Subgroups — C₉×#4794 #13141853

Order 1728.

342 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#947	C₂	1	1	yes	maximal, characteristic
2	C₉×#888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	1	yes	maximal, characteristic
6	C₉×#815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#809	C₂	1	1	yes	maximal, characteristic
8	C₃×#4794	C₃	1	1	yes	maximal, characteristic
9	C₉×#489	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	—	2	1	no	—
13	C₂×C₃×C₈×C₉	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×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#947	C₂×C₃	1	1	yes	characteristic
22	C₃×#888	C₂×C₃	1	1	yes	characteristic
23	C₃²×#103	C₂×C₃	1	1	yes	characteristic
24	C₃×#835	C₂×C₃	1	1	yes	characteristic
25	C₃×#821	C₂×C₃	1	1	yes	characteristic
26	C₃×#815	C₂×C₃	1	1	yes	characteristic
27	C₃×#809	C₂×C₃	1	1	yes	characteristic
28	C₉×C₄⋊Q₈	—	3	1	no	—
29	C₉×#124	—	3	1	no	—
30	C₉×#103	—	2	1	no	—
31	C₉×#103	—	3	1	no	—
32	C₉×#103	D₃	1	1	yes	characteristic, complemented
33	C₉×#102	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₉×#97	—	3	1	no	—
36	C₂²×D₃×C₉	—	2	1	no	—
37	C₃×Q₈×C₉	—	2	1	no	—
38	C₃×Q₈×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
44	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
45	C₉×D₁₂	—	2	1	no	—
46	C₉×D₁₂	—	2	1	no	—
47	C₉×Dic₆	—	4	2	no	—
48	C₃×C₈×C₉	—	2	1	no	—
49	C₉×C₃⋊C₈	—	2	1	no	—
50	C₃×#489	—	3	1	no	—
51	#4794	C₉	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×Dic₆	—	2	1	no	—
55	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
58	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
59	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
60	C₃×Dic₃⋊C₄	—	2	1	no	—
61	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×Q₈×C₉	—	2	1	no	—
64	C₂×Q₈×C₉	—	6	1	no	—
65	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
66	C₂×D₄×C₉	—	3	1	no	—
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	—	3	1	no	—
69	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	3	1	no	—
72	C₉×C₄⋊C₄	—	6	1	no	—
73	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₉×C₂²⋊C₄	—	6	1	no	—
75	C₉×C₂²⋊C₄	—	6	1	no	—
76	C₄²×C₉	—	3	1	no	—
77	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×D₃×C₉	—	4	1	no	—
79	C₂×D₃×C₉	—	4	1	no	—
80	C₂×D₃×C₉	—	4	1	no	—
81	C₃×C₄×C₉	—	4	1	no	—
82	C₃×C₄×C₉	—	4	1	no	—
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₃×C₄×C₉	#41	1	1	yes	characteristic
85	C₉×Dic₃	—	2	1	no	—
86	C₉×Dic₃	—	2	1	no	—
87	C₉×Dic₃	—	4	1	no	—
88	C₃×C₄⋊Q₈	—	3	1	no	—
89	C₃×#124	—	3	1	no	—
90	#947	C₂×C₉	1	1	yes	characteristic
91	#888	C₂×C₉	1	1	yes	characteristic, complemented
92	C₃×#103	—	2	1	no	—
93	C₃×#103	—	3	1	no	—
94	C₃×#103	C₂×C₉	1	1	yes	characteristic, complemented
95	C₃×#103	C₃×D₃	1	1	yes	characteristic
96	C₃×#102	—	3	1	no	—
97	C₃×#102	—	3	1	no	—
98	C₃×#97	—	3	1	no	—
99	#835	C₂×C₉	1	1	yes	characteristic, complemented
100	#821	C₂×C₉	1	1	yes	characteristic
101	#815	C₂×C₉	1	1	yes	characteristic, complemented
102	#809	C₂×C₉	1	1	yes	characteristic
103	C₂²×C₃×D₃	—	2	1	no	—
104	C₃²×Q₈	—	2	1	no	—
105	C₃²×Q₈	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	1	no	—
108	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	—	2	1	no	—
110	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
