Subgroups — C₃²×#137 #64186

Order 288.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Q₁₆	C₂	1	2	yes	maximal, complemented
4	C₃²×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#137	C₃	1	4	yes	maximal, direct product factor
7	C₃²×Q₈	—	2	1	no	—
8	C₃²×Q₈	C₂²	1	2	yes	—
9	C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₃²×D₄	—	2	1	no	—
11	C₃²×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄	—	2	1	no	—
13	C₂×C₃²×C₄	—	2	1	no	—
14	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₃²×C₈	C₂²	1	2	yes	—
16	C₃×#41	C₂×C₃	1	4	yes	—
17	C₂×C₃×Q₈	C₂×C₃	1	4	yes	complemented
18	C₃×Q₁₆	C₂×C₃	1	8	yes	complemented
19	C₃×SD₁₆	C₂×C₃	1	8	yes	complemented
20	C₃×M₁₆	C₂×C₃	1	4	yes	complemented
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	D₄	1	1	yes	characteristic
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	—	2	2	no	—
25	C₃²×C₄	D₄	1	1	yes	characteristic
26	C₃²×C₄	C₂³	1	1	yes	characteristic
27	#137	C₃²	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×Q₈	—	2	4	no	—
29	C₃×Q₈	C₂²×C₃	1	8	yes	—
30	C₃×Q₈	C₂²×C₃	1	4	yes	—
31	C₃×D₄	—	2	4	no	—
32	C₃×D₄	C₂²×C₃	1	4	yes	—
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
36	C₃×C₈	C₂²×C₃	1	8	yes	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	—	4	1	no	—
39	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
40	#41	C₂×C₃²	1	1	yes	characteristic
41	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
42	Q₁₆	C₂×C₃²	1	2	yes	complemented
43	SD₁₆	C₂×C₃²	1	2	yes	complemented
44	M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
45	C₂²×C₃	—	2	4	no	—
46	C₂²×C₃	C₃×D₄	1	4	yes	—
47	C₃×C₄	—	2	4	no	—
48	C₃×C₄	—	2	8	no	—
49	C₃×C₄	C₃×D₄	1	4	yes	—
50	C₃×C₄	C₂³×C₃	1	4	yes	—
51	C₃²	#137	1	1	yes	characteristic, direct product factor, Sylow
52	Q₈	—	2	1	no	—
53	Q₈	C₂²×C₃²	1	2	yes	—
54	Q₈	C₂²×C₃²	1	1	yes	characteristic
55	D₄	—	2	1	no	—
56	D₄	C₂²×C₃²	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
60	C₈	C₂²×C₃²	1	2	yes	—
61	C₂×C₃	—	2	4	no	—
62	C₂×C₃	—	4	4	no	—
63	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	C₃²×D₄	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2	2	no	—
68	C₄	C₃²×D₄	1	1	yes	characteristic
69	C₄	C₂³×C₃²	1	1	yes	characteristic
70	C₃	C₃×#137	1	4	yes	direct product factor
71	C₂	—	2	1	no	—
72	C₂	—	4	1	no	—
73	C₂	C₂×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃²
prime radical	C₃²×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×Q₈	C₂	semidirect
3. C₃²×Q₁₆	C₂	semidirect
4. C₃²×SD₁₆	C₂	semidirect
5. C₃²×M₁₆	C₂	semidirect
6. C₃×#137	C₃	direct (N × H)
17. C₂×C₃×Q₈	C₂×C₃	semidirect
18. C₃×Q₁₆	C₂×C₃	semidirect
19. C₃×SD₁₆	C₂×C₃	semidirect
20. C₃×M₁₆	C₂×C₃	semidirect
27. #137	C₃²	direct (N × H)
41. C₂×Q₈	C₂×C₃²	semidirect
42. Q₁₆	C₂×C₃²	semidirect
43. SD₁₆	C₂×C₃²	semidirect
44. M₁₆	C₂×C₃²	semidirect
51. C₃²	#137	direct (N × H)
70. C₃	C₃×#137	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
C₃²×Q₁₆	2
C₃²×SD₁₆	2
C₃²×M₁₆	2
C₃×#137	3

Subgroups — C32×#137 #64186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#137 #64186