Subgroups — #10589426

Order 576.

258 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	C₃⋊#338	—	3	4	no	maximal
4	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
5	C₃²×M₁₆	—	2	2	no	—
6	C₂×C₃²×C₈	C₄	1	2	yes	—
7	#3748	—	2	2	no	—
8	#3748	—	2	2	no	—
9	C₂×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	—
10	C₂×C₃×M₁₆	D₃	1	4	yes	—
11	C₂×#208	—	3	8	no	—
12	C₂³×C₃²	Q₈	1	1	yes	characteristic
13	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
14	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₂×C₃²×C₄	D₄	1	2	yes	—
17	C₃²×C₈	—	2	2	no	—
18	C₃⋊(C₃⋊C₈)	—	2	4	no	—
19	#338	—	9	1	no	Sylow
20	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
21	C₃×M₁₆	—	2	8	no	—
22	C₂×C₃×C₈	Dic₃	1	8	yes	—
23	#208	—	6	8	no	—
24	#208	—	6	8	no	—
25	C₂×C₃⋊C₈	—	3	16	no	—
26	C₂²×C₃²	—	4	1	no	—
27	C₂²×C₃²	C₄⋊C₄	1	2	yes	—
28	C₂²×C₃²	C₄⋊C₄	1	1	yes	characteristic
29	C₃²×C₄	C₄²	1	1	yes	characteristic
30	C₃²×C₄	C₂²⋊C₄	1	2	yes	—
31	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×M₁₆	—	9	2	no	—
33	C₂×M₁₆	C₃⋊D₃	1	1	yes	characteristic
34	C₂³×C₃	Dic₆	1	4	yes	—
35	C₂×C₃×C₄	C₄×D₃	1	8	yes	—
36	C₂×C₃×C₄	D₁₂	1	4	yes	—
37	C₂×C₃×C₄	C₂×Dic₃	1	4	yes	—
38	C₂×C₃×C₄	#67	1	8	yes	—
39	C₃×C₈	—	2	8	no	—
40	C₃⋊C₈	—	6	16	no	—
41	C₂×C₃²	—	2	2	no	—
42	C₂×C₃²	—	2	1	no	—
43	C₂×C₃²	#95	1	1	yes	characteristic
44	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
45	M₁₆	—	2	2	no	—
46	M₁₆	—	18	2	no	—
47	M₁₆	—	18	2	no	—
48	C₂×C₈	—	9	4	no	—
49	C₂×C₈	C₃⋊Dic₃	1	2	yes	—
50	C₂²×C₃	—	4	4	no	—
51	C₂²×C₃	Dic₃⋊C₄	1	8	yes	—
52	C₂²×C₃	C₄⋊Dic₃	1	4	yes	—
53	C₃×C₄	C₄×Dic₃	1	4	yes	—
54	C₃×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
55	C₃×C₄	#217	1	4	yes	—
56	C₃²	#338	1	1	yes	characteristic, complemented, Sylow
57	C₂³	#634	1	1	yes	characteristic
58	C₂×C₄	C₄×C₃⋊D₃	1	2	yes	—
59	C₂×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
60	C₂×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
61	C₂×C₄	#638	1	2	yes	—
62	C₈	—	2	2	no	—
63	C₈	—	18	4	no	—
64	C₂×C₃	—	2	8	no	—
65	C₂×C₃	—	2	4	no	—
66	C₂×C₃	C₃⋊#95	1	4	yes	—
67	C₂²	—	4	1	no	—
68	C₂²	#3750	1	2	yes	—
69	C₂²	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
70	C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
71	C₄	#3752	1	2	yes	—
72	C₄	#3757	1	1	yes	characteristic
73	C₃	C₃⋊#338	1	4	yes	complemented
74	C₂	—	2	2	no	—
75	C₂	—	2	1	no	—
76	C₂	#63658	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₃⋊Dic₃
commutator	C₃²×C₄	C₄²
Frattini	C₂²×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂×C₃²	#95
prime radical	C₂³×C₃²	Q₈
cosocle	C₂²×C₃²×C₄	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₃
hypocenter	C₃²	#338
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
56. C₃²	#338	semidirect
73. C₃	C₃⋊#338	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
C₃⋊#338	3