Subgroups — C₂³×#6308 #13181750

Order 1728.

11900 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	C₂²×#6308	C₂	1	28	yes	maximal, direct product factor
4	C₂³×#634	—	3	13	no	maximal
5	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
6	C₂²×C₃³×C₄	C₂²	1	14	yes	—
7	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	28	yes	—
8	C₂×#6308	C₂²	1	112	yes	direct product factor
9	C₂³×C₃²×C₄	D₃	1	13	yes	—
10	C₂³×C₃⋊Dic₃	—	3	26	no	—
11	C₂²×#634	—	3	364	no	—
12	C₂³×C₃³	Q₈	1	8	yes	complemented
13	C₂³×C₃³	C₂³	1	7	yes	—
14	C₂×C₃³×C₄	C₂³	1	28	yes	—
15	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	56	yes	—
16	#6308	C₂³	1	64	yes	direct product factor
17	C₂³×Dic₆	—	9	13	no	—
18	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
19	C₂²×C₃²×C₄	C₂×D₃	1	182	yes	—
20	C₂²×C₃⋊Dic₃	—	3	364	no	—
21	C₂×#634	—	3	1456	no	—
22	C₂²×C₃³	C₂×Q₈	1	28	yes	complemented
23	C₂²×C₃³	C₂⁴	1	7	yes	—
24	C₃³×C₄	C₂⁴	1	8	yes	—
25	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	16	yes	—
26	C₂³×C₃×C₄	C₃⋊D₃	1	13	yes	—
27	C₂³×Dic₃	—	9	26	no	—
28	C₂²×Dic₆	—	9	364	no	—
29	C₂³×C₃²	Dic₆	1	104	yes	complemented
30	C₂³×C₃²	C₂²×D₃	1	91	yes	—
31	C₂×C₃²×C₄	C₂²×D₃	1	364	yes	—
32	C₂×C₃⋊Dic₃	—	3	728	no	—
33	#634	—	3	832	no	—
34	C₂³×Q₈	—	27	1	no	Sylow
35	C₂×C₃³	C₂²×Q₈	1	14	yes	complemented
36	C₂×C₃³	C₂⁵	1	1	yes	characteristic
37	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
38	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	182	yes	—
39	C₂²×Dic₃	—	9	364	no	—
40	C₂×Dic₆	—	9	1456	no	—
41	C₂²×C₃²	C₂×Dic₆	1	364	yes	complemented
42	C₂²×C₃²	C₂³×D₃	1	91	yes	—
43	C₃²×C₄	C₂³×D₃	1	104	yes	—
44	C₃⋊Dic₃	—	3	208	no	—
45	C₂²×Q₈	—	27	28	no	—
46	C₂³×C₄	—	27	2	no	—
47	C₂³×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
48	C₃³	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₃	#634	1	104	yes	complemented
50	C₂³×C₃	C₂²×C₃⋊D₃	1	91	yes	—
51	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	364	yes	—
52	C₂×Dic₃	—	9	728	no	—
53	Dic₆	—	9	832	no	—
54	C₂×C₃²	C₂²×Dic₆	1	182	yes	complemented
55	C₂×C₃²	C₂⁴×D₃	1	13	yes	—
56	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
57	C₂×Q₈	—	27	112	no	—
58	C₂²×C₄	—	27	28	no	—
59	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	14	yes	—
60	C₂²×C₃	C₂×#634	1	364	yes	complemented
61	C₂²×C₃	C₂³×C₃⋊D₃	1	91	yes	—
62	C₃×C₄	C₂³×C₃⋊D₃	1	104	yes	—
63	Dic₃	—	9	208	no	—
64	C₃²	C₂³×Dic₆	1	13	yes	complemented
65	C₂³	#6308	1	8	yes	direct product factor
66	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	7	yes	—
67	Q₈	—	27	64	no	—
68	C₂×C₄	—	27	56	no	—
69	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	28	yes	—
70	C₂×C₃	C₂²×#634	1	182	yes	complemented
71	C₂×C₃	C₂⁴×C₃⋊D₃	1	13	yes	—
72	C₂²	C₂×#6308	1	28	yes	direct product factor
73	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	7	yes	—
74	C₄	—	27	16	no	—
75	C₄	C₂³×C₃⋊(C₃⋊D₃)	1	8	yes	—
76	C₃	C₂³×#634	1	13	yes	complemented
77	C₂	C₂²×#6308	1	14	yes	direct product factor
78	C₂	C₂⁴×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³	C₂⁵
Frattini	C₂	C₂⁴×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃³×C₄	C₂
socle, prime radical	C₂⁴×C₃³	C₂²
hypercenter	C₂³×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂³×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#6308	C₂	direct (N × H)
8. C₂×#6308	C₂²	direct (N × H)
12. C₂³×C₃³	Q₈	semidirect
16. #6308	C₂³	direct (N × H)
22. C₂²×C₃³	C₂×Q₈	semidirect
29. C₂³×C₃²	Dic₆	semidirect
35. C₂×C₃³	C₂²×Q₈	semidirect
41. C₂²×C₃²	C₂×Dic₆	semidirect
48. C₃³	C₂³×Q₈	semidirect
49. C₂³×C₃	#634	semidirect
54. C₂×C₃²	C₂²×Dic₆	semidirect
60. C₂²×C₃	C₂×#634	semidirect
64. C₃²	C₂³×Dic₆	semidirect
65. C₂³	#6308	direct (N × H)
70. C₂×C₃	C₂²×#634	semidirect
72. C₂²	C₂×#6308	direct (N × H)
76. C₃	C₂³×#634	semidirect
77. C₂	C₂²×#6308	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃³×C₄	2
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂²×#6308	2
C₂³×#634	3

Subgroups — C23×#6308 #13181750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6308 #13181750