Subgroups — C₂²×#3746 #10596213

Order 576.

774 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#634	C₂	1	2	yes	maximal
2	C₂²×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#3746	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₁₂	—	3	4	no	maximal
5	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊Dic₃	—	2	2	no	—
7	C₂×#634	—	2	6	no	—
8	C₂×#634	C₂²	1	12	yes	—
9	C₂×C₃²×C₈	C₂²	1	6	yes	—
10	#3746	C₂²	1	16	yes	direct product factor
11	C₂²×Dic₆	—	3	8	no	—
12	C₂²×C₃×C₈	D₃	1	4	yes	—
13	C₂×Dic₁₂	—	3	48	no	—
14	C₂³×C₃²	D₄	1	1	yes	characteristic
15	C₂×C₃²×C₄	D₄	1	3	yes	—
16	C₂×C₃²×C₄	C₂³	1	3	yes	—
17	C₂×C₃⋊Dic₃	—	2	12	no	—
18	#634	—	2	12	no	—
19	#634	C₂³	1	8	yes	—
20	C₃²×C₈	C₂³	1	4	yes	—
21	C₂²×Q₁₆	—	9	1	no	Sylow
22	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
23	C₂²×Dic₃	—	6	8	no	—
24	C₂×Dic₆	—	3	48	no	—
25	C₂×Dic₆	—	6	24	no	—
26	C₂×C₃×C₈	C₂×D₃	1	24	yes	—
27	Dic₁₂	—	3	64	no	—
28	C₂²×C₃²	Q₁₆	1	4	yes	complemented
29	C₂²×C₃²	C₂×D₄	1	3	yes	—
30	C₃²×C₄	C₂×D₄	1	3	yes	—
31	C₃²×C₄	C₂⁴	1	1	yes	characteristic
32	C₃⋊Dic₃	—	2	8	no	—
33	C₂²×Q₈	—	9	2	no	—
34	C₂×Q₁₆	—	9	12	no	—
35	C₂²×C₈	C₃⋊D₃	1	1	yes	characteristic
36	C₂³×C₃	D₁₂	1	4	yes	—
37	C₂×C₃×C₄	D₁₂	1	12	yes	—
38	C₂×C₃×C₄	C₂²×D₃	1	12	yes	—
39	C₂×Dic₃	—	6	48	no	—
40	Dic₆	—	3	32	no	—
41	Dic₆	—	6	48	no	—
42	C₃×C₈	C₂²×D₃	1	16	yes	—
43	C₂×C₃²	C₂×Q₁₆	1	6	yes	complemented
44	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
45	C₂×Q₈	—	9	12	no	—
46	C₂×Q₈	—	18	6	no	—
47	C₂²×C₄	—	18	2	no	—
48	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
49	Q₁₆	—	9	16	no	—
50	C₂×C₈	C₂×C₃⋊D₃	1	6	yes	—
51	C₂²×C₃	Dic₁₂	1	16	yes	complemented
52	C₂²×C₃	C₂×D₁₂	1	12	yes	—
53	C₃×C₄	C₂×D₁₂	1	12	yes	—
54	C₃×C₄	C₂³×D₃	1	4	yes	—
55	Dic₃	—	6	32	no	—
56	C₃²	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂³	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
58	Q₈	—	9	8	no	—
59	Q₈	—	18	12	no	—
60	C₂×C₄	—	18	12	no	—
61	C₂×C₄	C₄⋊(C₃⋊D₃)	1	3	yes	—
62	C₂×C₄	C₂²×C₃⋊D₃	1	3	yes	—
63	C₈	C₂²×C₃⋊D₃	1	4	yes	—
64	C₂×C₃	C₂×Dic₁₂	1	24	yes	complemented
65	C₂×C₃	C₂²×D₁₂	1	4	yes	—
66	C₂²	#3746	1	4	yes	direct product factor
67	C₂²	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
68	C₄	—	18	8	no	—
69	C₄	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
70	C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
71	C₃	C₂²×Dic₁₂	1	4	yes	complemented
72	C₂	C₂×#3746	1	6	yes	direct product factor
73	C₂	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂⁴
Frattini	C₄	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₈	C₂
socle, prime radical	C₂³×C₃²	D₄
hypercenter	C₂²×C₈	C₃⋊D₃
hypocenter	C₃²	C₂²×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#3746	C₂	direct (N × H)
10. #3746	C₂²	direct (N × H)
28. C₂²×C₃²	Q₁₆	semidirect
43. C₂×C₃²	C₂×Q₁₆	semidirect
51. C₂²×C₃	Dic₁₂	semidirect
56. C₃²	C₂²×Q₁₆	semidirect
64. C₂×C₃	C₂×Dic₁₂	semidirect
66. C₂²	#3746	direct (N × H)
71. C₃	C₂²×Dic₁₂	semidirect
72. C₂	C₂×#3746	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×#634	2
C₂²×C₃²×C₈	2
C₂×#3746	2
C₂²×Dic₁₂	3

Subgroups — C22×#3746 #10596213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3746 #10596213