Subgroups — C₉×#4898 #13142264

Order 1728.

342 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#909	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#908	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#204	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#901	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#824	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#823	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4898	C₃	1	1	yes	maximal, characteristic
9	C₉×#471	—	3	1	no	maximal
10	C₉×#235	C₂²	1	1	yes	characteristic
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₉×#208	C₂²	1	1	yes	characteristic
17	C₉×Dic₁₂	—	2	1	no	—
18	C₉×D₂₄	—	2	1	no	—
19	C₉×#204	—	2	1	no	—
20	C₉×#204	—	2	1	no	—
21	C₉×#204	—	2	1	no	—
22	C₉×#204	—	2	1	no	—
23	C₉×ΓL(1,𝔽₂₅)	—	2	1	no	—
24	D₃×C₈×C₉	—	2	1	no	—
25	C₃×#909	C₂×C₃	1	1	yes	characteristic
26	C₃×#908	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#204	C₂×C₃	1	1	yes	characteristic
28	C₃×#901	C₂×C₃	1	1	yes	characteristic
29	C₃²×#108	C₂×C₃	1	1	yes	characteristic
30	C₃×#824	C₂×C₃	1	1	yes	characteristic
31	C₃×#823	C₂×C₃	1	1	yes	characteristic
32	C₉×#137	—	3	1	no	—
33	C₉×AGL(1,ℤ/8)	—	3	1	no	—
34	C₂×C₉×SD₁₆	—	3	1	no	—
35	C₉×#131	—	3	1	no	—
36	C₉×#108	—	2	1	no	—
37	C₉×#108	D₃	1	1	yes	characteristic, complemented
38	C₉×#101	—	3	1	no	—
39	C₉×#100	—	3	1	no	—
40	C₂²×D₃×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₉×#67	—	2	1	no	—
43	C₂×C₉×Dic₃	—	2	1	no	—
44	C₉×D₁₂	—	2	1	no	—
45	C₉×D₁₂	—	2	1	no	—
46	C₉×D₁₂	D₄	1	1	yes	characteristic
47	C₄×D₃×C₉	—	2	1	no	—
48	C₉×Dic₆	—	2	1	no	—
49	C₉×Dic₆	—	2	1	no	—
50	C₉×Dic₆	D₄	1	1	yes	characteristic
51	C₃×C₈×C₉	—	2	1	no	—
52	C₃×C₈×C₉	—	2	1	no	—
53	C₃×C₈×C₉	D₄	1	1	yes	characteristic
54	C₃×C₈×C₉	D₄	1	1	yes	characteristic
55	C₉×C₃⋊C₈	—	2	1	no	—
56	C₃×#471	—	3	1	no	—
57	#4898	C₉	1	1	yes	characteristic, direct product factor
58	C₃×#235	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
61	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×#208	C₂²×C₃	1	1	yes	characteristic
65	C₃×Dic₁₂	—	2	1	no	—
66	C₃×D₂₄	—	2	1	no	—
67	C₃×#204	—	2	1	no	—
68	C₃×#204	—	2	1	no	—
69	C₃×#204	—	2	1	no	—
70	C₃×#204	—	2	1	no	—
71	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
72	C₃×D₃×C₈	—	2	1	no	—
73	C₉×#41	—	3	1	no	—
74	C₂×Q₈×C₉	—	3	1	no	—
75	C₂×D₄×C₉	—	3	1	no	—
76	C₉×Q₁₆	—	6	1	no	—
77	C₉×SD₁₆	—	6	1	no	—
78	C₉×SD₁₆	—	6	1	no	—
79	C₉×SD₁₆	—	6	1	no	—
80	C₉×SD₁₆	—	6	1	no	—
81	C₉×D₈	—	6	1	no	—
82	C₉×M₁₆	—	2	1	no	—
83	C₉×M₁₆	—	2	1	no	—
84	C₉×M₁₆	—	3	1	no	—
85	C₉×M₁₆	—	6	1	no	—
86	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
87	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
88	C₂×C₈×C₉	—	2	1	no	—
89	C₂×C₈×C₉	—	6	1	no	—
