Subgroups — C₃×#10588672 #13142591

Order 1728.

602 conjugacy classes of subgroups.

The subgroup lattice has 352 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, complemented
2	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	2	yes	maximal, complemented
5	C₃×C₉⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4992	—	3	1	no	maximal
7	#10588672	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃×D₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
12	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	—	2	2	no	—
14	C₃×(C₂×D₉)⋊C₄	—	2	4	no	—
15	C₃×(C₂×D₉)⋊C₄	—	2	2	no	—
16	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
18	C₂×C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
19	C₃×C₃⋊#95	—	3	2	no	—
20	C₃×C₃⋊#95	—	3	1	no	—
21	C₂×C₉×C₄⋊C₄	—	2	1	no	—
22	C₂×C₉×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	2	yes	—
24	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₉⋊#95	C₂×C₃	1	2	yes	complemented
26	C₉⋊#95	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂²×C₃×D₉	—	2	1	no	—
29	C₂²×C₃×D₉	—	2	1	no	—
30	C₂²×C₃×D₉	—	2	2	no	—
31	C₂²×C₃×D₉	—	2	2	no	—
32	C₂²×C₃×D₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	—	2	2	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	—	2	2	no	—
37	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
38	C₂×C₃×Dic₉	—	2	2	no	—
39	C₂×C₃×Dic₉	—	2	1	no	—
40	C₂×C₃×Dic₉	—	2	2	no	—
41	C₂×C₃×Dic₉	—	2	2	no	—
42	C₂×C₃×Dic₉	—	2	2	no	—
43	C₃×#397	—	9	1	no	—
44	#4992	—	3	1	no	—
45	C₂³×C₃×D₃	—	3	1	no	—
46	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
47	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	—	3	2	no	—
49	C₂²×C₃×Dic₃	—	3	1	no	—
50	C₂³×D₉	C₂²×C₃	1	1	yes	characteristic
51	C₃²×C₄⋊C₄	—	2	2	no	—
52	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
53	C₃×(C₂×D₃)⋊C₄	—	6	2	no	—
54	C₂²×C₄×C₉	—	2	1	no	—
55	C₂²×C₄×C₉	—	2	2	no	—
56	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
58	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
59	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
60	C₉×C₄⋊C₄	—	2	4	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	(C₂×D₉)⋊C₄	—	2	4	no	—
63	(C₂×D₉)⋊C₄	—	2	2	no	—
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	#41	1	1	yes	characteristic
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₂²×C₃×C₉	#41	1	2	yes	—
69	C₂×C₃×D₉	—	2	2	no	—
70	C₂×C₃×D₉	—	2	4	no	—
71	C₂×C₃×D₉	—	4	1	no	—
72	C₂×C₃×D₉	—	4	1	no	—
73	C₂×C₃×D₉	—	4	2	no	—
74	C₃×C₄×C₉	—	2	2	no	—
75	C₃×C₄×C₉	—	4	2	no	—
76	C₃×Dic₉	—	4	2	no	—
77	C₃×Dic₉	—	4	1	no	—
78	C₂×C₃×C₄⋊C₄	—	2	1	no	—
79	C₂×C₃×C₄⋊C₄	D₉	1	1	yes	characteristic, complemented
80	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
81	C₂×C₃×C₂²⋊C₄	—	9	2	no	—
82	C₂×C₃×C₂²⋊C₄	—	9	1	no	—
83	C₂×(C₂×D₃)⋊C₄	—	3	2	no	—
84	C₂×(C₂×D₃)⋊C₄	—	3	1	no	—
85	C₃×#95	—	9	2	no	—
86	C₃×#95	—	9	1	no	—
87	C₃⋊#95	—	3	2	no	—
88	C₃⋊#95	—	3	1	no	—
89	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
90	C₂²×C₃×D₃	—	3	2	no	—
91	C₂²×C₃×D₃	—	6	2	no	—
92	C₂²×C₃×D₃	—	6	1	no	—
93	C₂²×C₃×D₃	—	6	2	no	—
94	C₂²×C₃×D₃	—	6	1	no	—
95	C₂×C₃²×C₄	—	2	2	no	—
96	C₂×C₃²×C₄	—	2	2	no	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	#67	1	2	yes	—
100	C₂×C₃×Dic₃	—	6	2	no	—
101	C₂×C₃×Dic₃	—	6	2	no	—
102	C₂×C₃×Dic₃	—	6	2	no	—
103	C₂×C₃×Dic₃	—	6	2	no	—
104	C₂×C₃×Dic₃	—	6	1	no	—
105	C₂³×C₉	—	2	1	no	—
106	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
107	C₂²×D₉	—	2	1	no	—
