Subgroups — C₃×#2069 #72784

Order 384.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#430	C₂	1	3	yes	maximal, complemented
2	C₂×C₃×#100	C₂	1	3	yes	maximal
3	C₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	#2069	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×M₁₆	C₂²	1	3	yes	—
7	C₂×C₃×M₁₆	C₂²	1	3	yes	—
8	C₃×#108	—	2	6	no	—
9	C₃×#100	—	2	6	no	—
10	#430	C₂×C₃	1	3	yes	complemented
11	C₂×#100	C₂×C₃	1	3	yes	—
12	#338	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂⁴×C₃	Q₈	1	2	yes	—
14	C₂×C₃×D₄	—	2	3	no	—
15	C₂×C₃×D₄	—	2	3	no	—
16	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
17	C₃×M₁₆	—	2	3	no	—
18	C₃×M₁₆	—	2	3	no	—
19	C₃×M₁₆	—	2	3	no	—
20	C₃×M₁₆	D₄	1	6	yes	—
21	C₂×C₃×C₈	—	2	3	no	—
22	C₂×C₃×C₈	—	2	3	no	—
23	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
24	C₂×M₁₆	C₂²×C₃	1	3	yes	—
25	C₂×M₁₆	C₂²×C₃	1	3	yes	—
26	#108	—	2	6	no	—
27	#100	—	2	6	no	—
28	C₂³×C₃	—	2	6	no	—
29	C₂³×C₃	—	8	2	no	—
30	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
31	C₃×D₄	—	4	3	no	—
32	C₃×D₄	—	4	1	no	—
33	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
34	C₂×C₃×C₄	#41	1	3	yes	—
35	C₃×C₈	—	2	6	no	—
36	C₃×C₈	—	4	3	no	—
37	C₂⁴	C₃×Q₈	1	2	yes	—
38	C₂×D₄	—	2	3	no	—
39	C₂×D₄	—	2	3	no	—
40	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
41	M₁₆	—	2	3	no	—
42	M₁₆	—	2	3	no	—
43	M₁₆	—	2	3	no	—
44	M₁₆	C₃×D₄	1	6	yes	—
45	C₂×C₈	—	2	3	no	—
46	C₂×C₈	—	2	3	no	—
47	C₂²×C₃	—	4	1	no	—
48	C₂²×C₃	—	4	2	no	—
49	C₂²×C₃	—	8	6	no	—
50	C₂²×C₃	C₂²⋊Q₈	1	3	yes	—
51	C₃×C₄	#121	1	3	yes	—
52	C₃×C₄	#126	1	1	yes	characteristic
53	C₂³	—	2	6	no	—
54	C₂³	—	8	2	no	—
55	C₂³	C₂×C₃×Q₈	1	1	yes	characteristic
56	D₄	—	4	3	no	—
57	D₄	—	4	1	no	—
58	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
59	C₂×C₄	C₃×#41	1	3	yes	—
60	C₈	—	2	6	no	—
61	C₈	—	4	3	no	—
62	C₂×C₃	—	2	3	no	—
63	C₂×C₃	—	8	2	no	—
64	C₂×C₃	#396	1	1	yes	characteristic
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	8	6	no	—
68	C₂²	C₃×C₂²⋊Q₈	1	3	yes	—
69	C₄	C₃×#121	1	3	yes	—
70	C₄	C₃×#126	1	1	yes	characteristic
71	C₃	#2069	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	2	3	no	—
73	C₂	—	8	2	no	—
74	C₂	C₃×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#430	C₂	semidirect
3. C₃×#338	C₂	semidirect
4. #2069	C₃	direct (N × H)
10. #430	C₂×C₃	semidirect
12. #338	C₂×C₃	semidirect
71. C₃	#2069	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×#430	2
C₂×C₃×#100	2
C₃×#338	2
#2069	3

Subgroups — C3×#2069 #72784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2069 #72784