90	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₉	#41	1	1	yes	characteristic
92	C₂×D₃×C₉	—	2	1	no	—
93	C₂×D₃×C₉	—	4	1	no	—
94	C₂×D₃×C₉	—	8	1	no	—
95	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₉×Dic₃	—	2	1	no	—
98	C₉×Dic₃	—	4	1	no	—
99	C₃×#137	—	3	1	no	—
100	C₃×AGL(1,ℤ/8)	—	3	1	no	—
101	C₂×C₃×SD₁₆	—	3	1	no	—
102	C₃×#131	—	3	1	no	—
103	#909	C₂×C₉	1	1	yes	characteristic, complemented
104	#908	C₂×C₉	1	1	yes	characteristic
105	C₂×#204	C₂×C₉	1	1	yes	characteristic, complemented
106	#901	C₂×C₉	1	1	yes	characteristic, complemented
107	C₃×#108	—	2	1	no	—
108	C₃×#108	C₂×C₉	1	1	yes	characteristic, complemented
109	C₃×#108	C₃×D₃	1	1	yes	characteristic
110	C₃×#101	—	3	1	no	—
111	C₃×#100	—	3	1	no	—
112	#824	C₂×C₉	1	1	yes	characteristic, complemented
113	#823	C₂×C₉	1	1	yes	characteristic, complemented
114	C₂²×C₃×D₃	—	2	1	no	—
115	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₃×#67	—	2	1	no	—
117	C₂×C₃×Dic₃	—	2	1	no	—
118	C₃×D₁₂	—	2	1	no	—
119	C₃×D₁₂	—	2	1	no	—
120	C₃×D₁₂	C₃×D₄	1	1	yes	characteristic
121	C₃×C₄×D₃	—	2	1	no	—
122	C₃×Dic₆	—	2	1	no	—
123	C₃×Dic₆	—	2	1	no	—
124	C₃×Dic₆	C₃×D₄	1	1	yes	characteristic
125	C₂³×C₉	—	6	1	no	—
126	C₃²×C₈	—	2	1	no	—
127	C₃²×C₈	—	2	1	no	—
128	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
129	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
130	C₃×C₃⋊C₈	—	2	1	no	—
131	Q₈×C₉	—	3	1	no	—
132	Q₈×C₉	—	6	1	no	—
133	Q₈×C₉	—	6	1	no	—
134	D₄×C₉	—	3	1	no	—
135	D₄×C₉	—	6	1	no	—
136	D₄×C₉	—	6	1	no	—
137	D₄×C₉	—	6	1	no	—
138	C₂×C₄×C₉	—	2	1	no	—
139	C₂×C₄×C₉	—	6	1	no	—
140	C₂×C₄×C₉	—	6	1	no	—
141	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
142	C₈×C₉	—	2	1	no	—
143	C₈×C₉	—	2	1	no	—
144	C₈×C₉	—	2	1	no	—
145	C₈×C₉	—	2	1	no	—
146	C₈×C₉	—	4	1	no	—
147	C₈×C₉	—	4	1	no	—
148	C₈×C₉	—	6	1	no	—
149	C₈×C₉	D₁₂	1	1	yes	characteristic
150	C₈×C₉	D₁₂	1	1	yes	characteristic
151	#471	—	3	1	no	Sylow
152	C₂×C₃×C₉	—	2	1	no	—
153	C₂×C₃×C₉	#121	1	1	yes	characteristic
154	D₃×C₉	—	4	1	no	—
155	D₃×C₉	—	8	1	no	—
156	C₃×#41	—	3	1	no	—
157	C₂×C₃×Q₈	—	3	1	no	—
158	C₂×C₃×D₄	—	3	1	no	—
159	#235	C₂²×C₉	1	1	yes	characteristic
160	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
161	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
162	C₃×Q₁₆	—	6	1	no	—
163	C₃×SD₁₆	—	6	1	no	—
164	C₃×SD₁₆	—	6	1	no	—
165	C₃×SD₁₆	—	6	1	no	—
166	C₃×SD₁₆	—	6	1	no	—
167	C₃×D₈	—	6	1	no	—
168	C₃×M₁₆	—	2	1	no	—
169	C₃×M₁₆	—	2	1	no	—
170	C₃×M₁₆	—	3	1	no	—
171	C₃×M₁₆	—	6	1	no	—
172	C₃×M₁₆	C₂²×C₉	1	1	yes	characteristic
173	C₃×M₁₆	C₂²×C₉	1	1	yes	characteristic
174	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
175	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
176	C₂×C₃×C₈	—	2	1	no	—
177	C₂×C₃×C₈	—	6	1	no	—
178	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
