Subgroups — #10588911

Order 576.

303 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₉)⋊C₄	C₂	1	3	yes	maximal
3	C₉⋊#95	C₂	1	3	yes	maximal, complemented
4	#5231	—	3	1	no	maximal
5	C₂³×D₉	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₉	—	2	6	no	—
7	C₂²×C₄×C₉	C₂²	1	3	yes	—
8	C₂²×Dic₉	C₂²	1	3	yes	—
9	(C₂×D₉)⋊C₄	—	2	6	no	—
10	C₂²×C₃×Q₈	D₃	1	1	yes	characteristic
11	C₂×(C₂×D₃)⋊C₄	—	3	3	no	—
12	C₃⋊#95	—	3	3	no	—
13	C₂³×C₉	C₂³	1	1	yes	characteristic
14	C₂²×D₉	—	2	3	no	—
15	C₂²×D₉	—	2	3	no	—
16	C₂²×D₉	Q₈	1	2	yes	complemented
17	Q₈×C₉	—	2	8	no	—
18	C₂×C₄×C₉	—	2	6	no	—
19	C₂×C₄×C₉	D₄	1	6	yes	—
20	C₂×Dic₉	—	2	3	no	—
21	C₂×Dic₉	—	2	6	no	—
22	#393	—	9	1	no	Sylow
23	C₂³×D₃	—	3	1	no	—
24	C₂×C₃×Q₈	—	2	6	no	—
25	C₂²×C₃×C₄	C₂×D₃	1	3	yes	—
26	C₂²×Dic₃	—	3	3	no	—
27	(C₂×D₃)⋊C₄	—	6	6	no	—
28	C₂²×C₉	C₂×D₄	1	3	yes	—
29	C₂²×C₉	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₉	#41	1	3	yes	—
31	C₂×D₉	—	2	6	no	—
32	C₂×D₉	—	4	1	no	—
33	C₂×D₉	—	4	3	no	—
34	C₄×C₉	—	2	6	no	—
35	Dic₉	—	4	3	no	—
36	C₂²×Q₈	D₉	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₄	—	9	3	no	—
38	#95	—	9	3	no	—
39	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
40	C₂²×D₃	—	3	2	no	—
41	C₂²×D₃	—	6	3	no	—
42	C₂²×D₃	—	6	3	no	—
43	C₃×Q₈	—	2	8	no	—
44	C₂×C₃×C₄	—	2	6	no	—
45	C₂×C₃×C₄	#67	1	6	yes	—
46	C₂×Dic₃	—	6	6	no	—
47	C₂×Dic₃	—	6	3	no	—
48	C₂×C₉	#120	1	1	yes	characteristic
49	C₂×C₉	C₂²⋊Q₈	1	3	yes	—
50	C₂×C₉	#124	1	3	yes	—
51	D₉	—	4	2	no	—
52	C₂⁴	—	9	1	no	—
53	C₂×Q₈	—	2	6	no	—
54	C₂²×C₄	—	9	3	no	—
55	C₂²×C₄	C₂×D₉	1	3	yes	—
56	C₂²⋊C₄	—	18	6	no	—
57	C₂²×C₃	D₃×Q₈	1	1	yes	characteristic
58	C₂²×C₃	#239	1	3	yes	—
59	C₂²×C₃	C₂×#67	1	3	yes	—
60	C₂×D₃	—	6	6	no	—
61	C₂×D₃	—	12	3	no	—
62	C₂×D₃	—	12	1	no	—
63	C₃×C₄	—	2	6	no	—
64	Dic₃	—	12	3	no	—
65	C₉	#393	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	9	2	no	—
67	C₂³	—	18	3	no	—
68	C₂³	—	18	3	no	—
69	C₂³	C₂²×D₉	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	18	6	no	—
73	C₂×C₄	—	18	3	no	—
74	C₂×C₄	#611	1	6	yes	—
75	C₂×C₃	#946	1	3	yes	—
76	C₂×C₃	#947	1	3	yes	—
77	C₂×C₃	#953	1	1	yes	characteristic
78	D₃	—	12	2	no	—
79	C₂²	—	18	6	no	—
80	C₂²	—	36	3	no	—
81	C₂²	—	36	1	no	—
82	C₂²	Q₈×D₉	1	1	yes	characteristic
83	C₂²	#3701	1	3	yes	—
84	C₂²	C₂×#611	1	3	yes	—
85	C₄	—	2	6	no	—
86	C₄	—	36	3	no	—
87	C₃	#5231	1	1	yes	characteristic
88	C₂	—	36	2	no	—
89	C₂	#63508	1	3	yes	—
90	C₂	#63509	1	3	yes	—
91	C₂	#63515	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₉
commutator, cosocle	C₂³×C₉	C₂³
Frattini, socle	C₂³×C₃	C₂²×D₃
Fitting	C₂²×Q₈×C₉	C₂
prime radical	C₂³×D₉	C₂²
hypercenter	C₂²×Q₈	D₉
hypocenter	C₉	#393
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₉	C₂	semidirect
3. C₉⋊#95	C₂	semidirect
16. C₂²×D₉	Q₈	semidirect
36. C₂²×Q₈	D₉	semidirect
65. C₉	#393	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₉	2
C₂×(C₂×D₉)⋊C₄	2
C₉⋊#95	2
#5231	3