Subgroups — #72226172

Order 1536.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10709899	C₂	1	6	yes	maximal, complemented
2	C₃×C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
3	#123403	—	3	1	no	maximal, Sylow
4	#69060	—	2	12	no	—
5	#68062	C₂²	1	4	yes	—
6	C₃×C₈×C₁₆	C₂²	1	3	yes	—
7	#12515	—	3	6	no	—
8	C₁₆²	D₃	1	1	yes	characteristic, complemented
9	C₂×D₄₈	—	4	24	no	—
10	#4695	—	2	12	no	—
11	C₃×C₄×C₁₆	D₄	1	6	yes	—
12	C₃×C₈²	C₂³	1	1	yes	characteristic
13	#2246	—	6	12	no	—
14	#1712	—	3	4	no	—
15	C₈×C₁₆	C₂×D₃	1	3	yes	—
16	C₂×D₂₄	—	4	24	no	—
17	#874	—	4	4	no	—
18	C₂×C₃×C₁₆	D₈	1	12	yes	—
19	C₃×C₄×C₈	C₂×D₄	1	3	yes	—
20	D₄₈	—	8	48	no	—
21	C₂×D₁₆	—	12	24	no	—
22	#493	—	6	12	no	—
23	C₄×C₁₆	D₁₂	1	6	yes	—
24	C₈²	C₂²×D₃	1	1	yes	characteristic
25	C₂×D₁₂	—	8	12	no	—
26	C₂×C₃×C₈	C₂×D₈	1	6	yes	—
27	C₃×C₄²	#127	1	1	yes	characteristic
28	D₂₄	—	8	48	no	—
29	C₃×C₁₆	D₁₆	1	24	yes	complemented
30	C₂×D₈	—	12	24	no	—
31	#127	—	12	4	no	—
32	D₁₆	—	24	48	no	—
33	C₂×C₁₆	D₂₄	1	12	yes	—
34	C₄×C₈	C₂×D₁₂	1	3	yes	—
35	C₂²×D₃	—	16	4	no	—
36	C₂×C₃×C₄	#493	1	3	yes	—
37	D₁₂	—	16	24	no	—
38	C₃×C₈	C₂×D₁₆	1	12	yes	—
39	C₂×D₄	—	24	12	no	—
40	D₈	—	24	48	no	—
41	C₂×C₈	C₂×D₂₄	1	6	yes	—
42	C₄²	#874	1	1	yes	characteristic
43	C₁₆	D₄₈	1	24	yes	complemented
44	C₂²×C₃	#1712	1	1	yes	characteristic
45	C₂×D₃	—	32	12	no	—
46	C₃×C₄	#2246	1	6	yes	—
47	C₂³	—	48	4	no	—
48	D₄	—	48	24	no	—
49	C₂×C₄	#4695	1	3	yes	—
50	C₈	C₂×D₄₈	1	12	yes	—
51	C₂×C₃	#12515	1	3	yes	—
52	D₃	—	64	4	no	—
53	C₂²	—	96	12	no	—
54	C₂²	#68062	1	1	yes	characteristic
55	C₄	#69060	1	6	yes	—
56	C₃	#123403	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	192	4	no	—
58	C₂	#10709899	1	3	yes	—

	subgroup	G/H
center	C₂²	#68062
commutator, cosocle	C₃×C₈²	C₂³
Frattini	C₈²	C₂²×D₃
Fitting	C₃×C₁₆²	C₂
socle	C₂²×C₃	#1712
hypercenter	C₁₆²	D₃
hypocenter	C₃	#123403
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₁₆²

Complements

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

N	H ≅ G/N	type
1. #10709899	C₂	semidirect
2. C₃×C₁₆²	C₂	semidirect
8. C₁₆²	D₃	semidirect
29. C₃×C₁₆	D₁₆	semidirect
43. C₁₆	D₄₈	semidirect
56. C₃	#123403	semidirect

Maximal subgroups

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

subgroup	index
#10709899	2
C₃×C₁₆²	2
#123403	3