Subgroups — #10592824

Order 576.

306 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#3748	C₂	1	2	yes	maximal
3	#63646	C₂	1	4	yes	maximal, complemented
4	#5125	—	3	4	no	maximal
5	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₃²×M₁₆	C₄	1	4	yes	—
7	C₂×C₃²×C₈	C₂²	1	2	yes	—
8	#3748	—	2	2	no	—
9	#3748	C₂²	1	4	yes	—
10	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
11	C₂×C₃×M₁₆	D₃	1	4	yes	—
12	C₂×#208	—	3	8	no	—
13	#819	—	3	16	no	—
14	C₂³×C₃²	Q₈	1	1	yes	characteristic
15	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
16	C₂×C₃²×C₄	D₄	1	2	yes	—
17	C₂×C₃²×C₄	Q₈	1	1	yes	characteristic
18	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
19	C₃²×C₈	C₂×C₄	1	4	yes	—
20	C₃⋊(C₃⋊C₈)	—	2	4	no	—
21	#430	—	9	1	no	Sylow
22	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
23	C₃×M₁₆	Dic₃	1	16	yes	—
24	C₂×C₃×C₈	C₂×D₃	1	8	yes	—
25	#208	—	3	16	no	—
26	#208	—	6	8	no	—
27	C₂×C₃⋊C₈	—	6	8	no	—
28	C₂²×C₃²	—	4	1	no	—
29	C₂²×C₃²	C₄⋊C₄	1	2	yes	—
30	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
31	C₃²×C₄	C₄⋊C₄	1	2	yes	—
32	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×M₁₆	—	9	2	no	—
35	C₂×M₁₆	C₃⋊D₃	1	1	yes	characteristic
36	#108	—	9	4	no	—
37	C₂³×C₃	Dic₆	1	4	yes	—
38	C₂×C₃×C₄	Dic₆	1	4	yes	—
39	C₂×C₃×C₄	D₁₂	1	8	yes	—
40	C₂×C₃×C₄	C₂×Dic₃	1	8	yes	—
41	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
42	C₃×C₈	C₂×Dic₃	1	16	yes	—
43	C₃⋊C₈	—	6	16	no	—
44	C₂×C₃²	—	2	2	no	—
45	C₂×C₃²	—	2	1	no	—
46	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
48	M₁₆	—	9	4	no	—
49	M₁₆	—	18	2	no	—
50	M₁₆	C₃⋊Dic₃	1	4	yes	—
51	C₂×C₈	—	18	2	no	—
52	C₂×C₈	C₂×C₃⋊D₃	1	2	yes	—
53	C₂²×C₃	—	4	4	no	—
54	C₂²×C₃	C₄⋊Dic₃	1	8	yes	—
55	C₂²×C₃	C₂×Dic₆	1	4	yes	—
56	C₃×C₄	C₄⋊Dic₃	1	8	yes	—
57	C₃×C₄	C₂×D₁₂	1	4	yes	—
58	C₃×C₄	C₂²×Dic₃	1	4	yes	—
59	C₃²	#430	1	1	yes	characteristic, complemented, Sylow
60	C₂³	#634	1	1	yes	characteristic
61	C₂×C₄	#634	1	1	yes	characteristic
62	C₂×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
63	C₂×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
64	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
65	C₈	—	18	4	no	—
66	C₈	C₂×C₃⋊Dic₃	1	4	yes	—
67	C₂×C₃	—	2	8	no	—
68	C₂×C₃	—	2	4	no	—
69	C₂×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
70	C₂²	—	4	1	no	—
71	C₂²	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
72	C₂²	C₂×#634	1	1	yes	characteristic
73	C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
74	C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
75	C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
76	C₃	#5125	1	4	yes	complemented
77	C₂	—	2	2	no	—
78	C₂	—	2	1	no	—
79	C₂	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	Q₈
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	C₂×M₁₆	C₃⋊D₃
hypocenter	C₃²	#430
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #63646	C₂	semidirect
59. C₃²	#430	semidirect
76. C₃	#5125	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₁₆	2
C₂×#3748	2
#63646	2
#5125	3