Subgroups — #10589352

Order 576.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64078	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	#4486	—	3	4	no	maximal
5	C₂×#634	—	2	1	no	—
6	C₂×C₃²×C₈	—	2	1	no	—
7	C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
9	C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	#869	—	3	4	no	—
12	C₃×C₄⋊C₈	D₃	1	4	yes	—
13	C₄⋊(C₃⋊C₈)	—	3	4	no	—
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	1	yes	characteristic
17	C₂×C₃⋊Dic₃	—	2	2	no	—
18	#634	—	4	1	no	—
19	#634	—	4	1	no	—
20	C₃²×C₈	—	4	1	no	—
21	C₃⋊(C₃⋊C₈)	—	4	1	no	—
22	#332	—	9	1	no	Sylow
23	C₂×Dic₆	—	6	4	no	—
24	C₂×C₃×C₈	—	2	4	no	—
25	C₃×C₄²	C₂×D₃	1	4	yes	—
26	C₄⋊Dic₃	—	3	8	no	—
27	C₄⋊Dic₃	—	6	4	no	—
28	C₂×C₃⋊C₈	—	6	4	no	—
29	C₂²×C₃²	C₂²⋊C₄	1	1	yes	characteristic
30	C₃²×C₄	—	2	1	no	—
31	C₃²×C₄	D₈	1	1	yes	characteristic
32	C₃²×C₄	SD₁₆	1	1	yes	characteristic
33	C₃²×C₄	SD₁₆	1	1	yes	characteristic
34	C₃²×C₄	Q₁₆	1	1	yes	characteristic
35	C₃⋊Dic₃	—	4	2	no	—
36	C₄⋊Q₈	—	9	1	no	—
37	C₄⋊C₈	—	9	1	no	—
38	C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
39	C₂×C₃×C₄	C₄×D₃	1	4	yes	—
40	C₂×C₃×C₄	D₁₂	1	4	yes	—
41	C₂×C₃×C₄	#67	1	4	yes	—
42	C₂×Dic₃	—	6	8	no	—
43	Dic₆	—	12	4	no	—
44	Dic₆	—	12	4	no	—
45	C₃×C₈	—	4	4	no	—
46	C₃⋊C₈	—	12	4	no	—
47	C₂×C₃²	#101	1	1	yes	characteristic
48	C₂×C₃²	#102	1	1	yes	characteristic
49	C₂×C₃²	#103	1	1	yes	characteristic
50	C₂×Q₈	—	18	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	18	1	no	—
53	C₄⋊C₄	—	9	2	no	—
54	C₄⋊C₄	—	18	1	no	—
55	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂²×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
57	C₃×C₄	—	2	4	no	—
58	C₃×C₄	#204	1	4	yes	—
59	C₃×C₄	D₂₄	1	4	yes	—
60	C₃×C₄	#214	1	4	yes	—
61	C₃×C₄	#216	1	4	yes	—
62	Dic₃	—	12	8	no	—
63	C₃²	#332	1	1	yes	characteristic, complemented, Sylow
64	Q₈	—	36	1	no	—
65	Q₈	—	36	1	no	—
66	C₂×C₄	—	18	2	no	—
67	C₂×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
69	C₂×C₄	#638	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	36	1	no	—
72	C₂×C₃	#810	1	4	yes	—
73	C₂×C₃	#821	1	4	yes	—
74	C₂×C₃	#824	1	4	yes	—
75	C₂²	#3752	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	36	2	no	—
78	C₄	#3744	1	1	yes	characteristic
79	C₄	#3745	1	1	yes	characteristic
80	C₄	#3754	1	1	yes	characteristic
81	C₄	#3756	1	1	yes	characteristic
82	C₃	#4486	1	4	yes	complemented
83	C₂	#63637	1	1	yes	characteristic
84	C₂	#63648	1	1	yes	characteristic
85	C₂	#63651	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#3752
commutator	C₂×C₃²×C₄	C₂×C₄
Frattini	C₄²	C₂×C₃⋊D₃
Fitting	C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃²	C₂²⋊C₄
cosocle	C₃²×C₄²	C₂²
hypercenter	C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	#332
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
63. C₃²	#332	semidirect
82. C₃	#4486	semidirect

Maximal subgroups

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

subgroup	index
#64078	2
C₃²×C₄⋊C₈	2
C₄⋊(C₃⋊(C₃⋊C₈))	2
#4486	3