Subgroups — C₉×#4552 #13135651

Order 1728.

346 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	2	yes	maximal
3	C₃×#4552	C₃	1	1	yes	maximal, characteristic
4	C₉×#341	—	3	1	no	maximal
5	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₈×C₉	—	2	2	no	—
7	C₂×C₃×C₈×C₉	C₄	1	2	yes	—
8	C₉×#208	—	2	2	no	—
9	C₉×#208	C₄	1	4	yes	—
10	C₂×C₉×C₃⋊C₈	—	2	2	no	—
11	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×#208	C₂×C₃	1	2	yes	—
13	C₂×C₉×M₁₆	—	3	2	no	—
14	C₂²×C₈×C₉	—	2	1	no	—
15	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
16	C₂³×C₃×C₉	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
21	C₃×C₈×C₉	—	2	2	no	—
22	C₉×C₃⋊C₈	—	2	4	no	—
23	C₃×#341	—	3	1	no	—
24	#4552	C₉	1	1	yes	characteristic, direct product factor
25	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₃²×C₈	—	2	2	no	—
27	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
28	C₃×#208	—	2	2	no	—
29	C₃×#208	C₃×C₄	1	4	yes	—
30	C₂×C₃×C₃⋊C₈	—	2	2	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
33	C₉×M₁₆	—	3	4	no	—
34	C₉×M₁₆	—	6	2	no	—
35	C₂×C₈×C₉	—	2	4	no	—
36	C₂×C₈×C₉	—	2	2	no	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	6	2	no	—
39	C₂×C₈×C₉	Dic₃	1	2	yes	—
40	C₂²×C₃×C₉	—	2	2	no	—
41	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
43	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
44	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
45	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×M₁₆	—	3	2	no	—
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
50	C₂×#208	C₂×C₉	1	2	yes	—
51	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
53	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
56	C₂³×C₉	—	2	1	no	—
57	C₂³×C₉	Dic₆	1	1	yes	characteristic
58	C₃²×C₈	—	2	2	no	—
59	C₃×C₃⋊C₈	—	2	4	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
65	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₄×C₉	#67	1	2	yes	—
68	C₈×C₉	—	2	4	no	—
69	C₈×C₉	—	2	2	no	—
70	C₈×C₉	—	6	4	no	—
71	#341	—	3	1	no	Sylow
72	C₂×C₃×C₉	—	2	2	no	—
73	C₂×C₃×C₉	#95	1	1	yes	characteristic
74	C₂×C₃×C₉	#108	1	2	yes	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
78	C₃×M₁₆	—	3	4	no	—
79	C₃×M₁₆	—	6	2	no	—
80	C₂×C₃×C₈	—	2	4	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	6	2	no	—
85	C₂×C₃×C₈	C₄×C₉	1	2	yes	—
86	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
87	#208	—	2	2	no	—
88	#208	C₄×C₉	1	4	yes	—
89	C₂×C₃⋊C₈	—	2	2	no	—
90	C₂²×C₃²	—	2	2	no	—
91	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
92	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
93	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
94	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
95	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₉	—	2	4	no	—
97	C₂²×C₉	—	2	2	no	—
98	C₂²×C₉	—	2	2	no	—
99	C₂²×C₉	—	2	1	no	—
100	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
101	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
102	C₄×C₉	—	2	1	no	—
103	C₄×C₉	—	2	2	no	—
104	C₄×C₉	—	2	1	no	—
105	C₄×C₉	C₄×Dic₃	1	1	yes	characteristic
106	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
107	C₄×C₉	#217	1	1	yes	characteristic
108	C₂×M₁₆	—	3	2	no	—
109	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
110	C₃×C₉	#341	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	Q₈×C₉	1	1	yes	characteristic
113	C₂³×C₃	C₃×Dic₆	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
119	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
120	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
121	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
122	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
123	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
125	C₂×C₃×C₄	C₃×#67	1	2	yes	—
126	C₃×C₈	—	2	4	no	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	6	4	no	—
130	C₃⋊C₈	—	2	4	no	—
131	C₂×C₃²	—	2	2	no	—
132	C₂×C₃²	C₃×#95	1	1	yes	characteristic
133	C₂×C₃²	C₃×#108	1	2	yes	—
134	C₂×C₉	—	2	4	no	—
135	C₂×C₉	—	2	2	no	—
136	C₂×C₉	—	2	2	no	—
137	C₂×C₉	—	2	1	no	—
138	C₂×C₉	#819	1	2	yes	—
139	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
141	M₁₆	—	3	4	no	—
142	M₁₆	—	6	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	6	2	no	—
145	C₂×C₈	C₉×Dic₃	1	2	yes	—
146	C₂²×C₃	—	2	4	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
152	C₂²×C₃	C₉×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₃	C₃×Dic₃⋊C₄	1	2	yes	—
154	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	C₄²×C₉	1	1	yes	characteristic
159	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
160	C₃×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
161	C₃×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
162	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
163	C₃×C₄	C₃×#217	1	1	yes	characteristic
164	C₃²	C₃×#341	1	1	yes	characteristic
165	C₉	—	2	1	no	—
166	C₉	#4552	1	1	yes	characteristic, direct product factor
167	C₂³	C₉×Dic₆	1	1	yes	characteristic
168	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
169	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
170	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
171	C₂×C₄	C₉×#67	1	2	yes	—
172	C₈	—	2	2	no	—
173	C₈	—	6	4	no	—
174	C₂×C₃	—	2	4	no	—
175	C₂×C₃	—	2	2	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	—	2	2	no	—
179	C₂×C₃	C₉×#95	1	1	yes	characteristic
180	C₂×C₃	C₉×#108	1	2	yes	—
181	C₂×C₃	C₃×#819	1	2	yes	—
182	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	C₉×Dic₃⋊C₄	1	2	yes	—
185	C₂²	C₉×C₄⋊Dic₃	1	1	yes	characteristic
186	C₄	C₄×C₉×Dic₃	1	1	yes	characteristic
187	C₄	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
188	C₄	C₉×#217	1	1	yes	characteristic
189	C₃	—	2	1	no	—
190	C₃	C₉×#341	1	1	yes	characteristic, complemented
191	C₃	C₃×#4552	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	C₉×#819	1	2	yes	—
194	C₂	C₉×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	D₁₂
commutator	C₃×C₄	C₄²×C₉
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₃×C₄⋊C₄
prime radical	C₂³×C₃²	C₃×Q₈
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#341
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
24. #4552	C₉	direct (N × H)
110. C₃×C₉	#341	semidirect
166. C₉	#4552	direct (N × H)
190. C₃	C₉×#341	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₂×C₉×#208	2
C₃×#4552	3
C₉×#341	3

Subgroups — C9×#4552 #13135651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4552 #13135651