Subgroups — #64097

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#634	C₂	1	2	yes	maximal
2	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	#3750	C₂	1	2	yes	maximal
5	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	#888	—	3	4	no	maximal
7	C₂×C₃²×C₄	C₂²	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	#634	—	2	4	no	—
13	C₂×Dic₆	—	3	8	no	—
14	C₃×C₄⋊C₄	D₃	1	4	yes	—
15	C₄⋊Dic₃	—	3	4	no	—
16	Dic₃⋊C₄	—	3	8	no	—
17	C₄×Dic₃	—	3	4	no	—
18	C₂²×C₃²	C₂³	1	1	yes	characteristic
19	C₃²×C₄	—	2	2	no	—
20	C₃²×C₄	Q₈	1	2	yes	complemented
21	C₃⋊Dic₃	—	2	2	no	—
22	C₃⋊Dic₃	D₄	1	2	yes	—
23	C₃⋊Dic₃	Q₈	1	2	yes	complemented
24	C₄⋊Q₈	—	9	1	no	Sylow
25	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
26	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
27	C₂×Dic₃	—	3	4	no	—
28	C₂×Dic₃	—	3	4	no	—
29	C₂×Dic₃	—	3	8	no	—
30	Dic₆	—	6	16	no	—
31	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
34	C₂×Q₈	—	9	2	no	—
35	C₄⋊C₄	—	9	2	no	—
36	C₄⋊C₄	—	9	1	no	—
37	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
38	C₄²	—	9	1	no	—
39	C₂²×C₃	C₂²×D₃	1	4	yes	—
40	C₃×C₄	—	2	8	no	—
41	C₃×C₄	Dic₆	1	8	yes	complemented
42	Dic₃	—	3	8	no	—
43	Dic₃	—	3	8	no	—
44	Dic₃	—	6	8	no	—
45	C₃²	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
46	Q₈	—	18	4	no	—
47	C₂×C₄	—	9	1	no	—
48	C₂×C₄	—	9	1	no	—
49	C₂×C₄	—	9	2	no	—
50	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
51	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₂×C₃	C₂×Dic₆	1	4	yes	—
53	C₂×C₃	D₃×D₄	1	4	yes	—
54	C₂×C₃	D₃×Q₈	1	4	yes	—
55	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	9	2	no	—
58	C₄	—	9	2	no	—
59	C₄	—	18	2	no	—
60	C₄	#634	1	2	yes	complemented
61	C₃	#888	1	4	yes	complemented
62	C₂	C₂×#634	1	1	yes	characteristic
63	C₂	D₄×C₃⋊D₃	1	1	yes	characteristic
64	C₂	Q₈×C₃⋊D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
20. C₃²×C₄	Q₈	semidirect
23. C₃⋊Dic₃	Q₈	semidirect
41. C₃×C₄	Dic₆	semidirect
45. C₃²	C₄⋊Q₈	semidirect
60. C₄	#634	semidirect
61. C₃	#888	semidirect

Maximal subgroups

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

subgroup	index
C₂×#634	2
C₃²×C₄⋊C₄	2
C₄⋊(C₃⋊Dic₃)	2
#3750	2
C₄×C₃⋊Dic₃	2
#888	3