Subgroups — #63557

Order 288.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	(C₂×D₃)⋊C₈	—	3	2	no	maximal
4	C₂²×C₃⋊D₃	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₄×C₃⋊D₃	—	2	2	no	—
8	C₃×C₃⋊C₈	—	2	2	no	—
9	C₂×C₄×D₃	—	3	2	no	—
10	C₂×C₄×D₃	—	6	1	no	—
11	C₂×C₃×C₈	—	3	2	no	—
12	C₂×C₃⋊C₈	D₃	1	2	yes	complemented
13	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃⋊D₃	—	2	1	no	—
15	C₂×C₃⋊D₃	—	2	1	no	—
16	C₂×C₃⋊D₃	C₈	1	2	yes	complemented
17	C₃²×C₄	D₄	1	2	yes	—
18	C₃⋊Dic₃	—	2	1	no	—
19	C₂²⋊C₈	—	9	1	no	Sylow
20	C₂²×D₃	—	3	2	no	—
21	C₂²×D₃	—	6	1	no	—
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
24	C₂×Dic₃	—	3	2	no	—
25	C₂×Dic₃	—	6	1	no	—
26	C₄×D₃	—	6	4	no	—
27	C₄×D₃	—	6	2	no	—
28	C₄×D₃	—	6	2	no	—
29	C₃×C₈	—	6	2	no	—
30	C₃⋊C₈	—	2	2	no	—
31	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
33	C₂×C₃²	M₁₆	1	1	yes	characteristic
34	C₃⋊D₃	—	2	2	no	—
35	C₂²×C₄	—	9	1	no	—
36	C₂×C₈	—	3	2	no	—
37	C₂²×C₃	—	2	1	no	—
38	C₂²×C₃	C₄×D₃	1	2	yes	—
39	C₂×D₃	—	3	4	no	—
40	C₂×D₃	—	6	2	no	—
41	C₂×D₃	—	6	2	no	—
42	C₂×D₃	—	6	2	no	—
43	C₂×D₃	—	6	2	no	—
44	C₂×D₃	—	6	2	no	—
45	C₃×C₄	—	2	2	no	—
46	C₃×C₄	D₁₂	1	2	yes	—
47	C₃×C₄	#67	1	2	yes	—
48	Dic₃	—	6	2	no	—
49	Dic₃	—	6	2	no	—
50	C₃²	C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
51	C₂³	—	9	1	no	—
52	C₂×C₄	—	9	1	no	—
53	C₂×C₄	—	18	2	no	—
54	C₂×C₄	D₃²	1	1	yes	characteristic
55	C₈	—	6	2	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	D₃×C₈	1	2	yes	—
60	C₂×C₃	ΓL(1,𝔽₂₅)	1	2	yes	—
61	C₂×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
62	D₃	—	6	4	no	—
63	D₃	—	6	4	no	—
64	C₂²	—	9	2	no	—
65	C₂²	—	18	1	no	—
66	C₂²	—	18	1	no	—
67	C₂²	#624	1	1	yes	characteristic
68	C₄	—	18	1	no	—
69	C₄	#626	1	2	yes	—
70	C₃	—	2	1	no	—
71	C₃	(C₂×D₃)⋊C₈	1	2	yes	complemented
72	C₂	—	18	2	no	—
73	C₂	#3710	1	1	yes	characteristic
74	C₂	#3712	1	1	yes	characteristic
75	C₂	#3722	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃²
commutator	C₂×C₃²	C₂×C₈
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂²×C₃²	C₂×C₄
prime radical	C₂²×C₃⋊D₃	C₄
hypocenter	C₃²	C₂²⋊C₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊C₈	C₂	semidirect
12. C₂×C₃⋊C₈	D₃	semidirect
16. C₂×C₃⋊D₃	C₈	semidirect
50. C₃²	C₂²⋊C₈	semidirect
71. C₃	(C₂×D₃)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊D₃	2
C₂×C₃×C₃⋊C₈	2
(C₂×D₃)⋊C₈	3