Subgroups — #10592767

Order 576.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#64084	C₂	1	1	yes	maximal, characteristic
3	#63637	C₂	1	2	yes	maximal, complemented
4	#63636	C₂	1	2	yes	maximal
5	C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#5068	—	3	4	no	maximal
7	C₂×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
8	C₂×#634	C₂²	1	1	yes	characteristic
9	C₃²×C₄⋊C₄	—	2	2	no	—
10	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃²×C₄²	C₂²	1	1	yes	characteristic
12	#3752	—	2	2	no	—
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×#125	D₃	1	4	yes	complemented
15	#875	—	3	4	no	—
16	#810	—	3	8	no	—
17	#809	—	3	8	no	—
18	C₄×C₃⋊C₈	—	3	4	no	—
19	C₂²×C₃⋊D₃	—	2	1	no	—
20	C₂×C₃²×C₄	—	2	2	no	—
21	C₂×C₃²×C₄	D₄	1	2	yes	—
22	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₃⋊Dic₃	—	2	1	no	—
24	C₄⋊(C₃⋊D₃)	—	2	2	no	—
25	#634	—	2	2	no	—
26	C₃⋊(C₃⋊C₈)	—	2	2	no	—
27	#488	—	9	1	no	Sylow
28	C₂×D₁₂	—	3	4	no	—
29	C₂×Dic₆	—	3	4	no	—
30	C₃×C₄⋊C₄	—	2	8	no	—
31	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
32	C₃×C₄²	C₂×D₃	1	4	yes	—
33	(C₂×D₃)⋊C₄	—	6	8	no	—
34	C₂×C₃⋊C₈	—	3	8	no	—
35	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃⋊D₃	—	4	1	no	—
37	C₂×C₃⋊D₃	—	4	2	no	—
38	C₃²×C₄	—	2	2	no	—
39	C₃²×C₄	—	4	2	no	—
40	C₃²×C₄	#41	1	2	yes	—
41	C₃⋊Dic₃	—	4	1	no	—
42	#125	C₃⋊D₃	1	1	yes	characteristic, complemented
43	#124	—	9	1	no	—
44	#103	—	9	2	no	—
45	#102	—	9	2	no	—
46	C₄×C₈	—	9	1	no	—
47	C₂²×D₃	—	6	4	no	—
48	C₂×C₃×C₄	—	2	8	no	—
49	C₂×C₃×C₄	#67	1	8	yes	—
50	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
51	C₂×Dic₃	—	6	4	no	—
52	D₁₂	—	6	8	no	—
53	Dic₆	—	6	8	no	—
54	C₃⋊C₈	—	6	8	no	—
55	C₂×C₃²	#124	1	1	yes	characteristic
56	C₂×C₃²	#135	1	2	yes	—
57	C₃⋊D₃	—	8	1	no	—
58	C₂×Q₈	—	9	1	no	—
59	C₂×D₄	—	9	1	no	—
60	C₂×C₈	—	9	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
63	C₂²⋊C₄	—	18	2	no	—
64	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₂²×C₃	C₂×#67	1	4	yes	—
66	C₂×D₃	—	12	4	no	—
67	C₂×D₃	—	12	8	no	—
68	C₃×C₄	—	2	8	no	—
69	C₃×C₄	—	4	8	no	—
70	C₃×C₄	#239	1	8	yes	—
71	Dic₃	—	12	4	no	—
72	C₃²	#488	1	1	yes	characteristic, complemented, Sylow
73	C₂³	—	18	1	no	—
74	Q₈	—	18	2	no	—
75	D₄	—	18	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	18	1	no	—
78	C₂×C₄	#638	1	2	yes	—
79	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
80	C₈	—	18	2	no	—
81	C₂×C₃	#947	1	4	yes	—
82	C₂×C₃	#950	1	8	yes	—
83	D₃	—	24	4	no	—
84	C₂²	—	36	1	no	—
85	C₂²	—	36	2	no	—
86	C₂²	C₂×#638	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	4	2	no	—
89	C₄	—	36	1	no	—
90	C₄	#3832	1	2	yes	—
91	C₃	#5068	1	4	yes	complemented
92	C₂	—	72	1	no	—
93	C₂	#64156	1	1	yes	characteristic
94	C₂	#64159	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#125	C₂
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂²
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#488
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃²×#125	C₂	semidirect
3. #63637	C₂	semidirect
5. C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
14. C₃×#125	D₃	semidirect
42. #125	C₃⋊D₃	semidirect
72. C₃²	#488	semidirect
91. C₃	#5068	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#125	2
#64084	2
#63637	2
#63636	2
C₄×C₃⋊(C₃⋊C₈)	2
#5068	3