179	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
180	#208	C₂²×C₉	1	1	yes	characteristic
181	Dic₁₂	—	2	1	no	—
182	D₂₄	—	2	1	no	—
183	#204	—	2	1	no	—
184	#204	—	2	1	no	—
185	#204	—	2	1	no	—
186	#204	—	2	1	no	—
187	ΓL(1,𝔽₂₅)	—	2	1	no	—
188	D₃×C₈	—	2	1	no	—
189	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
190	C₂×C₃×D₃	—	2	1	no	—
191	C₂×C₃×D₃	—	4	1	no	—
192	C₂×C₃×D₃	—	8	1	no	—
193	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₃×Dic₃	—	2	1	no	—
196	C₃×Dic₃	—	4	1	no	—
197	C₂²×C₉	—	2	1	no	—
198	C₂²×C₉	—	6	1	no	—
199	C₂²×C₉	—	12	1	no	—
200	C₂²×C₉	—	24	1	no	—
201	C₂²×C₉	#239	1	1	yes	characteristic
202	C₄×C₉	—	2	1	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	6	1	no	—
205	C₄×C₉	—	12	1	no	—
206	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
207	C₄×C₉	D₃×D₄	1	1	yes	characteristic
208	#137	—	3	1	no	—
209	AGL(1,ℤ/8)	—	3	1	no	—
210	C₂×SD₁₆	—	3	1	no	—
211	#131	—	3	1	no	—
212	#108	D₃×C₉	1	1	yes	characteristic, complemented
213	#101	—	3	1	no	—
214	#100	—	3	1	no	—
215	C₃×C₉	#471	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₃	—	6	1	no	—
217	C₂²×D₃	—	2	1	no	—
218	C₃×Q₈	—	3	1	no	—
219	C₃×Q₈	—	6	1	no	—
220	C₃×Q₈	—	6	1	no	—
221	C₃×D₄	—	3	1	no	—
222	C₃×D₄	—	6	1	no	—
223	C₃×D₄	—	6	1	no	—
224	C₃×D₄	—	6	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	6	1	no	—
227	C₂×C₃×C₄	—	6	1	no	—
228	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
230	#67	—	2	1	no	—
231	C₂×Dic₃	—	2	1	no	—
232	D₁₂	—	2	1	no	—
233	D₁₂	—	2	1	no	—
234	D₁₂	D₄×C₉	1	1	yes	characteristic
235	C₄×D₃	—	2	1	no	—
236	Dic₆	—	2	1	no	—
237	Dic₆	—	2	1	no	—
238	Dic₆	D₄×C₉	1	1	yes	characteristic
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	—	2	1	no	—
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	2	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	4	1	no	—
246	C₃×C₈	—	4	1	no	—
247	C₃×C₈	—	6	1	no	—
248	C₃×C₈	D₄×C₉	1	1	yes	characteristic
249	C₃×C₈	D₄×C₉	1	1	yes	characteristic
250	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
251	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
252	C₃⋊C₈	—	2	1	no	—
253	C₂×C₃²	—	2	1	no	—
254	C₂×C₃²	C₃×#121	1	1	yes	characteristic
255	C₃×D₃	—	4	1	no	—
256	C₃×D₃	—	8	1	no	—
257	C₂×C₉	—	2	1	no	—
258	C₂×C₉	—	2	1	no	—
259	C₂×C₉	—	4	1	no	—
260	C₂×C₉	—	12	1	no	—
261	C₂×C₉	—	24	1	no	—
262	C₂×C₉	#895	1	1	yes	characteristic
263	#41	—	3	1	no	—
264	C₂×Q₈	—	3	1	no	—
265	C₂×D₄	—	3	1	no	—
266	Q₁₆	—	6	1	no	—
267	SD₁₆	—	6	1	no	—
268	SD₁₆	—	6	1	no	—
269	SD₁₆	—	6	1	no	—
270	SD₁₆	—	6	1	no	—
271	D₈	—	6	1	no	—
272	M₁₆	—	3	1	no	—
273	M₁₆	—	6	1	no	—
274	M₁₆	C₂×D₃×C₉	1	1	yes	characteristic
275	M₁₆	C₂×D₃×C₉	1	1	yes	characteristic
276	C₂×C₈	—	6	1	no	—
277	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	6	1	no	—
280	C₂²×C₃	—	12	1	no	—
281	C₂²×C₃	—	24	1	no	—
