Subgroups — #10592552

Order 576.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3746	C₂	1	1	yes	maximal, characteristic
2	#64096	C₂	1	2	yes	maximal
3	C₃²×#106	C₂	1	1	yes	maximal, characteristic
4	#63642	C₂	1	1	yes	maximal, characteristic
5	#63637	C₂	1	2	yes	maximal
6	#4853	—	3	4	no	maximal
7	C₂×#634	C₂²	1	2	yes	—
8	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	#3750	—	2	2	no	—
11	C₄×C₃⋊Dic₃	—	2	2	no	—
12	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#3746	C₄	1	4	yes	complemented
15	C₂×Dic₁₂	—	3	4	no	—
16	#887	—	3	8	no	—
17	C₃×#106	D₃	1	4	yes	—
18	#815	—	3	4	no	—
19	#810	—	3	8	no	—
20	C₂×C₃²×C₄	—	2	2	no	—
21	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	—	2	2	no	—
23	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
24	#634	—	2	2	no	—
25	#634	C₂×C₄	1	4	yes	—
26	C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₃⋊(C₃⋊C₈)	—	2	1	no	—
28	#441	—	9	1	no	Sylow
29	C₂×Dic₆	—	3	8	no	—
30	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
31	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
32	Dic₃⋊C₄	—	6	8	no	—
33	C₄×Dic₃	—	3	4	no	—
34	C₄×Dic₃	—	6	8	no	—
35	C₂×C₃⋊C₈	—	3	4	no	—
36	Dic₁₂	—	3	16	no	—
37	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
38	C₃²×C₄	—	4	2	no	—
39	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₃²×C₄	#41	1	1	yes	characteristic
41	C₃⋊Dic₃	—	2	4	no	—
42	C₃⋊Dic₃	—	2	2	no	—
43	C₂×Q₁₆	—	9	1	no	—
44	C₄×Q₈	—	9	2	no	—
45	#106	C₃⋊D₃	1	1	yes	characteristic
46	#103	—	9	2	no	—
47	#97	—	9	1	no	—
48	C₂×C₃×C₄	—	2	8	no	—
49	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
50	C₂×Dic₃	—	3	8	no	—
51	C₂×Dic₃	—	6	8	no	—
52	Dic₆	—	3	16	no	—
53	Dic₆	—	6	8	no	—
54	C₃×C₈	C₄×D₃	1	8	yes	—
55	C₃⋊C₈	—	6	4	no	—
56	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
57	C₂×C₃²	#137	1	2	yes	—
58	C₂×Q₈	—	9	2	no	—
59	Q₁₆	—	9	4	no	—
60	C₂×C₈	—	9	1	no	—
61	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
62	C₄⋊C₄	—	18	2	no	—
63	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
64	C₄²	—	9	1	no	—
65	C₄²	—	18	2	no	—
66	C₂²×C₃	D₃×D₄	1	4	yes	—
67	C₃×C₄	—	4	8	no	—
68	C₃×C₄	C₂×C₄×D₃	1	4	yes	—
69	C₃×C₄	#239	1	4	yes	—
70	Dic₃	—	6	16	no	—
71	Dic₃	—	6	8	no	—
72	C₃²	#441	1	1	yes	characteristic, complemented, Sylow
73	Q₈	—	9	4	no	—
74	Q₈	—	18	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	9	2	no	—
77	C₂×C₄	—	18	2	no	—
78	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
79	C₈	—	18	1	no	—
80	C₈	C₄×C₃⋊D₃	1	2	yes	—
81	C₂×C₃	#893	1	4	yes	—
82	C₂×C₃	#915	1	8	yes	—
83	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	18	4	no	—
86	C₄	—	18	2	no	—
87	C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
88	C₄	#3832	1	1	yes	characteristic
89	C₃	#4853	1	4	yes	complemented
90	C₂	#64102	1	1	yes	characteristic
91	C₂	#64124	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#106	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	#106	C₃⋊D₃
hypocenter	C₃²	#441
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
14. #3746	C₄	semidirect
72. C₃²	#441	semidirect
89. C₃	#4853	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3746	2
#64096	2
C₃²×#106	2
#63642	2
#63637	2
#4853	3