Subgroups — C₃×#2017 #72732

Order 384.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#575	C₂	1	3	yes	maximal, complemented
2	C₃×#408	C₂	1	3	yes	maximal, complemented
3	C₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	#2017	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×#41	C₂²	1	3	yes	—
6	C₃×#137	—	2	6	no	—
7	C₃×AGL(1,ℤ/8)	—	2	6	no	—
8	C₃×#135	—	2	6	no	—
9	C₂×C₃×M₁₆	C₂²	1	3	yes	—
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₂²⋊C₈	—	2	6	no	—
12	#575	C₂×C₃	1	3	yes	complemented
13	#408	C₂×C₃	1	3	yes	complemented
14	#338	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×#41	—	2	6	no	—
16	C₃×#41	—	2	6	no	—
17	C₂×C₃×Q₈	D₄	1	3	yes	—
18	C₂×C₃×D₄	—	2	3	no	—
19	C₂×C₃×D₄	D₄	1	3	yes	—
20	C₂²×C₃×C₄	—	2	3	no	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₃×Q₁₆	—	2	6	no	—
23	C₃×SD₁₆	—	2	12	no	—
24	C₃×D₈	—	2	6	no	—
25	C₃×M₁₆	—	2	6	no	—
26	C₂×C₃×C₈	—	2	3	no	—
27	C₂×C₃×C₈	D₄	1	6	yes	—
28	C₂×#41	C₂²×C₃	1	3	yes	—
29	#137	—	2	6	no	—
30	AGL(1,ℤ/8)	—	2	6	no	—
31	#135	—	2	6	no	—
32	C₂×M₁₆	C₂²×C₃	1	3	yes	—
33	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
34	C₂²⋊C₈	—	2	6	no	—
35	C₂³×C₃	—	2	3	no	—
36	C₂³×C₃	#41	1	1	yes	characteristic
37	C₃×Q₈	—	2	6	no	—
38	C₃×D₄	—	2	6	no	—
39	C₃×D₄	—	4	6	no	—
40	C₂×C₃×C₄	—	2	3	no	—
41	C₂×C₃×C₄	—	4	3	no	—
42	C₂×C₃×C₄	—	4	3	no	—
43	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
44	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
45	C₃×C₈	—	2	6	no	—
46	C₃×C₈	—	4	1	no	—
47	#41	—	2	6	no	—
48	#41	—	2	6	no	—
49	C₂×Q₈	C₃×D₄	1	3	yes	—
50	C₂×D₄	—	2	3	no	—
51	C₂×D₄	C₃×D₄	1	3	yes	—
52	C₂²×C₄	—	2	3	no	—
53	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
54	Q₁₆	—	2	6	no	—
55	SD₁₆	—	2	12	no	—
56	D₈	—	2	6	no	—
57	M₁₆	—	2	6	no	—
58	C₂×C₈	—	2	3	no	—
59	C₂×C₈	C₃×D₄	1	6	yes	—
60	C₂²×C₃	—	4	1	no	—
61	C₂²×C₃	—	4	6	no	—
62	C₂²×C₃	—	4	3	no	—
63	C₂²×C₃	#121	1	3	yes	—
64	C₃×C₄	—	4	3	no	—
65	C₃×C₄	#120	1	3	yes	—
66	C₃×C₄	#127	1	1	yes	characteristic
67	C₂³	—	2	3	no	—
68	C₂³	C₃×#41	1	1	yes	characteristic
69	Q₈	—	2	6	no	—
70	D₄	—	2	6	no	—
71	D₄	—	4	6	no	—
72	C₂×C₄	—	2	3	no	—
73	C₂×C₄	—	4	3	no	—
74	C₂×C₄	—	4	3	no	—
75	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
76	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	—	4	1	no	—
79	C₂×C₃	—	2	3	no	—
80	C₂×C₃	—	8	3	no	—
81	C₂×C₃	#392	1	1	yes	characteristic
82	C₂²	—	4	1	no	—
83	C₂²	—	4	6	no	—
84	C₂²	—	4	3	no	—
85	C₂²	C₃×#121	1	3	yes	—
86	C₄	—	4	3	no	—
87	C₄	C₃×#120	1	3	yes	—
88	C₄	C₃×#127	1	1	yes	characteristic
89	C₃	#2017	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	—	2	3	no	—
91	C₂	—	8	3	no	—
92	C₂	C₃×#392	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂×C₃	#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#575	C₂	semidirect
2. C₃×#408	C₂	semidirect
3. C₃×#338	C₂	semidirect
4. #2017	C₃	direct (N × H)
12. #575	C₂×C₃	semidirect
13. #408	C₂×C₃	semidirect
14. #338	C₂×C₃	semidirect
89. C₃	#2017	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×#575	2
C₃×#408	2
C₃×#338	2
#2017	3

Subgroups — C3×#2017 #72732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2017 #72732