Subgroups — C₂²×#638 #64369

Order 288.

708 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#638	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#67	—	3	4	no	maximal
6	C₂³×C₃²	—	2	4	no	—
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₂³×C₃²	C₂²	1	6	yes	complemented
9	C₂²×C₃⋊D₃	—	2	4	no	—
10	C₂²×C₃⋊D₃	C₂²	1	6	yes	complemented
11	#638	C₂²	1	16	yes	direct product factor
12	C₂×C₃⋊Dic₃	C₂²	1	6	yes	complemented
13	C₂⁴×C₃	D₃	1	4	yes	complemented
14	C₂³×D₃	—	3	4	no	—
15	C₂×#67	—	3	48	no	—
16	C₂²×Dic₃	—	3	4	no	—
17	C₂²×C₃²	—	2	12	no	—
18	C₂²×C₃²	D₄	1	4	yes	complemented
19	C₂²×C₃²	C₂³	1	3	yes	—
20	C₂²×C₃²	C₂³	1	4	yes	complemented
21	C₂×C₃⋊D₃	—	2	12	no	—
22	C₂×C₃⋊D₃	C₂³	1	4	yes	complemented
23	C₃⋊Dic₃	C₂³	1	4	yes	complemented
24	C₂²×D₄	—	9	1	no	Sylow
25	C₂³×C₃	—	2	16	no	—
26	C₂³×C₃	C₂×D₃	1	4	yes	—
27	C₂³×C₃	C₂×D₃	1	24	yes	complemented
28	C₂²×D₃	—	3	24	no	—
29	C₂²×D₃	—	6	16	no	—
30	#67	—	3	64	no	—
31	C₂×Dic₃	—	3	24	no	—
32	C₂×C₃²	—	2	4	no	—
33	C₂×C₃²	C₂×D₄	1	6	yes	complemented
34	C₂×C₃²	C₂⁴	1	1	yes	characteristic
35	C₃⋊D₃	—	2	4	no	—
36	C₂⁴	—	9	1	no	—
37	C₂⁴	C₃⋊D₃	1	1	yes	characteristic, complemented
38	C₂×D₄	—	9	12	no	—
39	C₂²×C₄	—	9	1	no	—
40	C₂²×C₃	—	2	48	no	—
41	C₂²×C₃	#67	1	16	yes	complemented
42	C₂²×C₃	C₂²×D₃	1	12	yes	—
43	C₂²×C₃	C₂²×D₃	1	16	yes	complemented
44	C₂×D₃	—	3	16	no	—
45	C₂×D₃	—	6	48	no	—
46	Dic₃	—	3	16	no	—
47	C₃²	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
48	C₂³	—	2	4	no	—
49	C₂³	—	9	6	no	—
50	C₂³	—	18	4	no	—
51	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₂³	C₂×C₃⋊D₃	1	6	yes	complemented
53	D₄	—	9	16	no	—
54	C₂×C₄	—	9	6	no	—
55	C₂×C₃	—	2	16	no	—
56	C₂×C₃	C₂×#67	1	24	yes	complemented
57	C₂×C₃	C₂³×D₃	1	4	yes	—
58	D₃	—	6	16	no	—
59	C₂²	—	2	12	no	—
60	C₂²	—	9	4	no	—
61	C₂²	—	18	12	no	—
62	C₂²	#638	1	4	yes	direct product factor
63	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
64	C₂²	C₂²×C₃⋊D₃	1	4	yes	complemented
65	C₄	—	9	4	no	—
66	C₃	C₂²×#67	1	4	yes	complemented
67	C₂	—	2	4	no	—
68	C₂	—	18	4	no	—
69	C₂	C₂×#638	1	6	yes	direct product factor
70	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²	C₂	semidirect
2. C₂³×C₃⋊D₃	C₂	semidirect
3. C₂×#638	C₂	direct (N × H)
4. C₂²×C₃⋊Dic₃	C₂	semidirect
8. C₂³×C₃²	C₂²	semidirect
10. C₂²×C₃⋊D₃	C₂²	semidirect
11. #638	C₂²	direct (N × H)
12. C₂×C₃⋊Dic₃	C₂²	semidirect
13. C₂⁴×C₃	D₃	semidirect
18. C₂²×C₃²	D₄	semidirect
20. C₂²×C₃²	C₂³	semidirect
22. C₂×C₃⋊D₃	C₂³	semidirect
23. C₃⋊Dic₃	C₂³	semidirect
27. C₂³×C₃	C₂×D₃	semidirect
33. C₂×C₃²	C₂×D₄	semidirect
37. C₂⁴	C₃⋊D₃	semidirect
41. C₂²×C₃	#67	semidirect
43. C₂²×C₃	C₂²×D₃	semidirect
47. C₃²	C₂²×D₄	semidirect
52. C₂³	C₂×C₃⋊D₃	semidirect
56. C₂×C₃	C₂×#67	semidirect
62. C₂²	#638	direct (N × H)
64. C₂²	C₂²×C₃⋊D₃	semidirect
66. C₃	C₂²×#67	semidirect
69. C₂	C₂×#638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²	2
C₂³×C₃⋊D₃	2
C₂×#638	2
C₂²×C₃⋊Dic₃	2
C₂²×#67	3

Subgroups — C22×#638 #64369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#638 #64369