108	C₂²×D₉	—	2	2	no	—
109	C₂²×D₉	—	2	1	no	—
110	C₂²×D₉	—	2	2	no	—
111	C₂²×D₉	C₃×D₄	1	2	yes	—
112	C₂×C₄×C₉	—	2	4	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	4	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	4	2	no	—
121	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
122	C₂×Dic₉	—	2	2	no	—
123	C₂×Dic₉	—	2	1	no	—
124	C₂×Dic₉	—	2	2	no	—
125	C₂×Dic₉	—	2	2	no	—
126	C₂×Dic₉	—	2	2	no	—
127	#397	—	9	1	no	Sylow
128	C₂×C₃×C₉	#121	1	1	yes	characteristic
129	C₂×C₃×C₉	#123	1	2	yes	—
130	C₂×C₃×C₉	#123	1	1	yes	characteristic
131	C₂×C₃×C₉	#124	1	1	yes	characteristic
132	C₂×C₃×C₉	#126	1	2	yes	—
133	C₃×D₉	—	4	2	no	—
134	C₂⁴×C₃	—	9	1	no	—
135	C₂³×D₃	—	3	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	—	2	2	no	—
138	C₂²×C₃×C₄	—	9	2	no	—
139	C₂²×C₃×C₄	—	9	1	no	—
140	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
141	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
143	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂²×Dic₃	—	3	2	no	—
145	C₂²×Dic₃	—	3	1	no	—
146	C₃×C₄⋊C₄	—	2	4	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₂²⋊C₄	—	18	4	no	—
150	C₃×C₂²⋊C₄	—	18	2	no	—
151	(C₂×D₃)⋊C₄	—	6	4	no	—
152	(C₂×D₃)⋊C₄	—	6	2	no	—
153	C₂²×C₃²	#235	1	2	yes	—
154	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
155	C₂²×C₃²	#237	1	1	yes	characteristic
156	C₂²×C₃²	#239	1	2	yes	—
157	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
158	C₂×C₃×D₃	—	6	4	no	—
159	C₂×C₃×D₃	—	6	2	no	—
160	C₂×C₃×D₃	—	12	2	no	—
161	C₂×C₃×D₃	—	12	1	no	—
162	C₂×C₃×D₃	—	12	1	no	—
163	C₃²×C₄	—	2	2	no	—
164	C₃²×C₄	—	4	2	no	—
165	C₃×Dic₃	—	12	2	no	—
166	C₃×Dic₃	—	12	1	no	—
167	C₂²×C₉	—	2	2	no	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	2	2	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂²×C₉	C₃×#41	1	2	yes	—
175	C₂²×C₉	C₃×#41	1	1	yes	characteristic
176	C₂²×C₉	C₃×#41	1	2	yes	—
177	C₂×D₉	—	2	2	no	—
178	C₂×D₉	—	2	4	no	—
179	C₂×D₉	—	4	1	no	—
180	C₂×D₉	—	4	1	no	—
181	C₂×D₉	—	4	2	no	—
182	C₄×C₉	—	2	4	no	—
183	C₄×C₉	—	2	2	no	—
184	C₄×C₉	—	4	4	no	—
185	C₄×C₉	—	4	2	no	—
186	Dic₉	—	4	2	no	—
187	Dic₉	—	4	1	no	—
188	C₂×C₄⋊C₄	C₃×D₉	1	1	yes	characteristic, complemented
189	C₂×C₂²⋊C₄	—	9	2	no	—
190	C₂×C₂²⋊C₄	—	9	1	no	—
191	#95	—	9	2	no	—
192	#95	—	9	1	no	—
193	C₃×C₉	#397	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	9	2	no	—
196	C₂³×C₃	—	18	2	no	—
197	C₂³×C₃	—	18	1	no	—
198	C₂³×C₃	—	18	2	no	—
199	C₂³×C₃	—	18	1	no	—
200	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
201	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
202	C₂²×D₃	—	3	2	no	—
203	C₂²×D₃	—	6	2	no	—
204	C₂²×D₃	—	6	1	no	—
205	C₂²×D₃	—	6	2	no	—
206	C₂²×D₃	—	6	1	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	4	2	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	18	2	no	—
221	C₂×C₃×C₄	—	18	2	no	—
222	C₂×C₃×C₄	—	18	2	no	—
223	C₂×C₃×C₄	—	18	2	no	—
224	C₂×C₃×C₄	—	18	1	no	—
225	C₂×C₃×C₄	#611	1	2	yes	—
226	C₂×C₃×C₄	C₃×#67	1	2	yes	—
227	C₂×Dic₃	—	6	2	no	—
228	C₂×Dic₃	—	6	2	no	—
229	C₂×Dic₃	—	6	2	no	—
230	C₂×Dic₃	—	6	2	no	—
231	C₂×Dic₃	—	6	1	no	—
232	C₂×C₃²	#894	1	2	yes	—
233	C₂×C₃²	#898	1	2	yes	—
234	C₂×C₃²	#929	1	1	yes	characteristic
235	C₂×C₃²	#938	1	1	yes	characteristic
236	C₂×C₃²	#947	1	1	yes	characteristic
237	C₃×D₃	—	12	2	no	—
238	C₂×C₉	—	2	2	no	—
239	C₂×C₉	—	2	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	C₃×#121	1	1	yes	characteristic
244	C₂×C₉	C₃×#123	1	2	yes	—
245	C₂×C₉	C₃×#123	1	1	yes	characteristic
