Subgroups — #63567

Order 288.

103 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#634	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#816	—	3	1	no	maximal
5	#810	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	—	2	1	no	—
8	#634	—	2	1	no	—
9	#634	C₄	1	2	yes	complemented
10	C₂×C₃×Dic₃	—	2	1	no	—
11	C₃×C₃⋊C₈	—	2	1	no	—
12	C₂×Dic₆	—	3	1	no	—
13	C₂×Dic₆	—	3	1	no	—
14	C₂×Dic₆	—	6	1	no	—
15	C₂×C₃×C₈	—	3	1	no	—
16	C₃×C₄⋊C₄	—	3	1	no	—
17	C₄⋊Dic₃	D₃	1	1	yes	characteristic
18	C₂×C₃⋊C₈	D₃	1	1	yes	characteristic
19	C₂²×C₃²	D₄	1	1	yes	characteristic
20	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₃²×C₄	D₄	1	1	yes	characteristic
22	C₃⋊Dic₃	—	2	2	no	—
23	C₃×Dic₃	—	4	1	no	—
24	#103	—	9	1	no	Sylow
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	—	6	1	no	—
27	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₃	—	2	1	no	—
30	C₂×Dic₃	—	6	2	no	—
31	C₂×Dic₃	—	6	1	no	—
32	C₂×Dic₃	—	6	1	no	—
33	Dic₆	—	3	2	no	—
34	Dic₆	—	3	2	no	—
35	Dic₆	—	6	2	no	—
36	Dic₆	—	6	2	no	—
37	Dic₆	—	6	1	no	—
38	Dic₆	—	6	1	no	—
39	C₃×C₈	—	6	1	no	—
40	C₃⋊C₈	—	2	1	no	—
41	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₃²	SD₁₆	1	1	yes	characteristic
43	C₂×C₃²	Q₁₆	1	1	yes	characteristic
44	C₂×Q₈	—	9	1	no	—
45	C₂×C₈	—	3	1	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	D₁₂	1	1	yes	characteristic
49	C₂²×C₃	#67	1	1	yes	characteristic
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	—	12	1	no	—
53	C₃×C₄	C₄×D₃	1	1	yes	characteristic
54	C₃×C₄	C₄×D₃	1	1	yes	characteristic
55	C₃×C₄	D₁₂	1	1	yes	characteristic
56	C₃×C₄	#67	1	1	yes	characteristic
57	Dic₃	—	4	1	no	—
58	Dic₃	—	6	4	no	—
59	Dic₃	—	6	2	no	—
60	Dic₃	—	6	2	no	—
61	C₃²	#103	1	1	yes	characteristic, complemented, Sylow
62	Q₈	—	9	2	no	—
63	Q₈	—	18	1	no	—
64	C₂×C₄	—	6	1	no	—
65	C₂×C₄	—	18	1	no	—
66	C₂×C₄	D₃²	1	1	yes	characteristic
67	C₈	—	6	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	—	2	1	no	—
71	C₂×C₃	#204	1	1	yes	characteristic
72	C₂×C₃	Dic₁₂	1	1	yes	characteristic
73	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
74	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
75	C₂×C₃	#214	1	1	yes	characteristic
76	C₂×C₃	#216	1	1	yes	characteristic
77	C₂²	#626	1	1	yes	characteristic
78	C₄	—	12	1	no	—
79	C₄	—	18	2	no	—
80	C₄	#624	1	1	yes	characteristic
81	C₄	#626	1	1	yes	characteristic
82	C₃	—	2	1	no	—
83	C₃	#810	1	1	yes	characteristic, complemented
84	C₃	#816	1	1	yes	characteristic, complemented
85	C₂	#3716	1	1	yes	characteristic
86	C₂	#3719	1	1	yes	characteristic
87	C₂	#3722	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#626
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	D₄
hypocenter	C₃²	#103
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
9. #634	C₄	semidirect
61. C₃²	#103	semidirect
83. C₃	#810	semidirect
84. C₃	#816	semidirect

Maximal subgroups

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

subgroup	index
C₂×#634	2
C₃×C₄⋊Dic₃	2
C₂×C₃×C₃⋊C₈	2
#816	3
#810	3