Subgroups — C₉×#5229 #13143591

Order 1728.

714 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₉×C₃⋊#95	C₂	1	4	yes	maximal
5	C₃×#5229	C₃	1	1	yes	maximal, characteristic
6	C₉×#391	—	3	1	no	maximal
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
14	C₉×C₄⋊Dic₃	—	2	2	no	—
15	C₄×C₉×Dic₃	—	2	2	no	—
16	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₂²×Q₈×C₉	—	2	1	no	—
20	C₂²×Q₈×C₉	D₃	1	1	yes	characteristic
21	C₃×C₃⋊#95	C₂×C₃	1	4	yes	—
22	C₂×C₉×C₄⋊C₄	—	3	1	no	—
23	C₂×C₄²×C₉	—	3	1	no	—
24	C₉×#95	—	3	4	no	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₃×Q₈×C₉	—	2	8	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	4	no	—
37	C₂×C₉×Dic₃	Q₈	1	4	yes	complemented
38	C₃×#391	—	3	1	no	—
39	#5229	C₉	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₂×C₃²×Q₈	C₃×C₄	1	4	yes	—
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	2	yes	—
46	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
47	C₃×C₄⋊Dic₃	—	2	2	no	—
48	C₃×C₄×Dic₃	—	2	2	no	—
49	C₂×Q₈×C₉	—	2	4	no	—
50	C₂×Q₈×C₉	—	2	4	no	—
51	C₂×Q₈×C₉	—	2	4	no	—
52	C₂×Q₈×C₉	—	2	2	no	—
53	C₂×Q₈×C₉	—	2	2	no	—
54	C₂×Q₈×C₉	Dic₃	1	4	yes	complemented
55	C₂²×C₄×C₉	—	2	2	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	3	2	no	—
58	C₂²×C₄×C₉	—	3	2	no	—
59	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
60	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
61	C₉×C₄⋊C₄	—	6	2	no	—
62	C₄²×C₉	—	6	2	no	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₃×C₄×C₉	—	2	4	no	—
69	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	—
70	C₉×Dic₃	—	2	4	no	—
71	C₉×Dic₃	—	4	2	no	—
72	C₂²×C₃×Q₈	—	2	1	no	—
73	C₂²×C₃×Q₈	C₂×C₉	1	1	yes	characteristic
74	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	—	3	1	no	—
76	C₂×C₃×C₄²	—	3	1	no	—
77	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic
78	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic
79	C₃×#95	—	3	4	no	—
80	C₃⋊#95	C₂×C₉	1	4	yes	—
81	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
82	C₃²×Q₈	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
86	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
88	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	—	2	2	no	—
92	C₂×C₃×Dic₃	—	2	4	no	—
93	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
96	Q₈×C₉	—	2	16	no	—
97	Q₈×C₉	—	2	8	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	4	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	4	no	—
107	C₂×C₄×C₉	—	6	2	no	—
108	C₂×C₄×C₉	—	6	2	no	—
109	C₂×C₄×C₉	—	6	4	no	—
110	C₂×C₄×C₉	—	6	2	no	—
111	C₂×C₄×C₉	C₂×Dic₃	1	4	yes	—
112	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
113	C₂×C₄×C₉	#67	1	2	yes	—
114	C₂×C₄×C₉	#67	1	2	yes	—
115	#391	—	3	1	no	Sylow
116	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
118	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₃×C₉	#124	1	1	yes	characteristic
120	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×Q₈	—	2	4	no	—
122	C₂×C₃×Q₈	—	2	4	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	2	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂×C₃×Q₈	—	2	4	no	—
128	C₂×C₃×Q₈	C₄×C₉	1	4	yes	complemented
129	C₂×C₃×Q₈	C₃×Dic₃	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	3	2	no	—
133	C₂²×C₃×C₄	—	3	2	no	—
134	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
135	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
139	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
140	C₃×C₄⋊C₄	—	6	2	no	—
141	C₃×C₄²	—	6	2	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₄×Dic₃	—	2	2	no	—
144	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
148	C₂²×C₃²	C₃×#41	1	2	yes	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
151	C₃×Dic₃	—	2	4	no	—
152	C₃×Dic₃	—	4	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	2	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	D₃×Q₈	1	2	yes	—
159	C₂²×C₉	#239	1	2	yes	—
160	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
161	C₂²×C₉	C₂×#67	1	1	yes	characteristic
162	C₂²×C₉	C₂×#67	1	1	yes	characteristic
163	C₄×C₉	—	2	8	no	—
164	C₄×C₉	—	2	4	no	—
165	C₄×C₉	—	2	4	no	—
166	C₄×C₉	—	6	4	no	—
167	C₄×C₉	—	12	2	no	—
168	C₄×C₉	#217	1	4	yes	—
169	C₂²×Q₈	D₃×C₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	3	1	no	—
171	C₂×C₄²	—	3	1	no	—
172	#95	—	3	4	no	—
173	C₃×C₉	#391	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
