Subgroups — C₂²×#624 #64324

Order 288.

539 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₂×#624	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×D₃	—	3	2	no	maximal
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊D₃	C₄	1	8	yes	complemented
7	C₂²×C₃⋊D₃	C₂²	1	6	yes	—
8	C₂×C₃×Dic₃	C₂²	1	12	yes	complemented
9	#624	C₂²	1	16	yes	direct product factor
10	C₂³×D₃	—	3	2	no	—
11	C₂³×D₃	—	6	1	no	—
12	C₂²×C₃×C₄	—	3	2	no	—
13	C₂²×Dic₃	D₃	1	2	yes	complemented
14	C₂×C₄×D₃	—	3	24	no	—
15	C₂²×C₃²	C₂×C₄	1	4	yes	complemented
16	C₂²×C₃²	C₂³	1	3	yes	—
17	C₂×C₃⋊D₃	C₂×C₄	1	24	yes	complemented
18	C₂×C₃⋊D₃	C₂³	1	4	yes	—
19	C₃×Dic₃	C₂³	1	8	yes	complemented
20	C₂³×C₄	—	9	1	no	Sylow
21	C₂³×C₃	—	2	1	no	—
22	C₂³×C₃	C₂×D₃	1	2	yes	—
23	C₂²×D₃	—	3	16	no	—
24	C₂²×D₃	—	3	12	no	—
25	C₂²×D₃	—	6	8	no	—
26	C₂²×D₃	—	6	6	no	—
27	C₂×C₃×C₄	—	3	12	no	—
28	C₂×Dic₃	C₂×D₃	1	12	yes	complemented
29	C₄×D₃	—	3	32	no	—
30	C₂×C₃²	C₂²×C₄	1	6	yes	complemented
31	C₂×C₃²	C₂⁴	1	1	yes	characteristic
32	C₃⋊D₃	C₂²×C₄	1	8	yes	complemented
33	C₂⁴	—	9	1	no	—
34	C₂²×C₄	—	3	2	no	—
35	C₂²×C₄	—	9	12	no	—
36	C₂²×C₃	—	2	4	no	—
37	C₂²×C₃	—	2	3	no	—
38	C₂²×C₃	C₄×D₃	1	8	yes	complemented
39	C₂²×C₃	C₂²×D₃	1	6	yes	—
40	C₂×D₃	—	3	48	no	—
41	C₂×D₃	—	3	8	no	—
42	C₂×D₃	—	6	24	no	—
43	C₂×D₃	—	6	4	no	—
44	C₃×C₄	—	3	8	no	—
45	Dic₃	C₂²×D₃	1	8	yes	complemented
46	C₃²	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
47	C₂³	—	9	8	no	—
48	C₂³	—	9	6	no	—
49	C₂³	D₃²	1	1	yes	characteristic
50	C₂×C₄	—	3	12	no	—
51	C₂×C₄	—	9	16	no	—
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	—	2	6	no	—
54	C₂×C₃	C₂×C₄×D₃	1	12	yes	complemented
55	C₂×C₃	C₂³×D₃	1	2	yes	—
56	D₃	—	3	16	no	—
57	D₃	—	6	8	no	—
58	C₂²	—	9	4	no	—
59	C₂²	—	9	24	no	—
60	C₂²	#624	1	4	yes	direct product factor
61	C₂²	C₂×D₃²	1	3	yes	—
62	C₄	—	3	8	no	—
63	C₃	—	2	1	no	—
64	C₃	C₂²×C₄×D₃	1	2	yes	complemented
65	C₂	—	9	8	no	—
66	C₂	C₂×#624	1	6	yes	direct product factor
67	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, hypocenter	C₃²	C₂³×C₄
Frattini	C₂	C₂²×D₃²
Fitting, socle	C₂³×C₃²	C₂²
prime radical	C₂³×C₃⋊D₃	C₂
cosocle	C₂×C₃²	C₂⁴
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₃	C₂	semidirect
3. C₂×#624	C₂	direct (N × H)
6. C₂²×C₃⋊D₃	C₄	semidirect
8. C₂×C₃×Dic₃	C₂²	semidirect
9. #624	C₂²	direct (N × H)
13. C₂²×Dic₃	D₃	semidirect
15. C₂²×C₃²	C₂×C₄	semidirect
17. C₂×C₃⋊D₃	C₂×C₄	semidirect
19. C₃×Dic₃	C₂³	semidirect
28. C₂×Dic₃	C₂×D₃	semidirect
30. C₂×C₃²	C₂²×C₄	semidirect
32. C₃⋊D₃	C₂²×C₄	semidirect
38. C₂²×C₃	C₄×D₃	semidirect
45. Dic₃	C₂²×D₃	semidirect
46. C₃²	C₂³×C₄	semidirect
54. C₂×C₃	C₂×C₄×D₃	semidirect
60. C₂²	#624	direct (N × H)
64. C₃	C₂²×C₄×D₃	semidirect
66. C₂	C₂×#624	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×C₃⋊D₃	2
C₂²×C₃×Dic₃	2
C₂×#624	2
C₂²×C₄×D₃	3

Subgroups — C22×#624 #64324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#624 #64324