111	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
112	C₃×D₁₂	—	2	1	no	—
113	C₃×D₁₂	—	2	1	no	—
114	C₃×Dic₆	—	4	2	no	—
115	C₂³×C₉	—	6	1	no	—
116	C₃²×C₈	—	2	1	no	—
117	C₃×C₃⋊C₈	—	2	1	no	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	2	1	no	—
120	Q₈×C₉	—	2	1	no	—
121	Q₈×C₉	—	4	1	no	—
122	Q₈×C₉	—	12	2	no	—
123	D₄×C₉	—	6	1	no	—
124	D₄×C₉	—	6	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	3	1	no	—
129	C₂×C₄×C₉	—	3	1	no	—
130	C₂×C₄×C₉	—	4	1	no	—
131	C₂×C₄×C₉	—	4	1	no	—
132	C₂×C₄×C₉	—	6	1	no	—
133	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
134	C₈×C₉	—	2	1	no	—
135	C₈×C₉	—	4	1	no	—
136	C₈×C₉	—	6	1	no	—
137	#489	—	3	1	no	Sylow
138	C₂×C₃×C₉	#124	1	1	yes	characteristic
139	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₃×C₉	#137	1	1	yes	characteristic
141	D₃×C₉	—	8	1	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	—	6	1	no	—
144	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
146	C₂×C₃×D₄	—	3	1	no	—
147	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
148	C₂×Dic₆	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	3	1	no	—
151	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
152	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	—	3	1	no	—
155	C₃×C₄⋊C₄	—	6	1	no	—
156	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
157	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
158	C₃×C₂²⋊C₄	—	6	1	no	—
159	C₃×C₂²⋊C₄	—	6	1	no	—
160	C₃×C₄²	—	3	1	no	—
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	(C₂×D₃)⋊C₄	—	2	1	no	—
163	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
164	Dic₃⋊C₄	—	2	1	no	—
165	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
167	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×D₃	—	4	1	no	—
169	C₂×C₃×D₃	—	4	1	no	—
170	C₂×C₃×D₃	—	4	1	no	—
171	C₃²×C₄	—	4	1	no	—
172	C₃²×C₄	—	4	1	no	—
173	C₃²×C₄	C₃×#41	1	1	yes	characteristic
174	C₃²×C₄	C₃×#41	1	1	yes	characteristic
175	C₃×Dic₃	—	2	1	no	—
176	C₃×Dic₃	—	2	1	no	—
177	C₃×Dic₃	—	4	1	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	12	1	no	—
180	C₂²×C₉	—	12	1	no	—
181	C₂²×C₉	—	12	1	no	—
182	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	4	2	no	—
186	C₄×C₉	—	4	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	—	6	1	no	—
189	C₄×C₉	—	6	1	no	—
190	C₄×C₉	—	8	1	no	—
191	C₄×C₉	—	12	1	no	—
192	C₄×C₉	#235	1	1	yes	characteristic
193	C₄×C₉	#237	1	1	yes	characteristic
194	C₄⋊Q₈	—	3	1	no	—
195	#124	—	3	1	no	—
196	#103	—	3	1	no	—
197	#103	D₃×C₉	1	1	yes	characteristic, complemented
198	#102	—	3	1	no	—
199	#102	—	3	1	no	—
200	#97	—	3	1	no	—
201	C₃×C₉	#489	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	6	1	no	—
203	C₂²×D₃	—	2	1	no	—
204	C₃×Q₈	—	2	2	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	2	1	no	—
207	C₃×Q₈	—	2	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	4	1	no	—
210	C₃×Q₈	—	12	2	no	—
211	C₃×D₄	—	6	1	no	—
212	C₃×D₄	—	6	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	3	1	no	—
219	C₂×C₃×C₄	—	3	1	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	6	1	no	—
223	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
227	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
228	D₁₂	—	2	1	no	—
229	D₁₂	—	2	1	no	—
230	Dic₆	—	4	2	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	6	1	no	—
235	C₃⋊C₈	—	2	1	no	—
