Subgroups — #63739

Order 288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#625	C₂	1	1	yes	maximal, characteristic
2	C₂×#622	C₂	1	1	yes	maximal, characteristic, complemented
3	#3724	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×D₃	—	2	1	no	—
5	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×Dic₃	—	2	1	no	—
7	#625	—	2	1	no	—
8	#625	C₄	1	2	yes	complemented
9	#622	—	2	1	no	—
10	C₂×#67	—	6	1	no	—
11	Dic₃⋊C₄	—	6	1	no	—
12	C₂²×C₃²	D₄	1	1	yes	characteristic
13	C₂×C₃×D₃	—	2	2	no	—
14	C₂×C₃×D₃	—	4	1	no	—
15	C₂×C₃×D₃	—	4	1	no	—
16	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
17	C₃⋊Dic₃	D₄	1	1	yes	characteristic
18	C₃×Dic₃	—	4	1	no	—
19	#102	—	9	1	no	maximal, Sylow
20	C₂³×C₃	—	6	1	no	—
21	C₂²×D₃	—	2	1	no	—
22	C₂×C₃×C₄	—	6	1	no	—
23	#67	—	6	2	no	—
24	#67	—	6	2	no	—
25	C₂×Dic₃	—	2	1	no	—
26	C₂×Dic₃	—	6	1	no	—
27	C₂×Dic₃	—	6	1	no	—
28	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃²	D₈	1	1	yes	characteristic
30	C₂×C₃²	SD₁₆	1	1	yes	characteristic
31	C₃×D₃	—	4	2	no	—
32	C₂×D₄	—	9	1	no	—
33	C₂×C₈	—	9	1	no	—
34	C₄⋊C₄	—	9	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	—	2	1	no	—
37	C₂²×C₃	—	6	2	no	—
38	C₂²×C₃	—	12	1	no	—
39	C₂²×C₃	—	12	1	no	—
40	C₂×D₃	—	2	2	no	—
41	C₂×D₃	—	4	1	no	—
42	C₂×D₃	—	4	1	no	—
43	C₃×C₄	—	12	1	no	—
44	Dic₃	—	4	1	no	—
45	Dic₃	—	6	1	no	—
46	Dic₃	—	6	1	no	—
47	Dic₃	—	6	1	no	—
48	Dic₃	—	6	1	no	—
49	C₃²	#102	1	1	yes	characteristic, complemented, Sylow
50	C₂³	—	6	1	no	—
51	D₄	—	9	2	no	—
52	D₄	—	18	1	no	—
53	C₂×C₄	—	6	1	no	—
54	C₂×C₄	—	9	1	no	—
55	C₈	—	18	1	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	12	2	no	—
63	D₃	—	4	2	no	—
64	C₂²	—	6	2	no	—
65	C₂²	—	12	1	no	—
66	C₂²	—	12	1	no	—
67	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
68	C₄	—	9	1	no	—
69	C₄	—	9	1	no	—
70	C₄	—	12	1	no	—
71	C₃	—	2	1	no	—
72	C₃	—	2	1	no	—
73	C₂	—	12	2	no	—
74	C₂	D₃²⋊C₄	1	1	yes	characteristic
75	C₂	#3774	1	1	yes	characteristic
76	C₂	#3775	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂×C₄
Fitting, socle	C₂²×C₃²	D₄
prime radical	C₂×#625	C₂
cosocle	C₂×C₃⋊Dic₃	C₂²
hypocenter	C₃²	#102
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#622	C₂	semidirect
3. #3724	C₂	semidirect
8. #625	C₄	semidirect
49. C₃²	#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×#625	2
C₂×#622	2
#3724	2
#102	9