Subgroups — C₉×#4881 #13142201

Order 1728.

390 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₉×#892	C₂	1	1	yes	maximal, characteristic
3	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#808	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4881	C₃	1	1	yes	maximal, characteristic
9	C₉×#428	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	—	2	1	no	—
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₄×C₉×Dic₃	—	2	1	no	—
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
22	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
23	C₃×#892	C₂×C₃	1	1	yes	characteristic
24	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₃²×#107	C₂×C₃	1	1	yes	characteristic
26	C₃×#817	C₂×C₃	1	1	yes	characteristic
27	C₃×#808	C₂×C₃	1	1	yes	characteristic
28	C₃×#807	C₂×C₃	1	1	yes	characteristic
29	C₂×C₉×M₁₆	—	3	1	no	—
30	C₉×#117	—	3	1	no	—
31	C₂×C₉×C₄⋊C₄	—	3	1	no	—
32	C₉×#107	—	2	1	no	—
33	C₉×#107	—	3	1	no	—
34	C₉×#107	D₃	1	1	yes	characteristic, complemented
35	C₉×#106	—	3	1	no	—
36	C₉×#106	—	3	1	no	—
37	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
44	C₄×D₃×C₉	—	4	2	no	—
45	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
46	C₄×D₃×C₉	Q₈	1	2	yes	—
47	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
48	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
49	C₃×#428	—	3	1	no	—
50	#4881	C₉	1	1	yes	characteristic, direct product factor
51	C₂×C₃×C₄×D₃	—	2	1	no	—
52	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
57	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₃×Dic₃⋊C₄	—	2	1	no	—
60	C₃×C₄×Dic₃	—	2	1	no	—
61	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	—
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₂²×C₄×C₉	—	6	1	no	—
65	C₉×M₁₆	—	3	4	no	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	3	1	no	—
68	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
69	C₉×C₄⋊C₄	—	2	1	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	3	1	no	—
72	C₉×C₄⋊C₄	—	3	1	no	—
73	C₉×C₄⋊C₄	—	6	1	no	—
74	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
75	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
76	C₉×C₂²⋊C₄	—	6	1	no	—
77	C₄²×C₉	—	6	1	no	—
78	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	—	2	1	no	—
81	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
82	C₃×C₄×C₉	—	4	1	no	—
83	C₃×C₄×C₉	—	4	1	no	—
84	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
86	C₉×Dic₃	—	4	1	no	—
87	C₉×Dic₃	—	4	1	no	—
88	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
89	C₂×C₃×M₁₆	—	3	1	no	—
90	C₃×#117	—	3	1	no	—
91	C₂×C₃×C₄⋊C₄	—	3	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	C₂×C₉	1	1	yes	characteristic
93	#892	C₂×C₉	1	1	yes	characteristic
94	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
95	C₃×#107	—	2	1	no	—
96	C₃×#107	—	3	1	no	—
97	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
98	C₃×#107	C₃×D₃	1	1	yes	characteristic
99	C₃×#106	—	3	1	no	—
100	C₃×#106	—	3	1	no	—
101	#817	C₂×C₉	1	1	yes	characteristic, complemented
102	#808	C₂×C₉	1	1	yes	characteristic, complemented
103	#807	C₂×C₉	1	1	yes	characteristic, complemented
104	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₃×Dic₃	—	2	1	no	—
109	C₂×C₃×Dic₃	—	2	1	no	—
110	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
111	C₃×C₄×D₃	—	4	2	no	—
112	C₃×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
113	C₃×C₄×D₃	C₃×Q₈	1	2	yes	—
114	C₂³×C₉	—	3	1	no	—
