Subgroups — C₉×#4983 #13142575

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×Dic₃⋊C₄	C₂	1	2	yes	maximal
4	C₉×C₃⋊#95	C₂	1	2	yes	maximal
5	C₉×C₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₃×#4983	C₃	1	1	yes	maximal, characteristic
7	C₉×#398	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	—	2	2	no	—
14	C₉×Dic₃⋊C₄	—	2	2	no	—
15	C₉×Dic₃⋊C₄	—	2	4	no	—
16	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
19	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
20	C₃×C₃⋊#95	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×C₄⋊C₄	—	2	1	no	—
22	C₂×C₉×C₄⋊C₄	—	3	2	no	—
23	C₂×C₉×C₄⋊C₄	—	3	1	no	—
24	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
25	C₉×#95	—	3	1	no	—
26	C₉×#95	—	3	2	no	—
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₉×Dic₃	—	2	1	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	—	2	1	no	—
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	—	2	1	no	—
38	C₂×C₉×Dic₃	—	2	2	no	—
39	C₂×C₉×Dic₃	D₄	1	2	yes	—
40	C₂×C₉×Dic₃	Q₈	1	4	yes	—
41	C₃×#398	—	3	1	no	—
42	#4983	C₉	1	1	yes	characteristic, direct product factor
43	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃×Dic₃⋊C₄	—	2	2	no	—
50	C₃×Dic₃⋊C₄	—	2	4	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	3	1	no	—
54	C₂²×C₄×C₉	—	3	2	no	—
55	C₂²×C₄×C₉	—	3	1	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	2	4	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	6	4	no	—
61	C₉×C₄⋊C₄	—	6	2	no	—
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₃×C₉	#41	1	1	yes	characteristic
66	C₂²×C₃×C₉	#41	1	2	yes	—
67	C₃×C₄×C₉	—	2	2	no	—
68	C₃×C₄×C₉	—	4	2	no	—
69	C₉×Dic₃	—	2	2	no	—
70	C₉×Dic₃	—	2	4	no	—
71	C₉×Dic₃	—	4	1	no	—
72	C₂×C₃×C₄⋊C₄	—	2	1	no	—
73	C₂×C₃×C₄⋊C₄	—	3	2	no	—
74	C₂×C₃×C₄⋊C₄	—	3	1	no	—
75	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
77	C₂×Dic₃⋊C₄	C₂×C₉	1	1	yes	characteristic
78	C₂×Dic₃⋊C₄	C₂×C₉	1	2	yes	—
79	C₃×#95	—	3	1	no	—
80	C₃×#95	—	3	2	no	—
81	C₃⋊#95	C₂×C₉	1	2	yes	—
82	C₃⋊#95	C₂×C₉	1	1	yes	characteristic
83	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	1	no	—
91	C₂×C₃×Dic₃	—	2	2	no	—
92	C₂×C₃×Dic₃	—	2	1	no	—
93	C₂×C₃×Dic₃	—	2	1	no	—
94	C₂×C₃×Dic₃	—	2	2	no	—
95	C₂×C₃×Dic₃	C₃×D₄	1	2	yes	—
96	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
97	C₂³×C₉	—	2	1	no	—
98	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	4	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	3	2	no	—
107	C₂×C₄×C₉	—	3	4	no	—
108	C₂×C₄×C₉	—	4	2	no	—
109	C₂×C₄×C₉	—	4	2	no	—
110	C₂×C₄×C₉	—	6	2	no	—
111	C₂×C₄×C₉	—	6	2	no	—
112	C₂×C₄×C₉	—	6	1	no	—
113	C₂×C₄×C₉	—	6	1	no	—
114	C₂×C₄×C₉	—	6	2	no	—
115	C₂×C₄×C₉	—	6	1	no	—
116	C₂×C₄×C₉	#67	1	2	yes	—
117	#398	—	3	1	no	Sylow
118	C₂×C₃×C₉	#121	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₃×C₉	#123	1	1	yes	characteristic
121	C₂×C₃×C₉	#125	1	2	yes	—
122	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	—	3	2	no	—
126	C₂²×C₃×C₄	—	3	1	no	—
127	C₂²×C₃×C₄	—	3	1	no	—
128	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
129	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
133	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
134	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	6	4	no	—
139	C₃×C₄⋊C₄	—	6	2	no	—
140	Dic₃⋊C₄	—	2	2	no	—
141	Dic₃⋊C₄	—	2	4	no	—
142	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
145	C₂²×C₃²	C₃×#41	1	2	yes	—
146	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
147	C₃²×C₄	—	2	2	no	—
148	C₃²×C₄	—	4	2	no	—
149	C₃×Dic₃	—	2	2	no	—
150	C₃×Dic₃	—	2	4	no	—
151	C₃×Dic₃	—	4	1	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	#235	1	2	yes	—
158	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
159	C₂²×C₉	#237	1	1	yes	characteristic
160	C₂²×C₉	D₃×Q₈	1	2	yes	—
161	C₂²×C₉	C₂×#67	1	1	yes	characteristic
162	C₄×C₉	—	2	4	no	—
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	4	4	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	—	6	4	no	—
