Subgroups — #10592635

Order 576.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	#63658	C₂	1	4	yes	maximal
4	#4936	—	3	4	no	maximal
5	C₂²×C₃²×C₄	C₂²	1	2	yes	—
6	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₃	C₂²	1	4	yes	—
8	C₃²×C₄²	C₄	1	4	yes	complemented
9	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
10	C₂×C₃×C₄²	D₃	1	4	yes	—
11	C₂×C₄⋊Dic₃	—	3	8	no	—
12	C₃⋊#95	—	3	16	no	—
13	C₂³×C₃²	C₂³	1	1	yes	characteristic
14	C₂×C₃²×C₄	—	2	4	no	—
15	C₂×C₃²×C₄	C₂×C₄	1	4	yes	—
16	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
17	C₂×C₃²×C₄	D₄	1	2	yes	—
18	C₂×C₃²×C₄	Q₈	1	2	yes	—
19	C₂×C₃⋊Dic₃	—	2	4	no	—
20	C₂×C₃⋊Dic₃	—	2	8	no	—
21	#382	—	9	1	no	Sylow
22	C₂²×C₃×C₄	C₂×D₃	1	8	yes	—
23	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
24	C₂²×Dic₃	—	3	16	no	—
25	C₃×C₄²	Dic₃	1	16	yes	complemented
26	C₄⋊Dic₃	—	6	16	no	—
27	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₃²	#41	1	4	yes	—
31	C₃²×C₄	—	2	4	no	—
32	C₃²×C₄	C₄⋊C₄	1	4	yes	—
33	C₃⋊Dic₃	—	4	4	no	—
34	C₂×C₄⋊C₄	—	9	2	no	—
35	C₂×C₄²	C₃⋊D₃	1	1	yes	characteristic
36	#95	—	9	4	no	—
37	C₂³×C₃	C₂²×D₃	1	4	yes	—
38	C₂×C₃×C₄	—	2	16	no	—
39	C₂×C₃×C₄	Dic₆	1	8	yes	—
40	C₂×C₃×C₄	D₁₂	1	8	yes	—
41	C₂×C₃×C₄	C₂×Dic₃	1	16	yes	—
42	C₂×C₃×C₄	C₂×Dic₃	1	8	yes	—
43	C₂×Dic₃	—	6	16	no	—
44	C₂×Dic₃	—	6	32	no	—
45	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃²	#117	1	2	yes	—
47	C₂×C₃²	#124	1	2	yes	—
48	C₂×C₃²	#125	1	2	yes	—
49	C₂²×C₄	—	9	4	no	—
50	C₂²×C₄	C₂×C₃⋊D₃	1	2	yes	—
51	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₄⋊C₄	—	18	4	no	—
53	C₄²	C₃⋊Dic₃	1	4	yes	complemented
54	C₂²×C₃	C₂×Dic₆	1	4	yes	—
55	C₂²×C₃	C₂×D₁₂	1	4	yes	—
56	C₂²×C₃	#235	1	16	yes	—
57	C₂²×C₃	C₂²×Dic₃	1	4	yes	—
58	C₃×C₄	—	2	16	no	—
59	C₃×C₄	C₄⋊Dic₃	1	16	yes	—
60	Dic₃	—	12	16	no	—
61	C₃²	#382	1	1	yes	characteristic, complemented, Sylow
62	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	18	4	no	—
65	C₂×C₄	—	18	8	no	—
66	C₂×C₄	#634	1	2	yes	—
67	C₂×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
68	C₂×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
69	C₂×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
70	C₂×C₃	#870	1	8	yes	—
71	C₂×C₃	#875	1	8	yes	—
72	C₂×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
73	C₂×C₃	#926	1	8	yes	—
74	C₂²	C₂×#634	1	1	yes	characteristic
75	C₂²	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
76	C₂²	#3828	1	4	yes	—
77	C₂²	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	36	4	no	—
80	C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
81	C₃	#4936	1	4	yes	complemented
82	C₂	#64079	1	2	yes	—
83	C₂	#64084	1	2	yes	—
84	C₂	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
85	C₂	#64135	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³
hypercenter	C₂×C₄²	C₃⋊D₃
hypocenter	C₃²	#382
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
8. C₃²×C₄²	C₄	semidirect
25. C₃×C₄²	Dic₃	semidirect
53. C₄²	C₃⋊Dic₃	semidirect
61. C₃²	#382	semidirect
81. C₃	#4936	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄²	2
C₂×C₄⋊(C₃⋊Dic₃)	2
#63658	2
#4936	3