Subgroups — #10589419

Order 576.

174 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	#63639	C₂	1	2	yes	maximal, complemented
3	#4553	—	3	4	no	maximal
4	C₂²×C₃²×C₄	C₄	1	1	yes	characteristic
5	C₃²×M₁₆	—	2	2	no	—
6	C₂×C₃²×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
9	C₂×C₃×M₁₆	D₃	1	4	yes	—
10	#812	—	3	8	no	—
11	C₂³×C₃²	C₈	1	1	yes	characteristic
12	C₂×C₃²×C₄	—	2	1	no	—
13	C₂×C₃²×C₄	—	2	1	no	—
14	C₂×C₃²×C₄	C₈	1	1	yes	characteristic
15	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₈	—	2	1	no	—
17	C₃²×C₈	D₄	1	2	yes	—
18	#349	—	9	1	no	Sylow
19	C₂²×C₃×C₄	Dic₃	1	4	yes	—
20	C₃×M₁₆	—	2	8	no	—
21	C₂×C₃×C₈	Dic₃	1	4	yes	—
22	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
23	C₃⋊C₁₆	—	6	8	no	—
24	C₂²×C₃²	—	2	1	no	—
25	C₂²×C₃²	—	4	1	no	—
26	C₂²×C₃²	C₂×C₈	1	1	yes	characteristic
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₃²×C₄	M₁₆	1	1	yes	characteristic
30	C₂×M₁₆	C₃⋊D₃	1	1	yes	characteristic
31	M₃₂	—	9	2	no	—
32	C₂³×C₃	C₃⋊C₈	1	4	yes	—
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	C₃⋊C₈	1	4	yes	—
36	C₂×C₃×C₄	C₂×Dic₃	1	4	yes	—
37	C₃×C₈	—	2	4	no	—
38	C₃×C₈	#67	1	8	yes	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	—	4	1	no	—
41	C₂×C₃²	C₂²⋊C₈	1	1	yes	characteristic
42	C₂²×C₄	C₃⋊Dic₃	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	C₂×C₈	C₃⋊Dic₃	1	1	yes	characteristic
45	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₁₆	—	18	2	no	—
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	—	4	4	no	—
49	C₂²×C₃	C₂×C₃⋊C₈	1	4	yes	—
50	C₃×C₄	—	2	4	no	—
51	C₃×C₄	#208	1	4	yes	—
52	C₃×C₄	#217	1	4	yes	—
53	C₃²	#349	1	1	yes	characteristic, complemented, Sylow
54	C₂³	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
58	C₂×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	#638	1	2	yes	—
61	C₂×C₃	—	2	4	no	—
62	C₂×C₃	—	4	4	no	—
63	C₂×C₃	#830	1	4	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	#3748	1	1	yes	characteristic
69	C₄	#3757	1	1	yes	characteristic
70	C₃	#4553	1	4	yes	complemented
71	C₂	—	2	1	no	—
72	C₂	—	4	1	no	—
73	C₂	#63657	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#3757
commutator	C₂²×C₃²	C₂×C₈
Frattini	C₂×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	C₈
cosocle	C₂×C₃²×C₈	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₃
hypocenter	C₃²	#349
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #63639	C₂	semidirect
53. C₃²	#349	semidirect
70. C₃	#4553	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
#63639	2
#4553	3