Subgroups — C₂×C₃²×#41 #64373

Order 288.

492 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×D₄	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃²×C₄	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#41	C₃	1	4	yes	maximal, direct product factor
6	C₂³×C₃²	C₂²	1	3	yes	—
7	C₃²×Q₈	C₂²	1	4	yes	complemented
8	C₃²×D₄	C₂²	1	12	yes	complemented
9	C₂×C₃²×C₄	C₂²	1	3	yes	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄	C₂²	1	12	yes	complemented
12	C₃×#41	C₂×C₃	1	32	yes	direct product factor
13	C₂×C₃×Q₈	C₂×C₃	1	4	yes	complemented
14	C₂×C₃×D₄	C₂×C₃	1	12	yes	complemented
15	C₂²×C₃×C₄	C₂×C₃	1	12	yes	complemented
16	C₂²×C₃²	—	2	6	no	—
17	C₂²×C₃²	C₂³	1	6	yes	—
18	C₂²×C₃²	C₂³	1	1	yes	characteristic
19	C₃²×C₄	C₂³	1	6	yes	—
20	C₃²×C₄	C₂³	1	2	yes	—
21	C₂×#41	C₃²	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	C₂²×C₃	1	12	yes	—
23	C₃×Q₈	C₂²×C₃	1	16	yes	complemented
24	C₃×D₄	C₂²×C₃	1	48	yes	complemented
25	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
26	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
27	C₂×C₃×C₄	C₂²×C₃	1	48	yes	complemented
28	C₂×C₃²	—	2	6	no	—
29	C₂×C₃²	#41	1	2	yes	direct product factor
30	C₂×C₃²	C₂⁴	1	1	yes	characteristic
31	#41	C₂×C₃²	1	8	yes	direct product factor
32	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
33	C₂×D₄	C₂×C₃²	1	3	yes	complemented
34	C₂²×C₄	C₂×C₃²	1	3	yes	complemented
35	C₂²×C₃	—	2	24	no	—
36	C₂²×C₃	C₂³×C₃	1	24	yes	—
37	C₂²×C₃	C₂³×C₃	1	4	yes	—
38	C₃×C₄	C₂³×C₃	1	8	yes	—
39	C₃×C₄	C₂³×C₃	1	24	yes	—
40	C₃²	C₂×#41	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³	C₂²×C₃²	1	3	yes	—
42	Q₈	C₂²×C₃²	1	4	yes	complemented
43	D₄	C₂²×C₃²	1	12	yes	complemented
44	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
45	C₂×C₄	C₂²×C₃²	1	3	yes	—
46	C₂×C₄	C₂²×C₃²	1	12	yes	complemented
47	C₂×C₃	—	2	24	no	—
48	C₂×C₃	C₃×#41	1	8	yes	direct product factor
49	C₂×C₃	C₂⁴×C₃	1	4	yes	—
50	C₂²	—	2	6	no	—
51	C₂²	C₂³×C₃²	1	6	yes	—
52	C₂²	C₂³×C₃²	1	1	yes	characteristic
53	C₄	C₂³×C₃²	1	2	yes	—
54	C₄	C₂³×C₃²	1	6	yes	—
55	C₃	C₂×C₃×#41	1	4	yes	direct product factor
56	C₂	—	2	6	no	—
57	C₂	C₃²×#41	1	2	yes	direct product factor
58	C₂	C₂⁴×C₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₃²
socle	C₂²×C₃²	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	direct (N × H)
2. C₂×C₃²×Q₈	C₂	semidirect
3. C₂×C₃²×D₄	C₂	semidirect
4. C₂²×C₃²×C₄	C₂	semidirect
5. C₂×C₃×#41	C₃	direct (N × H)
7. C₃²×Q₈	C₂²	semidirect
8. C₃²×D₄	C₂²	semidirect
11. C₂×C₃²×C₄	C₂²	semidirect
12. C₃×#41	C₂×C₃	direct (N × H)
13. C₂×C₃×Q₈	C₂×C₃	semidirect
14. C₂×C₃×D₄	C₂×C₃	semidirect
15. C₂²×C₃×C₄	C₂×C₃	semidirect
21. C₂×#41	C₃²	direct (N × H)
23. C₃×Q₈	C₂²×C₃	semidirect
24. C₃×D₄	C₂²×C₃	semidirect
27. C₂×C₃×C₄	C₂²×C₃	semidirect
29. C₂×C₃²	#41	direct (N × H)
31. #41	C₂×C₃²	direct (N × H)
32. C₂×Q₈	C₂×C₃²	semidirect
33. C₂×D₄	C₂×C₃²	semidirect
34. C₂²×C₄	C₂×C₃²	semidirect
40. C₃²	C₂×#41	direct (N × H)
42. Q₈	C₂²×C₃²	semidirect
43. D₄	C₂²×C₃²	semidirect
46. C₂×C₄	C₂²×C₃²	semidirect
48. C₂×C₃	C₃×#41	direct (N × H)
55. C₃	C₂×C₃×#41	direct (N × H)
57. C₂	C₃²×#41	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#41	2
C₂×C₃²×Q₈	2
C₂×C₃²×D₄	2
C₂²×C₃²×C₄	2
C₂×C₃×#41	3

Subgroups — C2×C32×#41 #64373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#41 #64373