Subgroups — #10592818

Order 576.

342 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	#64135	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#63641	C₂	1	4	yes	maximal, complemented
5	#5119	—	3	4	no	maximal
6	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₃²×M₁₆	—	2	2	no	—
8	C₂×C₃²×C₈	C₂²	1	2	yes	—
9	#3757	C₄	1	2	yes	—
10	C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
11	C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₃×M₁₆	D₃	1	4	yes	—
15	#926	—	3	4	no	—
16	C₂²×C₃⋊C₈	—	3	4	no	—
17	Dic₃⋊C₈	—	3	16	no	—
18	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₄	D₄	1	2	yes	—
21	C₂×C₃²×C₄	Q₈	1	2	yes	—
22	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
24	C₃²×C₈	—	2	2	no	—
25	C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	#424	—	9	1	no	Sylow
27	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
28	C₃×M₁₆	—	2	8	no	—
29	C₂×C₃×C₈	C₂×D₃	1	8	yes	—
30	#217	—	3	8	no	—
31	C₄⋊Dic₃	—	3	8	no	—
32	C₄×Dic₃	—	3	8	no	—
33	C₂×C₃⋊C₈	—	3	8	no	—
34	C₂×C₃⋊C₈	—	6	8	no	—
35	C₂²×C₃²	—	2	2	no	—
36	C₂²×C₃²	C₄⋊C₄	1	2	yes	—
37	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
38	C₃²×C₄	C₄⋊C₄	1	2	yes	—
39	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₃⋊Dic₃	—	2	4	no	—
42	C₂×M₁₆	C₃⋊D₃	1	1	yes	characteristic
43	C₂²×C₈	—	9	1	no	—
44	#117	—	9	1	no	—
45	C₄⋊C₈	—	9	4	no	—
46	C₂³×C₃	C₄×D₃	1	4	yes	—
47	C₂×C₃×C₄	Dic₆	1	8	yes	—
48	C₂×C₃×C₄	C₄×D₃	1	4	yes	—
49	C₂×C₃×C₄	#67	1	8	yes	—
50	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
51	C₂×Dic₃	—	3	16	no	—
52	C₃×C₈	—	2	8	no	—
53	C₃⋊C₈	—	6	8	no	—
54	C₂×C₃²	—	2	2	no	—
55	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃²	#131	1	2	yes	—
57	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
58	M₁₆	—	2	2	no	—
59	C₂×C₈	—	9	2	no	—
60	C₂×C₈	—	18	2	no	—
61	C₂×C₈	C₂×C₃⋊D₃	1	2	yes	—
62	C₄⋊C₄	—	9	2	no	—
63	C₂²⋊C₄	—	9	2	no	—
64	C₄²	—	9	2	no	—
65	C₂²×C₃	—	2	8	no	—
66	C₂²×C₃	Dic₃⋊C₄	1	8	yes	—
67	C₂²×C₃	C₂×C₄×D₃	1	4	yes	—
68	C₃×C₄	Dic₃⋊C₄	1	8	yes	—
69	C₃×C₄	C₂×Dic₆	1	4	yes	—
70	C₃×C₄	C₂×#67	1	4	yes	—
71	Dic₃	—	6	16	no	—
72	C₃²	#424	1	1	yes	characteristic, complemented, Sylow
73	C₂³	C₄×C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₄	—	9	4	no	—
75	C₂×C₄	#634	1	2	yes	—
76	C₂×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
77	C₂×C₄	#638	1	2	yes	—
78	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	18	2	no	—
81	C₂×C₃	—	2	8	no	—
82	C₂×C₃	#907	1	8	yes	—
83	C₂×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	#3750	1	2	yes	—
86	C₂²	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
87	C₄	—	18	4	no	—
88	C₄	#3750	1	2	yes	—
89	C₄	C₂×#634	1	1	yes	characteristic
90	C₄	C₂×#638	1	1	yes	characteristic
91	C₃	#5119	1	4	yes	complemented
92	C₂	—	2	2	no	—
93	C₂	#64116	1	2	yes	—
94	C₂	C₂×#3750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₄
Fitting	C₂×C₃²×M₁₆	C₂
prime radical	C₂³×C₃²	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	C₂×M₁₆	C₃⋊D₃
hypocenter	C₃²	#424
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #63641	C₂	semidirect
72. C₃²	#424	semidirect
91. C₃	#5119	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
#64135	2
C₂²×C₃⋊(C₃⋊C₈)	2
#63641	2
#5119	3