Subgroups — #72226175

Order 1536.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10709931	C₂	1	3	yes	maximal, complemented
2	#10709905	C₂	1	3	yes	maximal
3	C₃×C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
4	#123411	—	3	1	no	maximal, Sylow
5	#69061	—	2	6	no	—
6	#69046	—	2	6	no	—
7	#68065	C₂²	1	3	yes	—
8	#68062	C₂²	1	1	yes	characteristic
9	C₃×C₈×C₁₆	C₂²	1	3	yes	—
10	#12523	—	3	3	no	—
11	#12517	—	3	3	no	—
12	C₁₆²	D₃	1	1	yes	characteristic, complemented
13	C₂×#800	—	4	12	no	—
14	#4699	—	2	3	no	—
15	#4695	—	2	3	no	—
16	#4682	—	2	6	no	—
17	C₃×C₄×C₁₆	D₄	1	6	yes	—
18	C₃×C₈²	C₂³	1	1	yes	characteristic
19	#4505	—	4	12	no	—
20	#2254	—	6	6	no	—
21	#2248	—	6	6	no	—
22	#1713	—	3	3	no	—
23	#1712	—	3	1	no	—
24	C₈×C₁₆	C₂×D₃	1	3	yes	—
25	C₂×Dic₁₂	—	4	6	no	—
26	C₂×D₂₄	—	4	6	no	—
27	#874	—	4	1	no	—
28	#869	—	4	3	no	—
29	C₂×C₃×C₁₆	D₈	1	12	yes	—
30	C₃×C₄×C₈	C₂×D₄	1	3	yes	—
31	#818	—	4	12	no	—
32	#800	—	8	24	no	—
33	C₂×SD₃₂	—	12	12	no	—
34	#500	—	6	6	no	—
35	#494	—	6	3	no	—
36	#493	—	6	3	no	—
37	#367	—	12	12	no	—
38	C₄×C₁₆	D₁₂	1	6	yes	—
39	C₈²	C₂²×D₃	1	1	yes	characteristic
40	C₂×D₁₂	—	8	3	no	—
41	C₂×Dic₆	—	8	3	no	—
42	C₂×C₃×C₈	C₂×D₈	1	6	yes	—
43	C₃×C₄²	#127	1	1	yes	characteristic
44	C₄⋊Dic₃	—	8	6	no	—
45	Dic₁₂	—	8	12	no	—
46	D₂₄	—	8	12	no	—
47	C₃×C₁₆	SD₃₂	1	24	yes	complemented
48	C₂×Q₁₆	—	12	6	no	—
49	C₂×D₈	—	12	6	no	—
50	C₄⋊Q₈	—	12	3	no	—
51	#127	—	12	1	no	—
52	SD₃₂	—	24	24	no	—
53	C₂×C₁₆	D₂₄	1	12	yes	—
54	#107	—	12	12	no	—
55	C₄×C₈	C₂×D₁₂	1	3	yes	—
56	C₂²×D₃	—	16	1	no	—
57	C₂×C₃×C₄	#493	1	3	yes	—
58	C₂×Dic₃	—	16	3	no	—
59	D₁₂	—	16	6	no	—
60	Dic₆	—	16	6	no	—
61	C₃×C₈	C₂×SD₃₂	1	12	yes	—
62	C₂×Q₈	—	24	3	no	—
63	C₂×D₄	—	24	3	no	—
64	Q₁₆	—	24	12	no	—
65	D₈	—	24	12	no	—
66	C₂×C₈	C₂×D₂₄	1	6	yes	—
67	C₄⋊C₄	—	24	6	no	—
68	C₄²	#874	1	1	yes	characteristic
69	C₁₆	#800	1	24	yes	complemented
70	C₂²×C₃	#1712	1	1	yes	characteristic
71	C₂×D₃	—	32	3	no	—
72	C₃×C₄	#2248	1	6	yes	—
73	Dic₃	—	32	3	no	—
74	C₂³	—	48	1	no	—
75	Q₈	—	48	6	no	—
76	D₄	—	48	6	no	—
77	C₂×C₄	—	48	3	no	—
78	C₂×C₄	#4695	1	3	yes	—
79	C₈	C₂×#800	1	12	yes	—
80	C₂×C₃	#12517	1	3	yes	—
81	D₃	—	64	1	no	—
82	C₂²	—	96	3	no	—
83	C₂²	#68062	1	1	yes	characteristic
84	C₄	—	96	3	no	—
85	C₄	#69061	1	6	yes	—
86	C₃	#123411	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	192	1	no	—
88	C₂	#10709905	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
prime radical	#68062	C₂²
hypercenter	C₁₆²	D₃
hypocenter	C₃	#123411
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. #10709931	C₂	semidirect
3. C₃×C₁₆²	C₂	semidirect
12. C₁₆²	D₃	semidirect
47. C₃×C₁₆	SD₃₂	semidirect
69. C₁₆	#800	semidirect
86. C₃	#123411	semidirect

Maximal subgroups

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

subgroup	index
#10709931	2
#10709905	2
C₃×C₁₆²	2
#123411	3