167	C₄×C₉	—	6	2	no	—
168	C₄×C₉	—	12	1	no	—
169	C₂×C₄⋊C₄	—	3	2	no	—
170	C₂×C₄⋊C₄	—	3	1	no	—
171	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
172	#95	—	3	1	no	—
173	#95	—	3	2	no	—
174	C₃×C₉	#398	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	1	no	—
176	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
177	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	3	2	no	—
190	C₂×C₃×C₄	—	3	4	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	1	no	—
196	C₂×C₃×C₄	—	6	1	no	—
197	C₂×C₃×C₄	—	6	2	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
200	C₂×C₃×C₄	C₃×#67	1	2	yes	—
201	C₂×Dic₃	—	2	2	no	—
202	C₂×Dic₃	—	2	1	no	—
203	C₂×Dic₃	—	2	2	no	—
204	C₂×Dic₃	—	2	1	no	—
205	C₂×Dic₃	—	2	1	no	—
206	C₂×Dic₃	—	2	2	no	—
207	C₂×Dic₃	D₄×C₉	1	2	yes	—
208	C₂×Dic₃	Q₈×C₉	1	4	yes	—
209	C₂×C₃²	C₃×#121	1	1	yes	characteristic
210	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
211	C₂×C₃²	C₃×#123	1	1	yes	characteristic
212	C₂×C₃²	C₃×#125	1	2	yes	—
213	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₉	—	2	2	no	—
215	C₂×C₉	—	2	2	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	#889	1	2	yes	—
220	C₂×C₉	#896	1	2	yes	—
221	C₂×C₉	#929	1	1	yes	characteristic
222	C₂×C₉	#938	1	1	yes	characteristic
223	C₂×C₉	#944	1	1	yes	characteristic
224	C₂²×C₄	—	3	2	no	—
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
228	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	6	4	no	—
231	C₄⋊C₄	—	6	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
238	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
239	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
240	C₂²×C₃	C₉×#41	1	2	yes	—
241	C₂²×C₃	C₉×#41	1	1	yes	characteristic
242	C₂²×C₃	C₃×#235	1	2	yes	—
243	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
244	C₂²×C₃	C₃×#237	1	1	yes	characteristic
245	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
246	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	4	4	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	6	4	no	—
254	C₃×C₄	—	6	2	no	—
255	C₃×C₄	—	12	1	no	—
256	Dic₃	—	2	2	no	—
257	Dic₃	—	2	4	no	—
258	Dic₃	—	4	1	no	—
259	C₃²	C₃×#398	1	1	yes	characteristic
260	C₉	—	2	1	no	—
261	C₉	#4983	1	1	yes	characteristic, direct product factor
262	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	3	4	no	—
269	C₂×C₄	—	6	2	no	—
270	C₂×C₄	—	6	2	no	—
271	C₂×C₄	—	6	1	no	—
272	C₂×C₄	—	6	1	no	—
273	C₂×C₄	—	6	2	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	C₉×#67	1	2	yes	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	C₉×#121	1	1	yes	characteristic
282	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
283	C₂×C₃	C₉×#123	1	1	yes	characteristic
284	C₂×C₃	C₉×#125	1	2	yes	—
285	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
286	C₂×C₃	C₃×#889	1	2	yes	—
287	C₂×C₃	C₃×#896	1	2	yes	—
288	C₂×C₃	C₃×#929	1	1	yes	characteristic
289	C₂×C₃	C₃×#938	1	1	yes	characteristic
290	C₂×C₃	C₃×#944	1	1	yes	characteristic
291	C₂²	C₉×#235	1	2	yes	—
292	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
293	C₂²	C₉×#237	1	1	yes	characteristic
294	C₂²	D₃×Q₈×C₉	1	2	yes	—
295	C₂²	C₂×C₉×#67	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	6	2	no	—
299	C₄	—	6	4	no	—
300	C₄	—	12	1	no	—
301	C₃	—	2	1	no	—
302	C₃	C₉×#398	1	1	yes	characteristic, complemented
303	C₃	C₃×#4983	1	1	yes	characteristic
304	C₂	C₉×#889	1	2	yes	—
305	C₂	C₉×#896	1	2	yes	—
306	C₂	C₉×#929	1	1	yes	characteristic
307	C₂	C₉×#938	1	1	yes	characteristic
308	C₂	C₉×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
42. #4983	C₉	direct (N × H)
174. C₃×C₉	#398	semidirect
261. C₉	#4983	direct (N × H)
302. C₃	C₉×#398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₉×Dic₃⋊C₄	2
C₉×C₃⋊#95	2
C₃×#4983	3
C₉×#398	3

Subgroups — C9×#4983 #13142575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4983 #13142575