115	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
116	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	3	1	no	—
121	C₂×C₄×C₉	—	3	2	no	—
122	C₂×C₄×C₉	—	3	2	no	—
123	C₂×C₄×C₉	—	4	1	no	—
124	C₂×C₄×C₉	—	4	1	no	—
125	C₂×C₄×C₉	—	6	1	no	—
126	C₂×C₄×C₉	—	6	1	no	—
127	C₂×C₄×C₉	—	12	2	no	—
128	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
129	C₈×C₉	—	2	2	no	—
130	C₈×C₉	—	3	2	no	—
131	C₈×C₉	C₄×D₃	1	2	yes	complemented
132	#428	—	3	1	no	Sylow
133	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₃×C₉	#137	1	1	yes	characteristic
136	D₃×C₉	—	2	2	no	—
137	C₂²×C₃×C₄	—	3	1	no	—
138	C₂²×C₃×C₄	—	6	1	no	—
139	C₂×C₄×D₃	—	2	1	no	—
140	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
141	C₃×M₁₆	—	3	4	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	3	1	no	—
144	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	3	1	no	—
149	C₃×C₄⋊C₄	—	3	1	no	—
150	C₃×C₄⋊C₄	—	6	1	no	—
151	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
152	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
154	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
155	C₃×C₂²⋊C₄	—	6	1	no	—
156	C₃×C₄²	—	6	1	no	—
157	(C₂×D₃)⋊C₄	—	2	1	no	—
158	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
159	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
160	Dic₃⋊C₄	—	2	1	no	—
161	C₄×Dic₃	—	2	1	no	—
162	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
163	ΓL(1,𝔽₂₅)	C₄×C₉	1	4	yes	complemented
164	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×D₃	—	2	1	no	—
166	C₂×C₃×D₃	—	2	1	no	—
167	C₂×C₃×D₃	C₃×C₄⋊C₄	1	2	yes	—
168	C₃²×C₄	—	4	1	no	—
169	C₃²×C₄	—	4	1	no	—
170	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
172	C₃×Dic₃	—	4	1	no	—
173	C₃×Dic₃	—	4	1	no	—
174	C₃×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	3	2	no	—
177	C₂²×C₉	—	6	1	no	—
178	C₂²×C₉	—	6	1	no	—
179	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	3	2	no	—
183	C₄×C₉	—	4	1	no	—
184	C₄×C₉	—	4	1	no	—
185	C₄×C₉	—	8	1	no	—
186	C₄×C₉	—	8	1	no	—
187	C₄×C₉	—	12	1	no	—
188	C₄×C₉	—	12	1	no	—
189	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
190	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
191	C₂×M₁₆	—	3	1	no	—
192	#117	—	3	1	no	—
193	C₂×C₄⋊C₄	—	3	1	no	—
194	#107	—	3	1	no	—
195	#107	D₃×C₉	1	1	yes	characteristic, complemented
196	#106	—	3	1	no	—
197	#106	—	3	1	no	—
198	C₃×C₉	#428	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	3	1	no	—
200	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	3	1	no	—
207	C₂×C₃×C₄	—	3	2	no	—
208	C₂×C₃×C₄	—	3	2	no	—
209	C₂×C₃×C₄	—	4	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	6	1	no	—
212	C₂×C₃×C₄	—	6	1	no	—
213	C₂×C₃×C₄	—	12	2	no	—
214	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂×Dic₃	—	2	1	no	—
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
219	C₄×D₃	—	4	2	no	—
220	C₄×D₃	C₂×C₄×C₉	1	2	yes	—
221	C₄×D₃	Q₈×C₉	1	2	yes	—
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	—	3	2	no	—
224	C₃×C₈	C₂×C₄×C₉	1	2	yes	complemented
225	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
226	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	complemented
227	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂×C₃²	C₃×#137	1	1	yes	characteristic
230	C₃×D₃	—	2	2	no	—
231	C₂×C₉	—	2	1	no	—
232	C₂×C₉	—	2	1	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	6	2	no	—
235	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₉	#911	1	1	yes	characteristic
