Subgroups — C₄²×#1084 #13144253

Order 1728.

1419 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#1084	C₂	1	6	yes	maximal
3	C₃×C₄²×Dic₃	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
5	C₂²×#1084	C₂²	1	4	yes	—
6	C₄²×H(3,𝔽₃)	C₄	1	4	yes	complemented
7	C₄×#1084	C₄	1	24	yes	direct product factor
8	C₂×C₃²×C₄²	D₃	1	4	yes	—
9	C₂×C₃×C₄×Dic₃	—	3	24	no	—
10	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
11	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	12	yes	—
12	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	6	yes	—
13	C₂×#1084	C₂×C₄	1	24	yes	—
14	C₃×C₄³	—	9	1	no	—
15	C₄²×Dic₃	—	3	4	no	—
16	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
17	C₂²×C₃×Dic₃	—	3	16	no	—
18	C₃²×C₄²	Dic₃	1	16	yes	complemented
19	C₃×C₄×Dic₃	—	3	96	no	—
20	C₂²×H(3,𝔽₃)	C₂²×C₄	1	6	yes	—
21	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
22	C₄×H(3,𝔽₃)	C₄²	1	12	yes	complemented
23	#1084	C₄²	1	16	yes	direct product factor
24	C₂×C₃×C₄²	—	3	4	no	—
25	C₂×C₃×C₄²	—	9	6	no	—
26	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
27	C₂×C₄×Dic₃	—	3	24	no	—
28	C₂³×C₃²	C₂²×D₃	1	4	yes	—
29	C₂×C₃²×C₄	C₄×D₃	1	48	yes	—
30	C₂×C₃²×C₄	C₂×Dic₃	1	24	yes	—
31	C₂×C₃×Dic₃	—	3	96	no	—
32	C₄³	—	9	1	no	Sylow
33	C₂×H(3,𝔽₃)	C₂×C₄²	1	4	yes	—
34	C₂×H(3,𝔽₃)	C₂×C₄²	1	3	yes	—
35	C₂²×C₃×C₄	—	3	12	no	—
36	C₂²×C₃×C₄	—	9	4	no	—
37	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
38	C₂²×Dic₃	—	3	16	no	—
39	C₃×C₄²	—	3	16	no	—
40	C₃×C₄²	—	9	24	no	—
41	C₃×C₄²	C₃⋊Dic₃	1	4	yes	—
42	C₄×Dic₃	—	3	96	no	—
43	C₂²×C₃²	C₂×C₄×D₃	1	24	yes	—
44	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
45	C₃²×C₄	C₄×Dic₃	1	48	yes	complemented
46	C₃×Dic₃	—	3	64	no	—
47	C₂×C₄²	—	9	6	no	—
48	C₂×C₄²	#272	1	1	yes	characteristic
49	H(3,𝔽₃)	C₄³	1	1	yes	characteristic, complemented, Sylow
50	C₂³×C₃	—	3	4	no	—
51	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
52	C₂×C₃×C₄	—	3	48	no	—
53	C₂×C₃×C₄	—	3	24	no	—
54	C₂×C₃×C₄	—	9	24	no	—
55	C₂×C₃×C₄	C₄×C₃⋊D₃	1	12	yes	—
56	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
57	C₂×Dic₃	—	3	96	no	—
58	C₂×C₃²	C₄²×D₃	1	16	yes	—
59	C₂×C₃²	C₂×C₄×Dic₃	1	12	yes	—
60	C₂²×C₄	—	9	4	no	—
61	C₂²×C₄	C₂×#272	1	3	yes	—
62	C₄²	—	9	24	no	—
63	C₄²	#1084	1	4	yes	direct product factor
64	C₂²×C₃	—	3	24	no	—
65	C₂²×C₃	—	3	4	no	—
66	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	6	yes	—
67	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
68	C₃×C₄	—	3	48	no	—
69	C₃×C₄	—	9	16	no	—
70	C₃×C₄	C₄×C₃⋊Dic₃	1	12	yes	—
71	Dic₃	—	3	64	no	—
72	C₃²	C₄²×Dic₃	1	4	yes	complemented
73	C₂³	C₂²×#272	1	1	yes	characteristic
74	C₂×C₄	—	9	24	no	—
75	C₂×C₄	C₄×#272	1	12	yes	—
76	C₂×C₄	C₂×#1084	1	6	yes	—
77	C₂×C₃	—	3	12	no	—
78	C₂×C₃	—	3	16	no	—
79	C₂×C₃	C₄²×C₃⋊D₃	1	4	yes	—
80	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	3	yes	—
81	C₂²	C₂×C₄×#272	1	6	yes	—
82	C₂²	C₂²×#1084	1	1	yes	characteristic
83	C₄	—	9	16	no	—
84	C₄	C₄×#1084	1	12	yes	direct product factor
85	C₃	—	3	4	no	—
86	C₃	C₄²×C₃⋊Dic₃	1	1	yes	characteristic
87	C₂	C₄²×#272	1	4	yes	—
88	C₂	C₂×C₄×#1084	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄²	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₄³
Frattini, socle	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₄²×H(3,𝔽₃)	C₂
prime radical, cosocle	C₂³×H(3,𝔽₃)	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×H(3,𝔽₃)	C₄	semidirect
7. C₄×#1084	C₄	direct (N × H)
18. C₃²×C₄²	Dic₃	semidirect
22. C₄×H(3,𝔽₃)	C₄²	semidirect
23. #1084	C₄²	direct (N × H)
45. C₃²×C₄	C₄×Dic₃	semidirect
49. H(3,𝔽₃)	C₄³	semidirect
63. C₄²	#1084	direct (N × H)
72. C₃²	C₄²×Dic₃	semidirect
84. C₄	C₄×#1084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄²×H(3,𝔽₃)	2
C₂×C₄×#1084	2
C₃×C₄²×Dic₃	3

Subgroups — C42×#1084 #13144253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#1084 #13144253