Subgroups — C₂×C₃×#216 #64066

Order 288.

139 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#216	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×Q₁₆	—	3	1	no	maximal
6	C₂×#216	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃²×Q₈	—	2	1	no	—
8	C₃²×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	—	2	1	no	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₃	—	2	1	no	—
12	C₃×Dic₆	—	2	1	no	—
13	C₃×Dic₆	C₂²	1	2	yes	—
14	C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₂×C₃×Q₈	—	3	1	no	—
17	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
18	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×Dic₆	C₂×C₃	1	1	yes	characteristic
20	C₃×Q₁₆	—	3	4	no	—
21	C₂×C₃×C₈	—	3	1	no	—
22	#216	C₂×C₃	1	4	yes	direct product factor
23	C₂×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃²	D₄	1	1	yes	characteristic
25	C₃²×C₄	—	2	2	no	—
26	C₃²×C₄	D₄	1	1	yes	characteristic
27	C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃×Dic₃	—	2	2	no	—
29	C₂×Q₁₆	—	3	1	no	Sylow
30	C₃×Q₈	—	2	2	no	—
31	C₃×Q₈	—	2	1	no	—
32	C₃×Q₈	—	2	1	no	—
33	C₃×Q₈	—	2	2	no	—
34	C₃×Q₈	—	3	2	no	—
35	C₃×Q₈	—	6	1	no	—
36	C₃×Q₈	C₂×D₃	1	2	yes	—
37	C₃×Q₈	C₂²×C₃	1	2	yes	—
38	C₂×C₃×C₄	—	2	2	no	—
39	C₂×C₃×C₄	—	2	1	no	—
40	C₂×C₃×C₄	—	2	1	no	—
41	C₂×C₃×C₄	—	2	1	no	—
42	C₂×C₃×C₄	—	6	1	no	—
43	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂×Dic₃	—	2	1	no	—
46	Dic₆	—	2	1	no	—
47	Dic₆	C₂²×C₃	1	2	yes	—
48	C₃×C₈	—	3	2	no	—
49	C₃⋊C₈	C₂²×C₃	1	2	yes	—
50	C₂×C₃²	Q₁₆	1	2	yes	complemented
51	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈	—	3	1	no	—
53	C₂×Q₈	C₃×D₃	1	1	yes	characteristic
54	Q₁₆	—	3	4	no	—
55	C₂×C₈	—	3	1	no	—
56	C₂²×C₃	—	2	1	no	—
57	C₂²×C₃	#67	1	1	yes	characteristic
58	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
59	C₃×C₄	—	2	4	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	6	2	no	—
65	C₃×C₄	#67	1	1	yes	characteristic
66	C₃×C₄	C₃×D₄	1	1	yes	characteristic
67	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
69	Dic₃	—	2	2	no	—
70	C₃²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	2	1	no	—
72	Q₈	—	3	2	no	—
73	Q₈	—	6	1	no	—
74	Q₈	C₂×C₃×D₃	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	6	1	no	—
77	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
78	C₈	—	3	2	no	—
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	#216	1	2	yes	direct product factor
82	C₂×C₃	C₃×Q₁₆	1	2	yes	complemented
83	C₂×C₃	C₂×#67	1	1	yes	characteristic
84	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
85	C₂²	C₃×#67	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	6	2	no	—
88	C₄	C₃×#67	1	1	yes	characteristic
89	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
90	C₃	—	2	1	no	—
91	C₃	C₂×#216	1	1	yes	characteristic, direct product factor
92	C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic, complemented
93	C₂	C₃×#216	1	2	yes	direct product factor
94	C₂	C₂×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#67
commutator, cosocle	C₃×C₄	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×Q₈	C₂
socle, prime radical	C₂²×C₃²	D₄
hypercenter	C₂×C₃×Q₈	D₃
hypocenter	C₃	C₂×C₃×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
3. C₃×#216	C₂	direct (N × H)
6. C₂×#216	C₃	direct (N × H)
22. #216	C₂×C₃	direct (N × H)
50. C₂×C₃²	Q₁₆	semidirect
70. C₃²	C₂×Q₁₆	semidirect
81. C₂×C₃	#216	direct (N × H)
82. C₂×C₃	C₃×Q₁₆	semidirect
91. C₃	C₂×#216	direct (N × H)
92. C₃	C₂×C₃×Q₁₆	semidirect
93. C₂	C₃×#216	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×Q₈	2
C₂×C₃×Dic₆	2
C₃×#216	2
C₂×C₃×C₃⋊C₈	2
C₂×C₃×Q₁₆	3
C₂×#216	3

Subgroups — C2×C3×#216 #64066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#216 #64066