Subgroups — C₉×#4872 #13142210

Order 1728.

438 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₉×#893	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#809	C₂	1	2	yes	maximal, complemented
6	C₃×#4872	C₃	1	1	yes	maximal, characteristic
7	C₄×C₉×D₈	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₂×C₄×D₃×C₉	—	2	2	no	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₉×D₂₄	C₄	1	4	yes	complemented
16	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
17	C₃×#893	C₂×C₃	1	2	yes	—
18	C₃²×#107	C₂×C₃	1	1	yes	characteristic
19	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
20	C₃×#809	C₂×C₃	1	2	yes	—
21	C₂×C₉×D₈	—	3	1	no	—
22	C₄×D₄×C₉	—	3	2	no	—
23	C₉×#107	—	2	1	no	—
24	C₉×#107	D₃	1	1	yes	characteristic, complemented
25	C₉×#102	—	3	2	no	—
26	C₄×C₈×C₉	—	3	1	no	—
27	C₂²×D₃×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
31	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
32	C₉×D₁₂	—	2	2	no	—
33	C₉×D₁₂	C₂×C₄	1	4	yes	complemented
34	C₄×D₃×C₉	—	4	4	no	—
35	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
36	C₉×C₃⋊C₈	—	2	1	no	—
37	C₃×C₄×D₈	—	3	1	no	—
38	#4872	C₉	1	1	yes	characteristic, direct product factor
39	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
40	C₂×C₃×C₄×D₃	—	2	2	no	—
41	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
43	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×D₂₄	C₃×C₄	1	4	yes	—
47	C₂×D₄×C₉	—	3	2	no	—
48	C₂²×C₄×C₉	—	6	2	no	—
49	C₉×D₈	—	3	4	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	—	3	1	no	—
52	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
55	C₉×C₂²⋊C₄	—	6	2	no	—
56	C₄²×C₉	—	3	1	no	—
57	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂×D₃×C₉	—	2	4	no	—
59	C₂×D₃×C₉	—	4	2	no	—
60	C₂×D₃×C₉	—	4	2	no	—
61	C₃×C₄×C₉	—	4	2	no	—
62	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	#41	1	1	yes	characteristic
64	C₉×Dic₃	—	2	1	no	—
65	C₉×Dic₃	D₈	1	2	yes	complemented
66	C₂×C₃×D₈	—	3	1	no	—
67	C₃×C₄×D₄	—	3	2	no	—
68	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic
69	#893	C₂×C₉	1	2	yes	complemented
70	C₃×#107	—	2	1	no	—
71	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
72	C₃×#107	C₃×D₃	1	1	yes	characteristic
73	C₃×#102	—	3	2	no	—
74	C₃×C₄×C₈	—	3	1	no	—
75	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
76	#809	C₂×C₉	1	2	yes	complemented
77	C₂²×C₃×D₃	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
80	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
81	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
82	C₃×D₁₂	—	2	2	no	—
83	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	—
84	C₃×C₄×D₃	—	4	4	no	—
85	C₂³×C₉	—	6	2	no	—
86	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
87	C₃×C₃⋊C₈	—	2	1	no	—
88	D₄×C₉	—	3	4	no	—
89	D₄×C₉	—	6	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	3	1	no	—
93	C₂×C₄×C₉	—	3	1	no	—
94	C₂×C₄×C₉	—	4	2	no	—
95	C₂×C₄×C₉	—	12	4	no	—
96	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
97	C₈×C₉	—	2	2	no	—
98	C₈×C₉	—	6	1	no	—
99	C₈×C₉	C₄×D₃	1	2	yes	complemented
100	C₄×D₈	—	3	1	no	Sylow
101	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₃×C₉	#135	1	1	yes	characteristic
104	D₃×C₉	—	4	4	no	—
105	C₂×C₃×D₄	—	3	2	no	—
106	C₂²×C₃×C₄	—	6	2	no	—
107	C₂×D₁₂	C₂²×C₉	1	2	yes	—
108	C₂×C₄×D₃	—	2	2	no	—
109	C₃×D₈	—	3	4	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	3	1	no	—
112	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
116	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
117	C₃×C₂²⋊C₄	—	6	2	no	—
118	C₃×C₄²	—	3	1	no	—
119	(C₂×D₃)⋊C₄	—	2	2	no	—
120	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
122	D₂₄	C₄×C₉	1	4	yes	complemented
123	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₃×D₃	—	2	4	no	—
125	C₂×C₃×D₃	—	4	2	no	—
126	C₂×C₃×D₃	—	4	2	no	—
127	C₃²×C₄	—	4	2	no	—
128	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₃²×C₄	C₃×#41	1	1	yes	characteristic
130	C₃×Dic₃	—	2	1	no	—
131	C₃×Dic₃	C₃×D₈	1	2	yes	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	6	4	no	—
134	C₂²×C₉	—	12	2	no	—
135	C₂²×C₉	—	12	2	no	—
136	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	2	1	no	—
139	C₄×C₉	—	3	2	no	—
140	C₄×C₉	—	4	2	no	—
141	C₄×C₉	—	6	1	no	—
142	C₄×C₉	—	8	2	no	—
143	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
144	C₄×C₉	#239	1	1	yes	characteristic
145	C₂×D₈	—	3	1	no	—
146	C₄×D₄	—	3	2	no	—
147	#107	D₃×C₉	1	1	yes	characteristic, complemented
148	#102	—	3	2	no	—
149	C₄×C₈	—	3	1	no	—
150	C₃×C₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	—	6	2	no	—
152	C₂²×D₃	—	2	2	no	—
153	C₃×D₄	—	3	4	no	—
154	C₃×D₄	—	6	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	1	no	—
159	C₂×C₃×C₄	—	3	1	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	12	4	no	—
162	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
164	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
165	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
166	D₁₂	—	2	2	no	—
167	D₁₂	C₂×C₄×C₉	1	4	yes	complemented
168	C₄×D₃	—	4	4	no	—
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	—	6	1	no	—
171	C₃×C₈	C₂×C₄×C₉	1	2	yes	complemented
172	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
173	C₃⋊C₈	—	2	1	no	—
174	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
175	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
176	C₂×C₃²	C₃×#135	1	1	yes	characteristic
177	C₃×D₃	—	4	4	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	2	1	no	—
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	12	4	no	—
182	C₂×C₉	#893	1	1	yes	characteristic
183	C₂×C₉	D₃×D₈	1	1	yes	characteristic
184	C₂×C₉	#919	1	1	yes	characteristic
185	C₂×D₄	—	3	2	no	—
186	C₂²×C₄	—	6	2	no	—
187	D₈	—	3	4	no	—
188	C₂×C₈	—	3	1	no	—
189	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
190	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
191	C₂²⋊C₄	—	6	2	no	—
192	C₄²	—	3	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	6	4	no	—
195	C₂²×C₃	—	12	2	no	—
196	C₂²×C₃	—	12	2	no	—
197	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
198	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
199	C₂×D₃	—	2	4	no	—
200	C₂×D₃	—	4	2	no	—
201	C₂×D₃	—	4	2	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	3	2	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	—	6	1	no	—
208	C₃×C₄	—	8	2	no	—
209	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
210	C₃×C₄	C₉×#41	1	1	yes	characteristic
211	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
212	C₃×C₄	C₃×#239	1	1	yes	characteristic
213	Dic₃	—	2	1	no	—
214	Dic₃	C₉×D₈	1	2	yes	complemented
215	C₃²	C₃×C₄×D₈	1	1	yes	characteristic
216	C₉	—	2	1	no	—
217	C₉	#4872	1	1	yes	characteristic, direct product factor
218	C₂³	—	6	2	no	—
219	D₄	—	3	4	no	—
220	D₄	—	6	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	12	4	no	—
225	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
226	C₈	—	6	1	no	—
227	C₈	C₄×D₃×C₉	1	2	yes	complemented
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	12	4	no	—
232	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
233	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
234	C₂×C₃	C₉×#135	1	1	yes	characteristic
235	C₂×C₃	C₃×#893	1	1	yes	characteristic
236	C₂×C₃	C₃×D₃×D₈	1	1	yes	characteristic
237	C₂×C₃	C₃×#919	1	1	yes	characteristic
238	D₃	—	4	4	no	—
239	C₂²	—	6	4	no	—
240	C₂²	—	12	2	no	—
241	C₂²	—	12	2	no	—
242	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
243	C₄	—	3	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	6	1	no	—
246	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
247	C₄	C₉×#239	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	C₄×C₉×D₈	1	1	yes	characteristic, complemented
250	C₃	C₃×#4872	1	1	yes	characteristic
251	C₂	—	12	4	no	—
252	C₂	C₉×#893	1	1	yes	characteristic
253	C₂	D₃×C₉×D₈	1	1	yes	characteristic
254	C₂	C₉×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#107	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₂₄	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₄×C₉×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#107