282	C₂²×C₃	C₉×#41	1	1	yes	characteristic
283	C₂²×C₃	C₃×#239	1	1	yes	characteristic
284	C₂×D₃	—	2	1	no	—
285	C₂×D₃	—	4	1	no	—
286	C₂×D₃	—	8	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	6	1	no	—
290	C₃×C₄	—	12	1	no	—
291	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
292	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
293	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
294	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
295	Dic₃	—	2	1	no	—
296	Dic₃	—	4	1	no	—
297	C₃²	C₃×#471	1	1	yes	characteristic
298	C₉	—	2	1	no	—
299	C₉	#4898	1	1	yes	characteristic, direct product factor
300	C₂³	—	6	1	no	—
301	Q₈	—	3	1	no	—
302	Q₈	—	6	1	no	—
303	Q₈	—	6	1	no	—
304	D₄	—	3	1	no	—
305	D₄	—	6	1	no	—
306	D₄	—	6	1	no	—
307	D₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	6	1	no	—
314	C₈	C₉×D₁₂	1	1	yes	characteristic
315	C₈	C₉×D₁₂	1	1	yes	characteristic
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	1	no	—
320	C₂×C₃	—	12	1	no	—
321	C₂×C₃	—	24	1	no	—
322	C₂×C₃	C₉×#121	1	1	yes	characteristic
323	C₂×C₃	C₃×#895	1	1	yes	characteristic
324	D₃	—	4	1	no	—
325	D₃	—	8	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	—	12	1	no	—
328	C₂²	—	24	1	no	—
329	C₂²	C₉×#239	1	1	yes	characteristic
330	C₄	—	6	1	no	—
331	C₄	—	12	1	no	—
332	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
333	C₄	D₃×D₄×C₉	1	1	yes	characteristic
334	C₃	—	2	1	no	—
335	C₃	C₉×#471	1	1	yes	characteristic, complemented
336	C₃	C₃×#4898	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	12	1	no	—
339	C₂	—	24	1	no	—
340	C₂	C₉×#895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#895
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#108	C₂
socle	C₂×C₃²	C₃×#121
prime radical	C₃×#908	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#108	D₃
hypocenter	C₃	C₉×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#108

Complements

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

N	H ≅ G/N	type
1. C₉×#909	C₂	semidirect
3. C₂×C₉×#204	C₂	semidirect
4. C₉×#901	C₂	semidirect
5. C₃×C₉×#108	C₂	semidirect
6. C₉×#824	C₂	semidirect
7. C₉×#823	C₂	semidirect
37. C₉×#108	D₃	semidirect
57. #4898	C₉	direct (N × H)
103. #909	C₂×C₉	semidirect
105. C₂×#204	C₂×C₉	semidirect
106. #901	C₂×C₉	semidirect
108. C₃×#108	C₂×C₉	semidirect
112. #824	C₂×C₉	semidirect
113. #823	C₂×C₉	semidirect
212. #108	D₃×C₉	semidirect
215. C₃×C₉	#471	semidirect
299. C₉	#4898	direct (N × H)
335. C₃	C₉×#471	semidirect

Maximal subgroups

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

subgroup	index
C₉×#909	2
C₉×#908	2
C₂×C₉×#204	2
C₉×#901	2
C₃×C₉×#108	2
C₉×#824	2
C₉×#823	2
C₃×#4898	3
C₉×#471	3

Subgroups — C9×#4898 #13142264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4898 #13142264