Subgroups — #10592883

Order 576.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	#63662	C₂	1	2	yes	maximal
4	#63644	C₂	1	1	yes	maximal, characteristic
5	#63642	C₂	1	1	yes	maximal, characteristic
6	#5184	—	3	4	no	maximal
7	C₂×C₃²×Q₈	C₂²	1	2	yes	—
8	C₃²×Q₁₆	C₄	1	4	yes	complemented
9	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
11	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₄×C₃⋊Dic₃	—	2	2	no	—
13	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	D₃	1	4	yes	—
16	Q₈×Dic₃	—	3	8	no	—
17	#835	—	3	8	no	—
18	#817	—	3	4	no	—
19	#815	—	3	4	no	—
20	C₃²×Q₈	—	2	2	no	—
21	C₃²×Q₈	C₂×C₄	1	4	yes	—
22	C₂×C₃²×C₄	—	2	2	no	—
23	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊Dic₃	—	2	2	no	—
25	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
26	C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₃⋊(C₃⋊C₈)	—	2	1	no	—
28	#441	—	9	1	no	Sylow
29	C₂×C₃×Q₈	C₂×D₃	1	8	yes	—
30	C₃×Q₁₆	Dic₃	1	16	yes	complemented
31	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
32	C₄⋊Dic₃	—	3	8	no	—
33	C₄⋊Dic₃	—	6	8	no	—
34	C₄×Dic₃	—	3	4	no	—
35	C₄×Dic₃	—	6	8	no	—
36	C₂×C₃⋊C₈	—	3	4	no	—
37	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
38	C₃²×C₄	—	2	4	no	—
39	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₃²×C₄	#41	1	1	yes	characteristic
41	C₃⋊Dic₃	—	2	2	no	—
42	C₃⋊Dic₃	—	4	2	no	—
43	C₂×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
44	C₄×Q₈	—	9	2	no	—
45	#106	—	9	1	no	—
46	#103	—	9	2	no	—
47	#97	—	9	1	no	—
48	C₃×Q₈	—	2	8	no	—
49	C₃×Q₈	C₂×Dic₃	1	16	yes	—
50	C₂×C₃×C₄	—	2	8	no	—
51	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
52	C₂×Dic₃	—	3	8	no	—
53	C₂×Dic₃	—	6	8	no	—
54	C₃×C₈	C₂×Dic₃	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	C₂×C₃⋊D₃	1	2	yes	—
59	Q₁₆	C₃⋊Dic₃	1	4	yes	complemented
60	C₂×C₈	—	9	1	no	—
61	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
62	C₄⋊C₄	—	9	2	no	—
63	C₄⋊C₄	—	18	2	no	—
64	C₄²	—	9	1	no	—
65	C₄²	—	18	2	no	—
66	C₂²×C₃	D₃×D₄	1	4	yes	—
67	C₃×C₄	—	2	16	no	—
68	C₃×C₄	#237	1	4	yes	—
69	C₃×C₄	C₂²×Dic₃	1	4	yes	—
70	Dic₃	—	6	8	no	—
71	Dic₃	—	12	8	no	—
72	C₃²	#441	1	1	yes	characteristic, complemented, Sylow
73	Q₈	—	2	2	no	—
74	Q₈	C₂×C₃⋊Dic₃	1	4	yes	—
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	C₂×C₃⋊Dic₃	1	2	yes	—
81	C₂×C₃	#918	1	8	yes	—
82	C₂×C₃	D₄×Dic₃	1	4	yes	—
83	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
84	C₄	—	2	4	no	—
85	C₄	—	18	2	no	—
86	C₄	—	36	2	no	—
87	C₄	#3830	1	1	yes	characteristic
88	C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
89	C₃	#5184	1	4	yes	complemented
90	C₂	#64127	1	2	yes	—
91	C₂	D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×Q₁₆	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	C₂×Q₁₆	C₃⋊D₃
hypocenter	C₃²	#441
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
8. C₃²×Q₁₆	C₄	semidirect
30. C₃×Q₁₆	Dic₃	semidirect
59. Q₁₆	C₃⋊Dic₃	semidirect
72. C₃²	#441	semidirect
89. C₃	#5184	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₁₆	2
Q₈×C₃⋊Dic₃	2
#63662	2
#63644	2
#63642	2
#5184	3