236	C₂×C₃²	C₃×#124	1	1	yes	characteristic
237	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂×C₃²	C₃×#137	1	1	yes	characteristic
239	C₃×D₃	—	8	1	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	24	1	no	—
244	C₂×C₉	#885	1	1	yes	characteristic
245	C₂×C₉	#914	1	1	yes	characteristic
246	C₂×C₉	#918	1	1	yes	characteristic
247	C₂×Q₈	—	6	1	no	—
248	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
249	C₂×D₄	—	3	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
255	C₂²⋊C₄	—	6	1	no	—
256	C₂²⋊C₄	—	6	1	no	—
257	C₄²	—	3	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	12	1	no	—
260	C₂²×C₃	—	12	1	no	—
261	C₂²×C₃	—	12	1	no	—
262	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
263	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
264	C₂×D₃	—	4	1	no	—
265	C₂×D₃	—	4	1	no	—
266	C₂×D₃	—	4	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	4	1	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	—	6	1	no	—
276	C₃×C₄	—	8	1	no	—
277	C₃×C₄	—	12	1	no	—
278	C₃×C₄	C₉×#41	1	1	yes	characteristic
279	C₃×C₄	C₉×#41	1	1	yes	characteristic
280	C₃×C₄	C₃×#235	1	1	yes	characteristic
281	C₃×C₄	C₃×#237	1	1	yes	characteristic
282	Dic₃	—	2	1	no	—
283	Dic₃	—	2	1	no	—
284	Dic₃	—	4	1	no	—
285	C₃²	C₃×#489	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	#4794	1	1	yes	characteristic, direct product factor
288	C₂³	—	6	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	12	2	no	—
292	D₄	—	6	1	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	6	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	24	1	no	—
306	C₂×C₃	C₉×#124	1	1	yes	characteristic
307	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₃	C₉×#137	1	1	yes	characteristic
309	C₂×C₃	C₃×#885	1	1	yes	characteristic
310	C₂×C₃	C₃×#914	1	1	yes	characteristic
311	C₂×C₃	C₃×#918	1	1	yes	characteristic
312	D₃	—	8	1	no	—
313	C₂²	—	12	1	no	—
314	C₂²	—	12	1	no	—
315	C₂²	—	12	1	no	—
316	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	6	1	no	—
320	C₄	—	6	1	no	—
321	C₄	—	12	1	no	—
322	C₄	C₉×#235	1	1	yes	characteristic
323	C₄	C₉×#237	1	1	yes	characteristic
324	C₃	—	2	1	no	—
325	C₃	C₉×#489	1	1	yes	characteristic, complemented
326	C₃	C₃×#4794	1	1	yes	characteristic
327	C₂	—	24	1	no	—
328	C₂	C₉×#885	1	1	yes	characteristic
329	C₂	C₉×#914	1	1	yes	characteristic
330	C₂	C₉×#918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#103	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#103	D₃
hypocenter	C₃	C₉×#489
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#103

Complements

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

N	H ≅ G/N	type
2. C₉×#888	C₂	semidirect
3. C₃×C₉×#103	C₂	semidirect
4. C₉×#835	C₂	semidirect
6. C₉×#815	C₂	semidirect
32. C₉×#103	D₃	semidirect
51. #4794	C₉	direct (N × H)
91. #888	C₂×C₉	semidirect
94. C₃×#103	C₂×C₉	semidirect
99. #835	C₂×C₉	semidirect
101. #815	C₂×C₉	semidirect
197. #103	D₃×C₉	semidirect
201. C₃×C₉	#489	semidirect
287. C₉	#4794	direct (N × H)
325. C₃	C₉×#489	semidirect

Maximal subgroups

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

subgroup	index
C₉×#947	2
C₉×#888	2
C₃×C₉×#103	2
C₉×#835	2
C₉×#821	2
C₉×#815	2
C₉×#809	2
C₃×#4794	3
C₉×#489	3

Subgroups — C9×#4794 #13141853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4794 #13141853