Subgroups — #72254887

Order 2304.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11772234	C₂	1	2	yes	maximal
2	C₃²×#1397	C₂	1	1	yes	maximal, characteristic
3	#10682870	—	3	4	no	maximal
4	C₂×C₃²×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₃₂	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#10589324	—	2	2	no	—
8	C₃×#1397	D₃	1	4	yes	—
9	#67501	—	3	8	no	—
10	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	Q₈	1	1	yes	characteristic
12	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
13	C₄⋊(C₃⋊(C₃⋊C₈))	—	4	2	no	—
14	C₃²×C₃₂	—	2	1	no	—
15	C₃²×C₃₂	C₈	1	2	yes	complemented
16	#7463	—	9	1	no	Sylow
17	C₂×C₃×C₃₂	Dic₃	1	4	yes	—
18	C₂×C₃×C₃₂	Dic₃	1	4	yes	—
19	C₃×C₄×C₁₆	C₂×D₃	1	4	yes	—
20	#4458	—	6	8	no	—
21	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	Q₁₆	1	1	yes	characteristic
23	C₃²×C₄²	D₈	1	1	yes	characteristic
24	C₂×C₃⋊(C₃⋊C₈)	—	8	2	no	—
25	C₃²×C₁₆	—	2	1	no	—
26	C₃²×C₁₆	C₂×C₈	1	1	yes	characteristic
27	C₃²×C₁₆	M₁₆	1	1	yes	characteristic
28	#1397	C₃⋊D₃	1	1	yes	characteristic
29	#1372	—	9	2	no	—
30	C₂×C₃×C₁₆	Dic₆	1	4	yes	—
31	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
32	C₃×C₄×C₈	D₁₂	1	4	yes	—
33	C₄⋊(C₃⋊C₈)	—	12	8	no	—
34	C₃×C₃₂	—	2	4	no	—
35	C₃×C₃₂	C₃⋊C₈	1	8	yes	complemented
36	C₂×C₃²×C₄	#107	1	1	yes	characteristic
37	C₂×C₃²×C₄	SD₃₂	1	2	yes	—
38	C₃²×C₈	—	2	1	no	—
39	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
40	C₃²×C₈	#108	1	1	yes	characteristic
41	C₃⋊(C₃⋊C₈)	—	16	2	no	—
42	C₂×C₃₂	C₃⋊Dic₃	1	1	yes	characteristic
43	C₂×C₃₂	C₃⋊Dic₃	1	1	yes	characteristic
44	C₄×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
45	#335	—	18	2	no	—
46	C₂×C₃×C₈	Dic₁₂	1	4	yes	—
47	C₂×C₃×C₈	C₄⋊Dic₃	1	4	yes	—
48	C₃×C₄²	D₂₄	1	4	yes	—
49	C₂×C₃⋊C₈	—	24	8	no	—
50	C₃×C₁₆	—	2	4	no	—
51	C₃×C₁₆	C₂×C₃⋊C₈	1	4	yes	—
52	C₃×C₁₆	#208	1	4	yes	—
53	C₂²×C₃²	#367	1	1	yes	characteristic
54	C₃²×C₄	—	2	2	no	—
55	C₃²×C₄	#335	1	1	yes	characteristic
56	C₃²×C₄	#368	1	1	yes	characteristic
57	C₂×C₁₆	#634	1	1	yes	characteristic
58	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
59	C₄⋊C₈	—	36	2	no	—
60	C₄×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
61	C₃₂	—	2	1	no	—
62	C₃₂	C₃⋊(C₃⋊C₈)	1	2	yes	complemented
63	C₂×C₃×C₄	#800	1	8	yes	—
64	C₂×C₃×C₄	#818	1	4	yes	—
65	C₃×C₈	—	2	4	no	—
66	C₃×C₈	C₄⋊(C₃⋊C₈)	1	4	yes	—
67	C₃×C₈	#819	1	4	yes	—
68	C₃⋊C₈	—	48	8	no	—
69	C₂×C₃²	#1372	1	1	yes	characteristic
70	C₂×C₃²	#1426	1	2	yes	—
71	C₂×C₈	—	72	2	no	—
72	C₂×C₈	#3746	1	1	yes	characteristic
73	C₂×C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
74	C₄²	#3745	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
77	C₁₆	#3748	1	1	yes	characteristic
78	C₂²×C₃	#4505	1	4	yes	—
79	C₃×C₄	—	2	8	no	—
80	C₃×C₄	#4458	1	4	yes	—
81	C₃×C₄	#4506	1	4	yes	—
82	C₃²	#7463	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄	#63627	1	2	yes	—
84	C₂×C₄	#63645	1	1	yes	characteristic
85	C₈	—	2	1	no	—
86	C₈	—	144	2	no	—
87	C₈	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
88	C₈	#63646	1	1	yes	characteristic
89	C₂×C₃	#67501	1	4	yes	—
90	C₂×C₃	#67601	1	8	yes	—
91	C₂²	#10589371	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	#10589324	1	1	yes	characteristic
94	C₄	#10589372	1	1	yes	characteristic
95	C₃	#10682870	1	4	yes	complemented
96	C₂	#11772234	1	1	yes	characteristic
97	C₂	#11773071	1	2	yes	—

	subgroup	G/H
center	C₂²	#10589371
commutator	C₃²×C₁₆	C₂×C₈
Frattini	C₄×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#1397	C₂
socle, prime radical	C₂²×C₃²	#367
cosocle	C₃²×C₄×C₁₆	C₂²
hypercenter	#1397	C₃⋊D₃
hypocenter	C₃²	#7463
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
15. C₃²×C₃₂	C₈	semidirect
35. C₃×C₃₂	C₃⋊C₈	semidirect
62. C₃₂	C₃⋊(C₃⋊C₈)	semidirect
82. C₃²	#7463	semidirect
95. C₃	#10682870	semidirect

Maximal subgroups

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

subgroup	index
#11772234	2
C₃²×#1397	2
#10682870	3