Subgroups — #63570

Order 288.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3828	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	#825	—	3	1	no	maximal
5	#805	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	#638	—	2	1	no	—
8	C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic, complemented
9	C₄×C₃⋊D₃	—	2	1	no	—
10	#634	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃×Dic₃	—	2	1	no	—
12	C₃×C₃⋊C₈	—	2	1	no	—
13	#235	—	3	1	no	—
14	#235	—	3	1	no	—
15	#235	—	6	1	no	—
16	C₃×M₁₆	—	3	1	no	—
17	C₃×C₄²	—	3	1	no	—
18	C₄×Dic₃	D₃	1	1	yes	characteristic, complemented
19	#208	D₃	1	1	yes	characteristic, complemented
20	C₂²×C₃²	D₄	1	1	yes	characteristic
21	C₂×C₃⋊D₃	—	2	1	no	—
22	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃²×C₄	D₄	1	1	yes	characteristic
24	C₃⋊Dic₃	—	2	1	no	—
25	C₃×Dic₃	—	2	2	no	—
26	#104	—	9	1	no	Sylow
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	6	1	no	—
29	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
31	#67	—	6	2	no	—
32	#67	—	6	1	no	—
33	#67	—	6	1	no	—
34	C₂×Dic₃	—	2	1	no	—
35	D₁₂	—	3	1	no	—
36	D₁₂	—	3	1	no	—
37	D₁₂	—	6	1	no	—
38	C₄×D₃	—	6	2	no	—
39	C₄×D₃	—	6	1	no	—
40	C₄×D₃	—	6	1	no	—
41	Dic₆	—	3	1	no	—
42	Dic₆	—	3	1	no	—
43	Dic₆	—	6	1	no	—
44	C₃×C₈	—	6	1	no	—
45	C₃⋊C₈	—	2	1	no	—
46	C₂×C₃²	—	2	1	no	—
47	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
48	C₃⋊D₃	—	4	1	no	—
49	#41	—	9	1	no	—
50	M₁₆	—	3	1	no	—
51	C₄²	—	3	1	no	—
52	C₂²×C₃	—	2	1	no	—
53	C₂²×C₃	D₁₂	1	1	yes	characteristic
54	C₂²×C₃	#67	1	1	yes	characteristic
55	C₂×D₃	—	6	2	no	—
56	C₂×D₃	—	6	1	no	—
57	C₂×D₃	—	6	1	no	—
58	C₃×C₄	—	2	1	no	—
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	—	6	2	no	—
61	C₃×C₄	C₄×D₃	1	1	yes	characteristic
62	C₃×C₄	C₄×D₃	1	1	yes	characteristic
63	C₃×C₄	D₁₂	1	1	yes	characteristic
64	C₃×C₄	#67	1	1	yes	characteristic
65	Dic₃	—	2	2	no	—
66	Dic₃	—	6	2	no	—
67	Dic₃	—	6	1	no	—
68	Dic₃	—	6	1	no	—
69	C₃²	#104	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	9	1	no	—
71	D₄	—	9	1	no	—
72	D₄	—	18	1	no	—
73	C₂×C₄	—	6	1	no	—
74	C₂×C₄	—	18	1	no	—
75	C₂×C₄	D₃²	1	1	yes	characteristic
76	C₈	—	6	1	no	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	—	4	1	no	—
81	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
82	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
83	D₃	—	12	2	no	—
84	D₃	—	12	1	no	—
85	D₃	—	12	1	no	—
86	C₂²	—	18	1	no	—
87	C₂²	#626	1	1	yes	characteristic
88	C₄	—	6	2	no	—
89	C₄	—	18	1	no	—
90	C₄	#624	1	1	yes	characteristic
91	C₄	#626	1	1	yes	characteristic
92	C₃	—	2	1	no	—
93	C₃	#805	1	1	yes	characteristic, complemented
94	C₃	#825	1	1	yes	characteristic, complemented
95	C₂	—	2	1	no	—
96	C₂	—	36	1	no	—
97	C₂	#3722	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#626
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂×C₃²	C₂²⋊C₄
prime radical	#3828	C₂
hypocenter	C₃²	#104
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Dic₃	C₂	semidirect
3. C₃×#208	C₂	semidirect
8. C₄⋊(C₃⋊D₃)	C₄	semidirect
10. #634	C₄	semidirect
18. C₄×Dic₃	D₃	semidirect
19. #208	D₃	semidirect
69. C₃²	#104	semidirect
93. C₃	#805	semidirect
94. C₃	#825	semidirect

Maximal subgroups

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

subgroup	index
#3828	2
C₃×C₄×Dic₃	2
C₃×#208	2
#825	3
#805	3