Subgroups — C₃²×#120 #64169

Order 288.

390 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	C₂×C₃²×D₄	C₂	1	3	yes	maximal, complemented
3	C₃²×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₃×#120	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	—	2	6	no	—
6	C₂³×C₃²	C₂²	1	3	yes	—
7	C₂³×C₃²	C₂²	1	1	yes	characteristic, complemented
8	C₃²×D₄	—	2	6	no	—
9	C₂×C₃²×C₄	C₂²	1	3	yes	complemented
10	C₂⁴×C₃	C₂×C₃	1	4	yes	complemented
11	C₂×C₃×D₄	C₂×C₃	1	12	yes	complemented
12	C₃×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
13	C₂²×C₃²	—	2	8	no	—
14	C₂²×C₃²	—	2	6	no	—
15	C₂²×C₃²	—	2	3	no	—
16	C₂²×C₃²	D₄	1	6	yes	complemented
17	C₂²×C₃²	C₂³	1	1	yes	characteristic
18	C₃²×C₄	—	2	3	no	—
19	#120	C₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂³×C₃	—	2	24	no	—
21	C₂³×C₃	C₂²×C₃	1	12	yes	—
22	C₂³×C₃	C₂²×C₃	1	4	yes	complemented
23	C₃×D₄	—	2	24	no	—
24	C₂×C₃×C₄	C₂²×C₃	1	12	yes	complemented
25	C₂×C₃²	—	2	6	no	—
26	C₂×C₃²	—	4	1	no	—
27	C₂×C₃²	C₂×D₄	1	3	yes	—
28	C₂⁴	C₂×C₃²	1	1	yes	characteristic, complemented
29	C₂×D₄	C₂×C₃²	1	3	yes	complemented
30	C₂²⋊C₄	C₂×C₃²	1	3	yes	complemented
31	C₂²×C₃	—	2	32	no	—
32	C₂²×C₃	—	2	24	no	—
33	C₂²×C₃	—	2	12	no	—
34	C₂²×C₃	C₃×D₄	1	24	yes	complemented
35	C₂²×C₃	C₂³×C₃	1	4	yes	—
36	C₃×C₄	—	2	12	no	—
37	C₃²	#120	1	1	yes	characteristic, direct product factor, Sylow
38	C₂³	—	2	6	no	—
39	C₂³	C₂²×C₃²	1	3	yes	—
40	C₂³	C₂²×C₃²	1	1	yes	characteristic, complemented
41	D₄	—	2	6	no	—
42	C₂×C₄	C₂²×C₃²	1	3	yes	complemented
43	C₂×C₃	—	2	24	no	—
44	C₂×C₃	—	4	4	no	—
45	C₂×C₃	C₂×C₃×D₄	1	12	yes	—
46	C₂²	—	2	8	no	—
47	C₂²	—	2	6	no	—
48	C₂²	—	2	3	no	—
49	C₂²	C₃²×D₄	1	6	yes	complemented
50	C₂²	C₂³×C₃²	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₃	C₃×#120	1	4	yes	direct product factor
53	C₂	—	2	6	no	—
54	C₂	—	4	1	no	—
55	C₂	C₂×C₃²×D₄	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²	C₂	semidirect
2. C₂×C₃²×D₄	C₂	semidirect
3. C₃²×C₂²⋊C₄	C₂	semidirect
4. C₃×#120	C₃	direct (N × H)
7. C₂³×C₃²	C₂²	semidirect
9. C₂×C₃²×C₄	C₂²	semidirect
10. C₂⁴×C₃	C₂×C₃	semidirect
11. C₂×C₃×D₄	C₂×C₃	semidirect
12. C₃×C₂²⋊C₄	C₂×C₃	semidirect
16. C₂²×C₃²	D₄	semidirect
19. #120	C₃²	direct (N × H)
22. C₂³×C₃	C₂²×C₃	semidirect
24. C₂×C₃×C₄	C₂²×C₃	semidirect
28. C₂⁴	C₂×C₃²	semidirect
29. C₂×D₄	C₂×C₃²	semidirect
30. C₂²⋊C₄	C₂×C₃²	semidirect
34. C₂²×C₃	C₃×D₄	semidirect
37. C₃²	#120	direct (N × H)
40. C₂³	C₂²×C₃²	semidirect
42. C₂×C₄	C₂²×C₃²	semidirect
49. C₂²	C₃²×D₄	semidirect
52. C₃	C₃×#120	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
C₂×C₃²×D₄	2
C₃²×C₂²⋊C₄	2
C₃×#120	3

Subgroups — C32×#120 #64169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#120 #64169