Subgroups — C₂×#3744 #64111

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3744	C₂	1	4	yes	maximal, direct product factor
5	C₂×#204	—	3	4	no	maximal
6	C₂²×C₃⋊D₃	—	2	1	no	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₃	—	2	1	no	—
9	C₄⋊(C₃⋊D₃)	—	2	1	no	—
10	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	#634	—	2	1	no	—
12	#634	C₂²	1	2	yes	complemented
13	C₃²×C₈	C₂²	1	2	yes	complemented
14	C₂×D₁₂	—	3	4	no	—
15	C₂×Dic₆	—	3	4	no	—
16	C₂×C₃×C₈	D₃	1	4	yes	complemented
17	#204	—	3	16	no	—
18	C₂²×C₃²	D₄	1	1	yes	characteristic
19	C₂×C₃⋊D₃	—	2	2	no	—
20	C₂×C₃⋊D₃	—	4	2	no	—
21	C₃²×C₄	D₄	1	1	yes	characteristic
22	C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₃⋊Dic₃	—	2	2	no	—
24	C₂×SD₁₆	—	9	1	no	Sylow
25	C₂²×D₃	—	6	4	no	—
26	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
27	C₂×Dic₃	—	6	4	no	—
28	D₁₂	—	3	8	no	—
29	D₁₂	—	6	4	no	—
30	Dic₆	—	3	8	no	—
31	Dic₆	—	6	4	no	—
32	C₃×C₈	C₂×D₃	1	8	yes	complemented
33	C₂×C₃²	SD₁₆	1	2	yes	complemented
34	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
35	C₃⋊D₃	—	4	2	no	—
36	C₂×Q₈	—	9	1	no	—
37	C₂×D₄	—	9	1	no	—
38	SD₁₆	—	9	4	no	—
39	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
40	C₂²×C₃	D₁₂	1	4	yes	—
41	C₂×D₃	—	6	8	no	—
42	C₂×D₃	—	12	8	no	—
43	C₃×C₄	D₁₂	1	4	yes	—
44	C₃×C₄	C₂²×D₃	1	4	yes	—
45	Dic₃	—	6	8	no	—
46	C₃²	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂³	—	18	1	no	—
48	Q₈	—	9	2	no	—
49	Q₈	—	18	1	no	—
50	D₄	—	9	2	no	—
51	D₄	—	18	1	no	—
52	C₂×C₄	—	18	1	no	—
53	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
54	C₈	C₂×C₃⋊D₃	1	2	yes	complemented
55	C₂×C₃	#204	1	8	yes	complemented
56	C₂×C₃	C₂×D₁₂	1	4	yes	—
57	D₃	—	12	8	no	—
58	C₂²	—	18	2	no	—
59	C₂²	—	36	2	no	—
60	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
61	C₄	—	18	2	no	—
62	C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
63	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
64	C₃	C₂×#204	1	4	yes	complemented
65	C₂	—	36	2	no	—
66	C₂	#3744	1	2	yes	direct product factor
67	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	D₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×SD₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#634	C₂	semidirect
3. C₂×C₃²×C₈	C₂	semidirect
4. #3744	C₂	direct (N × H)
12. #634	C₂²	semidirect
13. C₃²×C₈	C₂²	semidirect
16. C₂×C₃×C₈	D₃	semidirect
32. C₃×C₈	C₂×D₃	semidirect
33. C₂×C₃²	SD₁₆	semidirect
39. C₂×C₈	C₃⋊D₃	semidirect
46. C₃²	C₂×SD₁₆	semidirect
54. C₈	C₂×C₃⋊D₃	semidirect
55. C₂×C₃	#204	semidirect
64. C₃	C₂×#204	semidirect
66. C₂	#3744	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×C₄⋊(C₃⋊D₃)	2
C₂×#634	2
C₂×C₃²×C₈	2
#3744	2
C₂×#204	3

Subgroups — C2×#3744 #64111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3744 #64111