Subgroups — C₃×#1722 #72437

Order 384.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#500	C₂	1	2	yes	maximal, complemented
2	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#492	C₂	1	1	yes	maximal, characteristic
4	C₃×#486	C₂	1	2	yes	maximal
5	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#1722	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₁₆	—	2	2	no	—
8	C₂×C₃×SD₁₆	—	2	2	no	—
9	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×#127	C₂²	1	1	yes	characteristic
12	C₃×#107	—	2	4	no	—
13	C₃×#102	—	2	4	no	—
14	C₃×C₄×C₈	C₂²	1	2	yes	—
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#500	C₂×C₃	1	2	yes	complemented
17	#494	C₂×C₃	1	1	yes	characteristic, complemented
18	#492	C₂×C₃	1	1	yes	characteristic
19	#486	C₂×C₃	1	2	yes	—
20	#322	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×Q₈	—	2	1	no	—
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×D₄	—	2	2	no	—
24	C₂×C₃×D₄	—	2	1	no	—
25	C₃×Q₁₆	—	4	4	no	—
26	C₃×SD₁₆	—	4	4	no	—
27	C₂×C₃×C₈	D₄	1	4	yes	—
28	C₂×C₃×C₈	D₄	1	2	yes	—
29	C₃×C₄⋊C₄	—	2	2	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄⋊C₄	—	2	2	no	—
32	C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×Q₁₆	—	2	2	no	—
34	C₂×SD₁₆	—	2	2	no	—
35	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
36	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
37	#127	C₂²×C₃	1	1	yes	characteristic
38	#107	—	2	4	no	—
39	#102	—	2	4	no	—
40	C₄×C₈	C₂²×C₃	1	2	yes	—
41	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂³×C₃	—	4	1	no	—
43	C₃×Q₈	—	4	2	no	—
44	C₃×Q₈	—	4	4	no	—
45	C₃×D₄	—	4	2	no	—
46	C₃×D₄	—	4	4	no	—
47	C₂×C₃×C₄	—	4	2	no	—
48	C₂×C₃×C₄	—	4	1	no	—
49	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
51	C₃×C₈	—	2	4	no	—
52	C₃×C₈	D₈	1	4	yes	—
53	C₂×Q₈	—	2	1	no	—
54	C₂×Q₈	—	2	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	—	2	1	no	—
57	Q₁₆	—	4	4	no	—
58	SD₁₆	—	4	4	no	—
59	C₂×C₈	C₃×D₄	1	4	yes	—
60	C₂×C₈	C₃×D₄	1	2	yes	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄²	C₂³×C₃	1	1	yes	characteristic
65	C₂²×C₃	—	8	1	no	—
66	C₂²×C₃	—	8	2	no	—
67	C₂²×C₃	#127	1	1	yes	characteristic
68	C₃×C₄	—	8	1	no	—
69	C₃×C₄	—	8	2	no	—
70	C₃×C₄	C₂×D₈	1	2	yes	—
71	C₃×C₄	#137	1	4	yes	—
72	C₂³	—	4	1	no	—
73	Q₈	—	4	2	no	—
74	Q₈	—	4	4	no	—
75	D₄	—	4	2	no	—
76	D₄	—	4	4	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	C₃×D₈	1	4	yes	—
83	C₂×C₃	—	16	1	no	—
84	C₂×C₃	#493	1	1	yes	characteristic
85	C₂×C₃	#497	1	2	yes	—
86	C₂²	—	8	1	no	—
87	C₂²	—	8	2	no	—
88	C₂²	C₃×#127	1	1	yes	characteristic
89	C₄	—	8	1	no	—
90	C₄	—	8	2	no	—
91	C₄	C₂×C₃×D₈	1	2	yes	—
92	C₄	C₃×#137	1	4	yes	—
93	C₃	#1722	1	1	yes	characteristic, direct product factor, Sylow
94	C₂	—	16	1	no	—
95	C₂	C₃×#493	1	1	yes	characteristic
96	C₂	C₃×#497	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#500	C₂	semidirect
2. C₃×#494	C₂	semidirect
5. C₃×#322	C₂	semidirect
6. #1722	C₃	direct (N × H)
16. #500	C₂×C₃	semidirect
17. #494	C₂×C₃	semidirect
20. #322	C₂×C₃	semidirect
93. C₃	#1722	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×#500	2
C₃×#494	2
C₃×#492	2
C₃×#486	2
C₃×#322	2
#1722	3

Subgroups — C3×#1722 #72437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1722 #72437