	2	yes	—
43	C₂×C₁₆	C₃×D₄	1	2	yes	—
44	#107	—	2	4	no	—
45	#107	—	2	4	no	—
46	#103	—	4	8	no	—
47	C₄×C₈	C₃×D₄	1	2	yes	—
48	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
49	C₃×Q₈	—	8	8	no	—
50	C₂×C₃×C₄	—	8	4	no	—
51	C₂×C₃×C₄	#127	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
53	C₃×C₈	—	2	4	no	—
54	C₃×C₈	C₂×Q₁₆	1	2	yes	—
55	C₃×C₈	C₂×Q₁₆	1	2	yes	—
56	C₂×Q₈	—	4	4	no	—
57	C₂×C₈	C₃×D₈	1	4	yes	—
58	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
59	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	—	4	4	no	—
62	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
63	C₁₆	C₃×Q₁₆	1	4	yes	—
64	C₁₆	C₃×Q₁₆	1	4	yes	—
65	C₂²×C₃	#493	1	1	yes	characteristic
66	C₃×C₄	—	16	4	no	—
67	C₃×C₄	#494	1	1	yes	characteristic
68	C₃×C₄	#494	1	1	yes	characteristic
69	C₃×C₄	#508	1	4	yes	—
70	Q₈	—	8	8	no	—
71	C₂×C₄	—	8	4	no	—
72	C₂×C₄	C₃×#127	1	1	yes	characteristic
73	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
74	C₈	—	2	4	no	—
75	C₈	C₂×C₃×Q₁₆	1	2	yes	—
76	C₈	C₂×C₃×Q₁₆	1	2	yes	—
77	C₂×C₃	#1713	1	1	yes	characteristic
78	C₂×C₃	#2249	1	2	yes	—
79	C₂²	C₃×#493	1	1	yes	characteristic
80	C₄	—	16	4	no	—
81	C₄	C₃×#494	1	1	yes	characteristic
82	C₄	C₃×#494	1	1	yes	characteristic
83	C₄	C₃×#508	1	4	yes	—
84	C₃	#12522	1	1	yes	characteristic, direct product factor, Sylow
85	C₂	C₃×#1713	1	1	yes	characteristic
86	C₂	C₃×#2249	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
5. #12522	C₃	direct (N × H)
84. C₃	#12522	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×#2252	2
C₃×#1710	2
C₃×C₈×C₁₆	2
#12522	3

Subgroups — C3×#12522 #10631129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12522 #10631129