Subgroups — #63729

Order 288.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3802	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#157	C₂	1	1	yes	maximal, characteristic, complemented
3	#3710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃²	—	2	1	no	—
5	C₂×#157	—	2	1	no	—
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃×D₁₂	—	2	1	no	—
8	#625	C₄	1	2	yes	complemented
9	C₃×C₃⋊C₈	—	2	1	no	—
10	D₃×D₄	—	6	1	no	—
11	D₃×C₈	—	6	1	no	—
12	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
13	C₂×C₃×D₃	—	2	2	no	—
14	D₃²	—	4	1	no	—
15	D₃²	—	4	1	no	—
16	#157	—	4	1	no	—
17	C₃²×C₄	D₄	1	1	yes	characteristic
18	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
19	#102	—	9	1	no	maximal, Sylow
20	C₂²×D₃	—	6	2	no	—
21	C₃×D₄	—	6	1	no	—
22	#67	—	6	2	no	—
23	D₁₂	—	2	1	no	—
24	C₄×D₃	—	6	1	no	—
25	C₄×D₃	—	6	1	no	—
26	C₃×C₈	—	6	1	no	—
27	C₃⋊C₈	—	2	1	no	—
28	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
29	C₃⋊D₃	D₈	1	1	yes	characteristic
30	C₃⋊D₃	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	—	6	2	no	—
36	C₂×D₃	—	2	2	no	—
37	C₂×D₃	—	6	1	no	—
38	C₂×D₃	—	6	1	no	—
39	C₂×D₃	—	12	2	no	—
40	C₂×D₃	—	12	2	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	2	1	no	—
43	Dic₃	—	6	1	no	—
44	Dic₃	—	6	1	no	—
45	C₃²	#102	1	1	yes	characteristic, complemented, Sylow
46	C₂³	—	18	1	no	—
47	D₄	—	6	1	no	—
48	D₄	—	9	2	no	—
49	C₂×C₄	—	9	1	no	—
50	C₂×C₄	—	18	1	no	—
51	C₈	—	6	1	no	—
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	12	2	no	—
55	D₃	—	4	2	no	—
56	D₃	—	6	1	no	—
57	D₃	—	6	1	no	—
58	D₃	—	6	1	no	—
59	D₃	—	6	1	no	—
60	C₂²	—	6	2	no	—
61	C₂²	—	9	1	no	—
62	C₂²	—	36	1	no	—
63	C₂²	—	36	1	no	—
64	C₄	—	9	1	no	—
65	C₄	—	36	1	no	—
66	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
67	C₃	—	2	1	no	—
68	C₃	—	2	1	no	—
69	C₂	—	9	1	no	—
70	C₂	—	9	1	no	—
71	C₂	—	12	2	no	—
72	C₂	D₃²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃²⋊C₄
commutator	C₃⋊Dic₃	C₂×C₄
Frattini, hypercenter	C₄	GO⁺(4,𝔽₂)
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂²⋊C₄
prime radical	#3802	C₂
cosocle	C₄×C₃⋊D₃	C₂²
hypocenter	C₃²	#102
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊#157	C₂	semidirect
3. #3710	C₂	semidirect
8. #625	C₄	semidirect
45. C₃²	#102	semidirect

Maximal subgroups

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

subgroup	index
#3802	2
C₄⋊#157	2
#3710	2
#102	9