Subgroups — #10592408

Order 576.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64081	C₂	1	1	yes	maximal, characteristic
2	C₃²×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	#63647	C₂	1	2	yes	maximal
4	#63641	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#4709	—	3	4	no	maximal
7	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₈	C₂²	1	2	yes	—
9	C₃²×C₄²	C₂²	1	1	yes	characteristic
10	#3752	C₄	1	2	yes	—
11	#3750	C₄	1	2	yes	—
12	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	#872	—	3	4	no	—
15	C₃×#97	D₃	1	4	yes	complemented
16	(C₂×D₃)⋊C₈	—	3	8	no	—
17	Dic₃⋊C₈	—	3	8	no	—
18	C₄×C₃⋊C₈	—	3	4	no	—
19	C₂²×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
21	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
23	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₃⋊D₃	—	2	2	no	—
25	C₃²×C₈	—	2	2	no	—
26	C₃⋊(C₃⋊C₈)	—	2	2	no	—
27	#433	—	9	1	no	Sylow
28	C₂×C₄×D₃	—	3	4	no	—
29	C₂×C₃×C₈	C₂×D₃	1	8	yes	—
30	C₃×C₄²	C₂×D₃	1	4	yes	—
31	(C₂×D₃)⋊C₄	—	3	8	no	—
32	Dic₃⋊C₄	—	3	8	no	—
33	C₄×Dic₃	—	3	4	no	—
34	C₂×C₃⋊C₈	—	3	8	no	—
35	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃⋊D₃	—	2	1	no	—
37	C₂×C₃⋊D₃	—	2	2	no	—
38	C₃²×C₄	—	2	2	no	—
39	C₃²×C₄	#41	1	2	yes	—
40	C₃⋊Dic₃	—	2	1	no	—
41	C₃⋊Dic₃	—	2	2	no	—
42	#117	—	9	1	no	—
43	C₄⋊C₈	—	9	2	no	—
44	C₂²⋊C₈	—	9	2	no	—
45	#97	C₃⋊D₃	1	1	yes	characteristic, complemented
46	C₄×C₈	—	9	1	no	—
47	C₂²×D₃	—	3	4	no	—
48	C₂×C₃×C₄	C₄×D₃	1	8	yes	—
49	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
50	C₂×Dic₃	—	3	8	no	—
51	C₂×Dic₃	—	3	4	no	—
52	C₄×D₃	—	6	8	no	—
53	C₃×C₈	—	2	8	no	—
54	C₃⋊C₈	—	6	8	no	—
55	C₂×C₃²	#117	1	1	yes	characteristic
56	C₂×C₃²	#131	1	2	yes	—
57	C₃⋊D₃	—	4	1	no	—
58	C₂²×C₄	—	9	1	no	—
59	C₂×C₈	—	9	2	no	—
60	C₂×C₈	C₂×C₃⋊D₃	1	2	yes	—
61	C₄⋊C₄	—	9	2	no	—
62	C₂²⋊C₄	—	9	2	no	—
63	C₄²	—	9	1	no	—
64	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₂²×C₃	C₂×C₄×D₃	1	4	yes	—
66	C₂×D₃	—	6	4	no	—
67	C₂×D₃	—	6	8	no	—
68	C₃×C₄	—	2	8	no	—
69	C₃×C₄	#235	1	8	yes	—
70	Dic₃	—	6	4	no	—
71	Dic₃	—	6	8	no	—
72	C₃²	#433	1	1	yes	characteristic, complemented, Sylow
73	C₂³	—	9	1	no	—
74	C₂×C₄	—	9	2	no	—
75	C₂×C₄	—	9	1	no	—
76	C₂×C₄	—	18	2	no	—
77	C₂×C₄	C₄×C₃⋊D₃	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	#872	1	4	yes	—
82	C₂×C₃	#907	1	8	yes	—
83	D₃	—	12	4	no	—
84	C₂²	—	18	1	no	—
85	C₂²	—	18	2	no	—
86	C₂²	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	18	1	no	—
89	C₄	—	18	2	no	—
90	C₄	#3828	1	2	yes	—
91	C₃	#4709	1	4	yes	complemented
92	C₂	—	36	1	no	—
93	C₂	#64081	1	1	yes	characteristic
94	C₂	#64116	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₄
Fitting	C₃²×#97	C₂
prime radical	C₂²×C₃⋊D₃	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	#97	C₃⋊D₃
hypocenter	C₃²	#433
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₃²×#97	C₂	semidirect
4. #63641	C₂	semidirect
5. C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
15. C₃×#97	D₃	semidirect
45. #97	C₃⋊D₃	semidirect
72. C₃²	#433	semidirect
91. C₃	#4709	semidirect

Maximal subgroups

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

subgroup	index
#64081	2
C₃²×#97	2
#63647	2
#63641	2
C₄×C₃⋊(C₃⋊C₈)	2
#4709	3