Subgroups — C₂²×#634 #64355

Order 288.

468 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×#634	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₆	—	3	4	no	maximal
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₂²	1	6	yes	—
7	C₂×C₃⋊Dic₃	C₂²	1	12	yes	—
8	#634	C₂²	1	16	yes	direct product factor
9	C₂²×C₃×C₄	D₃	1	4	yes	—
10	C₂²×Dic₃	—	3	8	no	—
11	C₂×Dic₆	—	3	48	no	—
12	C₂²×C₃²	Q₈	1	4	yes	complemented
13	C₂²×C₃²	C₂³	1	3	yes	—
14	C₃²×C₄	C₂³	1	4	yes	—
15	C₃⋊Dic₃	C₂³	1	8	yes	—
16	C₂²×Q₈	—	9	1	no	Sylow
17	C₂³×C₃	C₂×D₃	1	4	yes	—
18	C₂×C₃×C₄	C₂×D₃	1	24	yes	—
19	C₂×Dic₃	—	3	48	no	—
20	Dic₆	—	3	64	no	—
21	C₂×C₃²	C₂×Q₈	1	6	yes	complemented
22	C₂×C₃²	C₂⁴	1	1	yes	characteristic
23	C₂×Q₈	—	9	12	no	—
24	C₂²×C₄	—	9	2	no	—
25	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
26	C₂²×C₃	Dic₆	1	16	yes	complemented
27	C₂²×C₃	C₂²×D₃	1	12	yes	—
28	C₃×C₄	C₂²×D₃	1	16	yes	—
29	Dic₃	—	3	32	no	—
30	C₃²	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
31	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
32	Q₈	—	9	16	no	—
33	C₂×C₄	—	9	12	no	—
34	C₂×C₄	C₂×C₃⋊D₃	1	6	yes	—
35	C₂×C₃	C₂×Dic₆	1	24	yes	complemented
36	C₂×C₃	C₂³×D₃	1	4	yes	—
37	C₂²	#634	1	4	yes	direct product factor
38	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
39	C₄	—	9	8	no	—
40	C₄	C₂²×C₃⋊D₃	1	4	yes	—
41	C₃	C₂²×Dic₆	1	4	yes	complemented
42	C₂	C₂×#634	1	6	yes	direct product factor
43	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#634	C₂	direct (N × H)
8. #634	C₂²	direct (N × H)
12. C₂²×C₃²	Q₈	semidirect
21. C₂×C₃²	C₂×Q₈	semidirect
26. C₂²×C₃	Dic₆	semidirect
30. C₃²	C₂²×Q₈	semidirect
35. C₂×C₃	C₂×Dic₆	semidirect
37. C₂²	#634	direct (N × H)
41. C₃	C₂²×Dic₆	semidirect
42. C₂	C₂×#634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
C₂²×C₃⋊Dic₃	2
C₂×#634	2
C₂²×Dic₆	3

Subgroups — C22×#634 #64355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#634 #64355