Subgroups — C₃²×#135 #64184

Order 288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	2	yes	maximal, complemented
2	C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#135	C₃	1	4	yes	maximal, direct product factor
7	C₃²×Q₈	C₂²	1	2	yes	—
8	C₃²×D₄	—	2	2	no	—
9	C₃²×D₄	C₂²	1	2	yes	—
10	C₂×C₃²×C₄	—	2	2	no	—
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×#41	C₂×C₃	1	8	yes	complemented
15	C₃×Q₁₆	C₂×C₃	1	4	yes	complemented
16	C₃×SD₁₆	C₂×C₃	1	8	yes	complemented
17	C₃×D₈	C₂×C₃	1	4	yes	complemented
18	C₂×C₃×C₈	C₂×C₃	1	4	yes	complemented
19	C₂²×C₃²	—	2	2	no	—
20	C₂²×C₃²	D₄	1	1	yes	characteristic
21	C₃²×C₄	—	2	2	no	—
22	C₃²×C₄	D₄	1	1	yes	characteristic
23	C₃²×C₄	C₂³	1	1	yes	characteristic
24	#135	C₃²	1	1	yes	characteristic, direct product factor, Sylow
25	C₃×Q₈	C₂²×C₃	1	8	yes	—
26	C₃×D₄	—	2	8	no	—
27	C₃×D₄	C₂²×C₃	1	8	yes	—
28	C₂×C₃×C₄	—	2	8	no	—
29	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
30	C₃×C₈	C₂²×C₃	1	4	yes	—
31	C₃×C₈	C₂²×C₃	1	4	yes	—
32	C₂×C₃²	—	2	1	no	—
33	C₂×C₃²	—	4	2	no	—
34	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
35	#41	C₂×C₃²	1	2	yes	complemented
36	Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
37	SD₁₆	C₂×C₃²	1	2	yes	complemented
38	D₈	C₂×C₃²	1	1	yes	characteristic, complemented
39	C₂×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	2	8	no	—
41	C₂²×C₃	C₃×D₄	1	4	yes	—
42	C₃×C₄	—	2	8	no	—
43	C₃×C₄	C₃×D₄	1	4	yes	—
44	C₃×C₄	C₂³×C₃	1	4	yes	—
45	C₃²	#135	1	1	yes	characteristic, direct product factor, Sylow
46	Q₈	C₂²×C₃²	1	2	yes	—
47	D₄	—	2	2	no	—
48	D₄	C₂²×C₃²	1	2	yes	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
51	C₈	C₂²×C₃²	1	1	yes	characteristic
52	C₈	C₂²×C₃²	1	1	yes	characteristic
53	C₂×C₃	—	2	4	no	—
54	C₂×C₃	—	4	8	no	—
55	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
56	C₂²	—	2	2	no	—
57	C₂²	C₃²×D₄	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	C₃²×D₄	1	1	yes	characteristic
60	C₄	C₂³×C₃²	1	1	yes	characteristic
61	C₃	C₃×#135	1	4	yes	direct product factor
62	C₂	—	2	1	no	—
63	C₂	—	4	2	no	—
64	C₂	C₂×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃²
socle	C₂×C₃²	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	semidirect
2. C₃²×Q₁₆	C₂	semidirect
3. C₃²×SD₁₆	C₂	semidirect
4. C₃²×D₈	C₂	semidirect
5. C₂×C₃²×C₈	C₂	semidirect
6. C₃×#135	C₃	direct (N × H)
14. C₃×#41	C₂×C₃	semidirect
15. C₃×Q₁₆	C₂×C₃	semidirect
16. C₃×SD₁₆	C₂×C₃	semidirect
17. C₃×D₈	C₂×C₃	semidirect
18. C₂×C₃×C₈	C₂×C₃	semidirect
24. #135	C₃²	direct (N × H)
35. #41	C₂×C₃²	semidirect
36. Q₁₆	C₂×C₃²	semidirect
37. SD₁₆	C₂×C₃²	semidirect
38. D₈	C₂×C₃²	semidirect
39. C₂×C₈	C₂×C₃²	semidirect
45. C₃²	#135	direct (N × H)
61. C₃	C₃×#135	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
C₃²×Q₁₆	2
C₃²×SD₁₆	2
C₃²×D₈	2
C₂×C₃²×C₈	2
C₃×#135	3

Subgroups — C32×#135 #64184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#135 #64184