Subgroups — #63802

Order 288.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3797	C₂	1	2	yes	maximal
2	C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₁₂	C₂	1	2	yes	maximal, complemented
4	#3718	C₂	1	2	yes	maximal, complemented
5	D₃×Q₁₆	—	3	2	no	maximal
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃×Dic₆	C₂²	1	4	yes	—
8	#627	—	2	2	no	—
9	#624	—	2	2	no	—
10	C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	D₃×Q₈	—	3	4	no	—
13	C₃×Q₁₆	—	3	2	no	—
14	#216	—	3	4	no	—
15	Dic₁₂	D₃	1	2	yes	complemented
16	D₃×C₈	—	3	2	no	—
17	D₃×C₈	—	6	1	no	—
18	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
19	C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₃⋊Dic₃	D₄	1	1	yes	characteristic
21	C₃×Dic₃	—	2	4	no	—
22	C₂×Q₁₆	—	9	1	no	Sylow
23	C₃×Q₈	—	3	4	no	—
24	C₄×D₃	—	3	2	no	—
25	C₄×D₃	—	6	1	no	—
26	C₄×D₃	—	6	4	no	—
27	Dic₆	—	6	4	no	—
28	Dic₆	C₂×D₃	1	4	yes	—
29	C₃×C₈	—	2	1	no	—
30	C₃×C₈	C₂×D₃	1	2	yes	—
31	C₃⋊C₈	—	3	2	no	—
32	C₃⋊C₈	—	6	1	no	—
33	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
34	C₃⋊D₃	Q₁₆	1	2	yes	complemented
35	C₂×Q₈	—	9	2	no	—
36	Q₁₆	—	3	2	no	—
37	Q₁₆	—	9	2	no	—
38	C₂×C₈	—	9	1	no	—
39	C₂×D₃	—	3	2	no	—
40	C₂×D₃	—	6	1	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	6	4	no	—
43	C₃×C₄	C₂²×D₃	1	2	yes	—
44	Dic₃	—	2	4	no	—
45	Dic₃	—	3	2	no	—
46	Dic₃	—	6	1	no	—
47	C₃²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
48	Q₈	—	3	4	no	—
49	Q₈	—	18	2	no	—
50	C₂×C₄	—	9	1	no	—
51	C₂×C₄	—	18	2	no	—
52	C₈	—	9	1	no	—
53	C₈	D₃²	1	1	yes	characteristic
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	D₃×D₄	1	2	yes	—
56	D₃	—	3	4	no	—
57	D₃	—	6	2	no	—
58	C₂²	—	9	1	no	—
59	C₄	—	6	4	no	—
60	C₄	—	9	1	no	—
61	C₄	C₂×D₃²	1	1	yes	characteristic
62	C₃	—	2	1	no	—
63	C₃	D₃×Q₁₆	1	2	yes	complemented
64	C₂	—	9	2	no	—
65	C₂	#3802	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3802
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂×D₃²
Fitting	C₃²×C₈	C₂²
socle	C₂×C₃²	C₂×D₄
prime radical	C₂×C₃⋊D₃	D₄
hypercenter	C₈	D₃²
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
3. C₃×Dic₁₂	C₂	semidirect
4. #3718	C₂	semidirect
15. Dic₁₂	D₃	semidirect
34. C₃⋊D₃	Q₁₆	semidirect
47. C₃²	C₂×Q₁₆	semidirect
63. C₃	D₃×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3797	2
C₈×C₃⋊D₃	2
C₃×Dic₁₂	2
#3718	2
D₃×Q₁₆	3