Subgroups — #64222

Order 288.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3797	C₂	1	2	yes	maximal
2	C₄×#157	C₂	1	1	yes	maximal, characteristic
3	#3773	C₂	1	4	yes	maximal
4	C₂×#157	C₂²	1	2	yes	—
5	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃×Dic₆	—	2	2	no	—
7	#627	—	2	2	no	—
8	#624	C₂²	1	4	yes	—
9	D₃×Q₈	—	6	2	no	—
10	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
11	#157	Q₈	1	4	yes	complemented
12	C₃²×C₄	D₄	1	1	yes	characteristic
13	C₃⋊Dic₃	D₄	1	1	yes	characteristic
14	C₃×Dic₃	—	2	4	no	—
15	C₄⋊Q₈	—	9	1	no	maximal, Sylow
16	C₃×Q₈	—	6	2	no	—
17	C₄×D₃	—	6	4	no	—
18	C₄×D₃	—	6	2	no	—
19	Dic₆	—	2	2	no	—
20	Dic₆	—	6	4	no	—
21	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
22	C₃⋊D₃	C₂×Q₈	1	2	yes	—
23	C₂×Q₈	—	9	2	no	—
24	C₄⋊C₄	—	9	4	no	—
25	C₄²	—	9	1	no	—
26	C₂×D₃	—	6	2	no	—
27	C₃×C₄	—	2	2	no	—
28	C₃×C₄	—	6	4	no	—
29	Dic₃	—	2	4	no	—
30	Dic₃	—	6	2	no	—
31	C₃²	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈	—	6	2	no	—
33	Q₈	—	18	2	no	—
34	C₂×C₄	—	9	4	no	—
35	C₂×C₄	—	9	2	no	—
36	C₂×C₄	—	9	1	no	—
37	C₂×C₃	—	2	2	no	—
38	D₃	—	6	4	no	—
39	C₂²	—	9	1	no	—
40	C₄	—	6	4	no	—
41	C₄	—	9	4	no	—
42	C₄	—	9	1	no	—
43	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
44	C₃	—	2	2	no	—
45	C₂	—	9	2	no	—
46	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×GO⁺(4,𝔽₂)
commutator, prime radical, cosocle	C₂×C₃⋊D₃	C₂³
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂×D₄
hypercenter	C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	C₄⋊Q₈
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

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

N	H ≅ G/N	type
11. #157	Q₈	semidirect
31. C₃²	C₄⋊Q₈	semidirect

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