176	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
177	C₃×Q₈	—	2	16	no	—
178	C₃×Q₈	—	2	8	no	—
179	C₃×Q₈	—	2	8	no	—
180	C₂×C₃×C₄	—	2	4	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	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	3	4	no	—
191	C₂×C₃×C₄	—	6	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	4	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
196	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
197	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
198	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
200	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
201	C₂×C₃×C₄	C₃×#67	1	2	yes	—
202	C₂×C₃×C₄	C₃×#67	1	2	yes	—
203	C₂×Dic₃	—	2	2	no	—
204	C₂×Dic₃	—	2	2	no	—
205	C₂×Dic₃	—	2	2	no	—
206	C₂×Dic₃	—	2	4	no	—
207	C₂×Dic₃	Q₈×C₉	1	4	yes	complemented
208	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃²	C₃×C₄×Q₈	1	2	yes	—
210	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
211	C₂×C₃²	C₃×#124	1	1	yes	characteristic
212	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
213	C₂×C₉	—	2	1	no	—
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	#944	1	1	yes	characteristic
219	C₂×C₉	Q₈×Dic₃	1	2	yes	—
220	C₂×C₉	#946	1	2	yes	—
221	C₂×C₉	#947	1	1	yes	characteristic
222	C₂×C₉	C₂×#217	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	C₉×Dic₃	1	4	yes	complemented
226	C₂²×C₄	—	3	2	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
229	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
230	C₄⋊C₄	—	6	2	no	—
231	C₄²	—	6	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
238	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
239	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
240	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
241	C₂²×C₃	C₉×#41	1	2	yes	—
242	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
243	C₂²×C₃	C₃×#239	1	2	yes	—
244	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
245	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
247	C₃×C₄	—	2	8	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	6	4	no	—
252	C₃×C₄	—	12	2	no	—
253	C₃×C₄	C₉×C₂²⋊C₄	1	4	yes	—
254	C₃×C₄	C₃×#217	1	4	yes	—
255	Dic₃	—	2	4	no	—
256	Dic₃	—	4	2	no	—
257	C₃²	C₃×#391	1	1	yes	characteristic
258	C₉	—	2	1	no	—
259	C₉	#5229	1	1	yes	characteristic, direct product factor
260	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	3	4	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	6	2	no	—
269	C₂×C₄	C₂×C₉×Dic₃	1	4	yes	—
270	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
271	C₂×C₄	C₉×#67	1	2	yes	—
272	C₂×C₄	C₉×#67	1	2	yes	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₃	C₄×Q₈×C₉	1	2	yes	—
280	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₃	C₉×#124	1	1	yes	characteristic
282	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
283	C₂×C₃	C₃×#944	1	1	yes	characteristic
284	C₂×C₃	C₃×Q₈×Dic₃	1	2	yes	—
285	C₂×C₃	C₃×#946	1	2	yes	—
286	C₂×C₃	C₃×#947	1	1	yes	characteristic
287	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
288	C₂²	D₃×Q₈×C₉	1	2	yes	—
289	C₂²	C₉×#239	1	2	yes	—
290	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
291	C₂²	C₂×C₉×#67	1	1	yes	characteristic
292	C₂²	C₂×C₉×#67	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	6	4	no	—
295	C₄	—	12	2	no	—
296	C₄	C₉×#217	1	4	yes	—
297	C₃	—	2	1	no	—
298	C₃	C₉×#391	1	1	yes	characteristic, complemented
299	C₃	C₃×#5229	1	1	yes	characteristic
300	C₂	C₉×#944	1	1	yes	characteristic
301	C₂	Q₈×C₉×Dic₃	1	2	yes	—
302	C₂	C₉×#946	1	2	yes	—
303	C₂	C₉×#947	1	1	yes	characteristic
304	C₂	C₂×C₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂²×Q₈×C₉	D₃
hypocenter	C₃	C₉×#391
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂²×Q₈×C₉

Complements

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

N	H ≅ G/N	type
9. C₂×C₃×Q₈×C₉	C₄	semidirect
37. C₂×C₉×Dic₃	Q₈	semidirect
39. #5229	C₉	direct (N × H)
54. C₂×Q₈×C₉	Dic₃	semidirect
128. C₂×C₃×Q₈	C₄×C₉	semidirect
173. C₃×C₉	#391	semidirect
207. C₂×Dic₃	Q₈×C₉	semidirect
225. C₂×Q₈	C₉×Dic₃	semidirect
259. C₉	#5229	direct (N × H)
298. C₃	C₉×#391	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₄×C₉×Dic₃	2
C₉×C₃⋊#95	2
C₃×#5229	3
C₉×#391	3

Subgroups — C9×#5229 #13143591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5229 #13143591