Subgroups — #64259

Order 288.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×A₄	C₂	1	1	yes	maximal, characteristic
2	C₃⋊SL(2,ℤ/4)	C₂	1	2	yes	maximal
3	A₄⋊Q₈	—	3	3	no	maximal
4	#924	—	3	1	no	maximal
5	C₂×C₃×A₄	C₂²	1	1	yes	characteristic
6	#634	—	4	1	no	maximal
7	C₂²×C₃×C₄	D₃	1	1	yes	characteristic
8	C₂×Dic₆	—	3	1	no	—
9	C₄×A₄	D₃	1	3	yes	—
10	SL(2,ℤ/4)	—	3	6	no	—
11	#217	—	3	2	no	—
12	C₄⋊Dic₃	—	3	1	no	—
13	Dic₃⋊C₄	—	3	2	no	—
14	C₃×A₄	Q₈	1	1	yes	characteristic, complemented
15	C₃²×C₄	—	4	1	no	—
16	C₃⋊Dic₃	—	4	2	no	—
17	C₂²⋊Q₈	—	9	1	no	Sylow
18	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
19	C₂×A₄	C₂×D₃	1	3	yes	—
20	C₂×C₃×C₄	—	3	1	no	—
21	C₂×C₃×C₄	—	3	1	no	—
22	C₂×Dic₃	—	3	2	no	—
23	C₂×Dic₃	—	3	2	no	—
24	Dic₆	—	6	1	no	—
25	Dic₆	—	6	1	no	—
26	Dic₆	—	12	3	no	—
27	C₂×C₃²	—	4	1	no	—
28	C₂×Q₈	—	9	1	no	—
29	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
30	C₄⋊C₄	—	9	2	no	—
31	C₄⋊C₄	—	9	1	no	—
32	C₂²⋊C₄	—	9	2	no	—
33	C₂²×C₃	—	3	1	no	—
34	C₂²×C₃	—	3	1	no	—
35	C₂²×C₃	Dic₆	1	1	yes	characteristic, complemented
36	A₄	Dic₆	1	3	yes	complemented
37	C₃×C₄	—	3	1	no	—
38	C₃×C₄	—	4	3	no	—
39	C₃×C₄	S₄	1	1	yes	characteristic
40	Dic₃	—	6	2	no	—
41	Dic₃	—	6	2	no	—
42	Dic₃	—	12	6	no	—
43	C₃²	—	4	1	no	Sylow
44	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
45	Q₈	—	18	1	no	—
46	Q₈	—	18	1	no	—
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	3	1	no	—
49	C₂×C₄	—	9	2	no	—
50	C₂×C₄	—	9	2	no	—
51	C₂×C₃	—	3	1	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	4	3	no	—
54	C₂×C₃	C₂×S₄	1	1	yes	characteristic
55	C₂²	—	3	1	no	—
56	C₂²	—	3	1	no	—
57	C₂²	#634	1	1	yes	characteristic, complemented
58	C₄	—	3	1	no	—
59	C₄	—	18	2	no	—
60	C₄	—	18	2	no	—
61	C₄	C₃⋊S₄	1	1	yes	characteristic
62	C₃	—	4	3	no	—
63	C₃	A₄⋊Q₈	1	1	yes	characteristic, complemented
64	C₂	—	3	1	no	—
65	C₂	—	3	1	no	—
66	C₂	C₂×C₃⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊S₄
commutator, prime radical, cosocle	C₂×C₃×A₄	C₂²
Fitting	C₂²×C₃×C₄	D₃
socle	C₂³×C₃	C₂×D₃
hypercenter	C₄	C₃⋊S₄
hypocenter	C₃×A₄	Q₈
LCS	G ▷ C₂×C₃×A₄ ▷ C₃×A₄
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
14. C₃×A₄	Q₈	semidirect
35. C₂²×C₃	Dic₆	semidirect
36. A₄	Dic₆	semidirect
57. C₂²	#634	semidirect
63. C₃	A₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×A₄	2
C₃⋊SL(2,ℤ/4)	2
A₄⋊Q₈	3
#924	3
#634	4