246	C₂×C₉	C₃×#124	1	1	yes	characteristic
247	C₂×C₉	C₃×#126	1	2	yes	—
248	D₉	—	4	2	no	—
249	C₂⁴	—	9	1	no	—
250	C₂²×C₄	—	9	2	no	—
251	C₂²×C₄	—	9	1	no	—
252	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
253	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	18	4	no	—
256	C₂²⋊C₄	—	18	2	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	18	2	no	—
263	C₂²×C₃	—	18	4	no	—
264	C₂²×C₃	—	36	1	no	—
265	C₂²×C₃	—	36	1	no	—
266	C₂²×C₃	—	36	2	no	—
267	C₂²×C₃	#3697	1	2	yes	—
268	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
269	C₂²×C₃	#3699	1	1	yes	characteristic
270	C₂²×C₃	#3701	1	2	yes	—
271	C₂²×C₃	C₂×#611	1	1	yes	characteristic
272	C₂²×C₃	C₃×#235	1	2	yes	—
273	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
274	C₂²×C₃	C₃×#237	1	1	yes	characteristic
275	C₂²×C₃	C₃×#239	1	2	yes	—
276	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
277	C₂×D₃	—	6	4	no	—
278	C₂×D₃	—	6	2	no	—
279	C₂×D₃	—	12	2	no	—
280	C₂×D₃	—	12	1	no	—
281	C₂×D₃	—	12	1	no	—
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	4	4	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	—	36	1	no	—
289	C₃×C₄	—	36	2	no	—
290	Dic₃	—	12	2	no	—
291	Dic₃	—	12	1	no	—
292	C₃²	#4992	1	1	yes	characteristic
293	C₉	—	2	1	no	—
294	C₉	C₃×#397	1	1	yes	characteristic, complemented
295	C₂³	—	9	2	no	—
296	C₂³	—	18	2	no	—
297	C₂³	—	18	1	no	—
298	C₂³	—	18	2	no	—
299	C₂³	—	18	1	no	—
300	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	18	2	no	—
306	C₂×C₄	—	18	2	no	—
307	C₂×C₄	—	18	2	no	—
308	C₂×C₄	—	18	2	no	—
309	C₂×C₄	—	18	1	no	—
310	C₂×C₄	C₃×#611	1	2	yes	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	36	2	no	—
317	C₂×C₃	#63456	1	2	yes	—
318	C₂×C₃	#63460	1	2	yes	—
319	C₂×C₃	#63491	1	1	yes	characteristic
320	C₂×C₃	#63500	1	1	yes	characteristic
321	C₂×C₃	#63509	1	1	yes	characteristic
322	C₂×C₃	C₃×#894	1	2	yes	—
323	C₂×C₃	C₃×#898	1	2	yes	—
324	C₂×C₃	C₃×#929	1	1	yes	characteristic
325	C₂×C₃	C₃×#938	1	1	yes	characteristic
326	C₂×C₃	C₃×#947	1	1	yes	characteristic
327	D₃	—	12	2	no	—
328	C₂²	—	18	2	no	—
329	C₂²	—	18	4	no	—
330	C₂²	—	36	1	no	—
331	C₂²	—	36	1	no	—
332	C₂²	—	36	2	no	—
333	C₂²	C₃×#3697	1	2	yes	—
334	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
335	C₂²	C₃×#3699	1	1	yes	characteristic
336	C₂²	C₃×#3701	1	2	yes	—
337	C₂²	C₂×C₃×#611	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	36	1	no	—
341	C₄	—	36	2	no	—
342	C₃	—	2	1	no	—
343	C₃	#10588672	1	1	yes	characteristic, direct product factor
344	C₃	C₃×#4992	1	1	yes	characteristic
345	C₂	—	36	2	no	—
346	C₂	C₃×#63456	1	2	yes	—
347	C₂	C₃×#63460	1	2	yes	—
348	C₂	C₃×#63491	1	1	yes	characteristic
349	C₂	C₃×#63500	1	1	yes	characteristic
350	C₂	C₃×#63509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂³×C₃×D₉	C₂²
hypercenter	C₂×C₃×C₄⋊C₄	D₉
hypocenter	C₉	C₃×#397
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
4. C₃×C₉⋊#95	C₂	semidirect
5. C₃×C₉⋊#95	C₂	semidirect
7. #10588672	C₃	direct (N × H)
22. C₂×C₉×C₄⋊C₄	C₂×C₃	semidirect
25. C₉⋊#95	C₂×C₃	semidirect
26. C₉⋊#95	C₂×C₃	semidirect
79. C₂×C₃×C₄⋊C₄	D₉	semidirect
188. C₂×C₄⋊C₄	C₃×D₉	semidirect
193. C₃×C₉	#397	semidirect
294. C₉	C₃×#397	semidirect
343. C₃	#10588672	direct (N × H)

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×C₃×(C₂×D₉)⋊C₄	2
C₃×C₉⋊#95	2
C₃×#4992	3
#10588672	3

Subgroups — C3×#10588672 #13142591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588672 #13142591