Subgroups — C₂²×C₃×#622 #11724506

Order 864.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#622	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#622	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂³×C₃³	C₄	1	1	yes	characteristic
5	C₂×C₃×C₃⋊Dic₃	C₄	1	3	yes	—
6	C₂×C₃×C₃⋊Dic₃	C₂²	1	3	yes	—
7	C₃×#622	C₂²	1	4	yes	direct product factor
8	C₂²×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₃×Dic₃	—	6	2	no	—
10	C₂×#622	C₂×C₃	1	6	yes	direct product factor
11	C₂²×C₃³	C₈	1	4	yes	complemented
12	C₂²×C₃³	C₂×C₄	1	3	yes	—
13	C₃×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
14	C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
15	C₂²×C₃×C₈	—	9	1	no	maximal
16	C₂³×C₃²	—	2	2	no	—
17	C₂³×C₃²	—	4	2	no	—
18	C₂³×C₃²	C₃×C₄	1	1	yes	characteristic
19	C₂×C₃⋊Dic₃	C₃×C₄	1	3	yes	—
20	C₂×C₃⋊Dic₃	C₂²×C₃	1	3	yes	—
21	C₂×C₃×Dic₃	—	6	6	no	—
22	C₂×C₃×Dic₃	—	6	6	no	—
23	#622	C₂²×C₃	1	4	yes	direct product factor
24	C₂×C₃³	C₂×C₈	1	6	yes	complemented
25	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	—	9	1	no	—
27	C₂²×Dic₃	—	6	2	no	—
28	C₂×C₃×C₈	—	9	6	no	—
29	C₂²×C₃²	—	2	8	no	—
30	C₂²×C₃²	—	2	6	no	—
31	C₂²×C₃²	—	4	8	no	—
32	C₂²×C₃²	—	4	6	no	—
33	C₂²×C₃²	C₃×C₈	1	4	yes	complemented
34	C₂²×C₃²	C₂×C₃×C₄	1	3	yes	—
35	C₃⋊Dic₃	C₂×C₃×C₄	1	3	yes	—
36	C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
37	C₃×Dic₃	—	6	6	no	—
38	C₃×Dic₃	—	6	2	no	—
39	C₂²×C₈	—	9	1	no	Sylow
40	C₃³	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
41	C₂³×C₃	—	2	2	no	—
42	C₂³×C₃	—	4	2	no	—
43	C₂³×C₃	#157	1	1	yes	characteristic
44	C₂×C₃×C₄	—	9	3	no	—
45	C₂×C₃×C₄	—	9	3	no	—
46	C₂×Dic₃	—	6	6	no	—
47	C₂×Dic₃	—	6	6	no	—
48	C₃×C₈	—	9	4	no	—
49	C₂×C₃²	—	2	2	no	—
50	C₂×C₃²	—	2	12	no	—
51	C₂×C₃²	—	4	12	no	—
52	C₂×C₃²	—	4	2	no	—
53	C₂×C₃²	C₂×C₃×C₈	1	6	yes	complemented
54	C₂×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
55	C₂²×C₄	—	9	1	no	—
56	C₂×C₈	—	9	6	no	—
57	C₂²×C₃	—	2	8	no	—
58	C₂²×C₃	—	2	6	no	—
59	C₂²×C₃	—	4	8	no	—
60	C₂²×C₃	—	4	6	no	—
61	C₂²×C₃	#622	1	4	yes	direct product factor
62	C₂²×C₃	C₂×#157	1	3	yes	—
63	C₃×C₄	—	9	1	no	—
64	C₃×C₄	—	9	3	no	—
65	Dic₃	—	6	6	no	—
66	Dic₃	—	6	2	no	—
67	C₃²	—	2	2	no	—
68	C₃²	—	4	2	no	—
69	C₃²	C₂²×C₃×C₈	1	1	yes	characteristic, complemented
70	C₂³	C₃×#157	1	1	yes	characteristic
71	C₂×C₄	—	9	3	no	—
72	C₂×C₄	—	9	3	no	—
73	C₈	—	9	4	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	—	2	12	no	—
76	C₂×C₃	—	4	2	no	—
77	C₂×C₃	—	4	12	no	—
78	C₂×C₃	C₂×#622	1	6	yes	direct product factor
79	C₂×C₃	C₂²×#157	1	1	yes	characteristic
80	C₂²	C₃×#622	1	4	yes	direct product factor
81	C₂²	C₂×C₃×#157	1	3	yes	—
82	C₄	—	9	1	no	—
83	C₄	—	9	3	no	—
84	C₃	—	2	2	no	—
85	C₃	—	4	2	no	—
86	C₃	C₂²×#622	1	1	yes	characteristic, direct product factor
87	C₂	C₂×C₃×#622	1	6	yes	direct product factor
88	C₂	C₂²×C₃×#157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₃	#157
commutator, hypocenter	C₃²	C₂²×C₃×C₈
Frattini	C₂	C₂²×C₃×#157
Fitting, socle, prime radical	C₂³×C₃³	C₄
cosocle	C₃⋊Dic₃	C₂³×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#622	C₂	direct (N × H)
3. C₂²×#622	C₃	direct (N × H)
7. C₃×#622	C₂²	direct (N × H)
10. C₂×#622	C₂×C₃	direct (N × H)
11. C₂²×C₃³	C₈	semidirect
23. #622	C₂²×C₃	direct (N × H)
24. C₂×C₃³	C₂×C₈	semidirect
33. C₂²×C₃²	C₃×C₈	semidirect
40. C₃³	C₂²×C₈	semidirect
53. C₂×C₃²	C₂×C₃×C₈	semidirect
61. C₂²×C₃	#622	direct (N × H)
69. C₃²	C₂²×C₃×C₈	semidirect
78. C₂×C₃	C₂×#622	direct (N × H)
80. C₂²	C₃×#622	direct (N × H)
86. C₃	C₂²×#622	direct (N × H)
87. C₂	C₂×C₃×#622	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₃⋊Dic₃	2
C₂×C₃×#622	2
C₂²×#622	3
C₂²×C₃×C₈	9

Subgroups — C22×C3×#622 #11724506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#622 #11724506