Subgroups — C₃²×#121 #64170

Order 288.

282 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#121	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²	C₂²	1	2	yes	—
8	C₂³×C₃²	C₂²	1	1	yes	characteristic
9	C₃²×D₄	—	2	4	no	—
10	C₃²×D₄	—	2	2	no	—
11	C₂×C₃²×C₄	—	2	2	no	—
12	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₄	C₂²	1	2	yes	—
14	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₄	C₂×C₃	1	8	yes	complemented
16	C₂×C₃×D₄	C₂×C₃	1	4	yes	complemented
17	C₂²×C₃×C₄	C₂×C₃	1	4	yes	complemented
18	C₃×C₄⋊C₄	C₂×C₃	1	4	yes	complemented
19	C₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
20	C₂²×C₃²	—	2	1	no	—
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	—	2	2	no	—
23	C₂²×C₃²	—	2	2	no	—
24	C₂²×C₃²	—	2	2	no	—
25	C₂²×C₃²	D₄	1	2	yes	complemented
26	C₂²×C₃²	C₂³	1	1	yes	characteristic
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	—	2	2	no	—
29	C₃²×C₄	D₄	1	2	yes	complemented
30	#121	C₃²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂³×C₃	C₂²×C₃	1	8	yes	—
32	C₂³×C₃	C₂²×C₃	1	4	yes	—
33	C₃×D₄	—	2	16	no	—
34	C₃×D₄	—	2	8	no	—
35	C₂×C₃×C₄	—	2	8	no	—
36	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
37	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
38	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
39	C₂×C₃²	—	2	2	no	—
40	C₂×C₃²	—	4	2	no	—
41	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃²	#41	1	1	yes	characteristic
44	C₂×D₄	C₂×C₃²	1	2	yes	complemented
45	C₂×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
46	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
47	C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	—	2	8	no	—
52	C₂²×C₃	—	2	8	no	—
53	C₂²×C₃	—	2	8	no	—
54	C₂²×C₃	C₃×D₄	1	8	yes	complemented
55	C₂²×C₃	C₂³×C₃	1	4	yes	—
56	C₃×C₄	—	2	4	no	—
57	C₃×C₄	—	2	8	no	—
58	C₃×C₄	C₃×D₄	1	8	yes	complemented
59	C₃²	#121	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³	C₂²×C₃²	1	2	yes	—
61	C₂³	C₂²×C₃²	1	1	yes	characteristic
62	D₄	—	2	4	no	—
63	D₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₃²	1	2	yes	—
67	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
68	C₂×C₃	—	2	8	no	—
69	C₂×C₃	—	4	8	no	—
70	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
71	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
72	C₂×C₃	C₃×#41	1	4	yes	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₃²×D₄	1	2	yes	complemented
79	C₂²	C₂³×C₃²	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	C₃²×D₄	1	2	yes	complemented
83	C₃	C₃×#121	1	4	yes	direct product factor
84	C₂	—	2	2	no	—
85	C₂	—	4	2	no	—
86	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
87	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
88	C₂	C₃²×#41	1	1	yes	characteristic

	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. C₂×C₃²×D₄	C₂	semidirect
2. C₂×C₃²×D₄	C₂	semidirect
3. C₂²×C₃²×C₄	C₂	semidirect
4. C₃²×C₄⋊C₄	C₂	semidirect
5. C₃²×C₂²⋊C₄	C₂	semidirect
6. C₃×#121	C₃	direct (N × H)
15. C₂×C₃×D₄	C₂×C₃	semidirect
16. C₂×C₃×D₄	C₂×C₃	semidirect
17. C₂²×C₃×C₄	C₂×C₃	semidirect
18. C₃×C₄⋊C₄	C₂×C₃	semidirect
19. C₃×C₂²⋊C₄	C₂×C₃	semidirect
25. C₂²×C₃²	D₄	semidirect
29. C₃²×C₄	D₄	semidirect
30. #121	C₃²	direct (N × H)
44. C₂×D₄	C₂×C₃²	semidirect
45. C₂×D₄	C₂×C₃²	semidirect
46. C₂²×C₄	C₂×C₃²	semidirect
47. C₄⋊C₄	C₂×C₃²	semidirect
48. C₂²⋊C₄	C₂×C₃²	semidirect
54. C₂²×C₃	C₃×D₄	semidirect
58. C₃×C₄	C₃×D₄	semidirect
59. C₃²	#121	direct (N × H)
78. C₂²	C₃²×D₄	semidirect
82. C₄	C₃²×D₄	semidirect
83. C₃	C₃×#121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₄	2
C₂²×C₃²×C₄	2
C₃²×C₄⋊C₄	2
C₃²×C₂²⋊C₄	2
C₃×#121	3

Subgroups — C32×#121 #64170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#121 #64170