237	C₂×C₉	#918	1	1	yes	characteristic
238	C₂²×C₄	—	3	1	no	—
239	C₂²×C₄	—	6	1	no	—
240	M₁₆	—	3	4	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
243	C₄⋊C₄	—	3	1	no	—
244	C₄⋊C₄	—	3	1	no	—
245	C₄⋊C₄	—	6	1	no	—
246	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	6	1	no	—
249	C₄²	—	6	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	3	2	no	—
252	C₂²×C₃	—	6	1	no	—
253	C₂²×C₃	—	6	1	no	—
254	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
255	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	2	1	no	—
258	C₂×D₃	C₉×C₄⋊C₄	1	2	yes	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	3	2	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	—	4	1	no	—
266	C₃×C₄	—	8	1	no	—
267	C₃×C₄	—	8	1	no	—
268	C₃×C₄	—	12	1	no	—
269	C₃×C₄	—	12	1	no	—
270	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
271	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
272	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
273	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
274	Dic₃	—	4	1	no	—
275	Dic₃	—	4	1	no	—
276	Dic₃	C₉×C₄⋊C₄	1	2	yes	—
277	C₃²	C₃×#428	1	1	yes	characteristic
278	C₉	—	2	1	no	—
279	C₉	#4881	1	1	yes	characteristic, direct product factor
280	C₂³	—	3	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	3	1	no	—
284	C₂×C₄	—	3	2	no	—
285	C₂×C₄	—	3	2	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
290	C₈	—	3	2	no	—
291	C₈	C₄×D₃×C₉	1	2	yes	complemented
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	6	2	no	—
296	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
297	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂×C₃	C₉×#137	1	1	yes	characteristic
299	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
300	C₂×C₃	C₃×#911	1	1	yes	characteristic
301	C₂×C₃	C₃×#918	1	1	yes	characteristic
302	D₃	—	2	2	no	—
303	C₂²	—	3	2	no	—
304	C₂²	—	6	1	no	—
305	C₂²	—	6	1	no	—
306	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
307	C₄	—	3	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	12	1	no	—
311	C₄	—	12	1	no	—
312	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
313	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
314	C₃	—	2	1	no	—
315	C₃	C₉×#428	1	1	yes	characteristic, complemented
316	C₃	C₃×#4881	1	1	yes	characteristic
317	C₂	—	6	2	no	—
318	C₂	D₃×C₉×C₄⋊C₄	1	1	yes	characteristic
319	C₂	C₉×#911	1	1	yes	characteristic
320	C₂	C₉×#918	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	C₂²×C₃×D₃	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#428
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
4. C₃×C₉×#107	C₂	semidirect
5. C₉×#817	C₂	semidirect
6. C₉×#808	C₂	semidirect
7. C₉×#807	C₂	semidirect
21. C₉×ΓL(1,𝔽₂₅)	C₄	semidirect
34. C₉×#107	D₃	semidirect
47. C₃×C₈×C₉	C₂×C₄	semidirect
48. C₉×C₃⋊C₈	C₂×C₄	semidirect
50. #4881	C₉	direct (N × H)
97. C₃×#107	C₂×C₉	semidirect
101. #817	C₂×C₉	semidirect
102. #808	C₂×C₉	semidirect
103. #807	C₂×C₉	semidirect
131. C₈×C₉	C₄×D₃	semidirect
163. ΓL(1,𝔽₂₅)	C₄×C₉	semidirect
195. #107	D₃×C₉	semidirect
198. C₃×C₉	#428	semidirect
224. C₃×C₈	C₂×C₄×C₉	semidirect
226. C₃⋊C₈	C₂×C₄×C₉	semidirect
279. C₉	#4881	direct (N × H)
291. C₈	C₄×D₃×C₉	semidirect
315. C₃	C₉×#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×ΓL(1,𝔽₂₅)	2
C₉×#892	2
D₃×C₉×C₄⋊C₄	2
C₃×C₉×#107	2
C₉×#817	2
C₉×#808	2
C₉×#807	2
C₃×#4881	3
C₉×#428	3

Subgroups — C9×#4881 #13142201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4881 #13142201