Complements

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

N	H ≅ G/N	type
2. C₉×#893	C₂	semidirect
3. C₃×C₉×#107	C₂	semidirect
4. C₈×C₉×Dic₃	C₂	semidirect
5. C₉×#809	C₂	semidirect
15. C₉×D₂₄	C₄	semidirect
24. C₉×#107	D₃	semidirect
33. C₉×D₁₂	C₂×C₄	semidirect
35. C₃×C₈×C₉	C₂×C₄	semidirect
38. #4872	C₉	direct (N × H)
65. C₉×Dic₃	D₈	semidirect
69. #893	C₂×C₉	semidirect
71. C₃×#107	C₂×C₉	semidirect
75. C₈×Dic₃	C₂×C₉	semidirect
76. #809	C₂×C₉	semidirect
99. C₈×C₉	C₄×D₃	semidirect
122. D₂₄	C₄×C₉	semidirect
147. #107	D₃×C₉	semidirect
150. C₃×C₉	C₄×D₈	semidirect
167. D₁₂	C₂×C₄×C₉	semidirect
171. C₃×C₈	C₂×C₄×C₉	semidirect
214. Dic₃	C₉×D₈	semidirect
217. C₉	#4872	direct (N × H)
227. C₈	C₄×D₃×C₉	semidirect
249. C₃	C₄×C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×D₂₄	2
C₉×#893	2
C₃×C₉×#107	2
C₈×C₉×Dic₃	2
C₉×#809	2
C₃×#4872	3
C₄×C₉×D₈	3

Subgroups — C9×#4872 #13142210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4872 #13142210