Subgroups — #13144399

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720571	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720571	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720516	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592710	—	3	1	no	maximal
9	#10588688	—	3	3	no	maximal
10	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₃×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₃×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	#88532	—	2	2	no	—
18	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
24	C₂×#3748	—	3	1	no	—
25	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
26	#63657	—	3	1	no	—
27	#63657	—	3	1	no	—
28	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
29	#63630	—	3	1	no	—
30	C₄×D₄×C₉	D₃	1	3	yes	—
31	C₂×#3667	—	3	3	no	—
32	C₂²×C₉⋊C₈	—	3	3	no	—
33	#63389	—	3	3	no	—
34	#63389	—	3	3	no	—
35	C₄⋊(C₉⋊C₈)	—	3	3	no	—
36	#63362	—	3	3	no	—
37	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
39	C₃×D₄×C₉	—	2	2	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	—	2	1	no	—
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₃⋊(C₉⋊C₈)	—	2	1	no	—
50	C₃⋊(C₉⋊C₈)	—	2	1	no	—
51	C₃⋊(C₉⋊C₈)	—	2	1	no	—
52	C₃⋊(C₉⋊C₈)	D₄	1	2	yes	complemented
53	#5011	—	9	3	no	—
54	#5011	—	9	1	no	—
55	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
56	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
59	C₃²×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
60	C₃²×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
61	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
62	#3748	—	6	2	no	—
63	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
64	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
65	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
67	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
68	C₂×D₄×C₉	Dic₃	1	3	yes	—
69	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
70	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
71	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
72	C₉×C₂²⋊C₄	Dic₃	1	3	yes	—
73	C₉×C₂²⋊C₄	Dic₃	1	3	yes	—
74	C₄²×C₉	C₂×D₃	1	3	yes	—
75	#3667	—	6	6	no	—
76	C₂×C₉⋊C₈	—	3	3	no	—
77	C₂×C₉⋊C₈	—	3	3	no	—
78	C₂×C₉⋊C₈	—	3	3	no	—
79	C₂×C₉⋊C₈	—	3	3	no	—
80	C₂×C₉⋊C₈	—	6	6	no	—
81	C₂²×C₃×C₉	—	2	1	no	—
82	C₂²×C₃×C₉	—	2	1	no	—
83	C₂²×C₃×C₉	—	2	1	no	—
84	C₂²×C₃×C₉	—	2	1	no	—
85	C₂²×C₃×C₉	—	2	1	no	—
86	C₂²×C₃×C₉	M₁₆	1	2	yes	complemented
87	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄×C₉	—	2	1	no	—
89	C₃×C₄×C₉	—	2	1	no	—
90	C₃×C₄×C₉	—	2	1	no	—
91	C₃×C₄×C₉	—	2	1	no	—
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₃×C₄×C₉	#41	1	1	yes	characteristic
94	C₃×C₄×D₄	D₉	1	3	yes	—
95	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
96	C₂×#208	—	9	3	no	—
97	C₂×#208	—	9	1	no	—
98	C₂²×C₃⋊C₈	—	9	3	no	—
99	C₂²×C₃⋊C₈	—	9	1	no	—
100	#830	—	9	3	no	—
101	#830	—	9	3	no	—
102	#830	—	9	1	no	—
103	#830	—	9	1	no	—
104	C₄⋊(C₃⋊C₈)	—	9	3	no	—
105	C₄⋊(C₃⋊C₈)	—	9	1	no	—
106	#803	—	9	3	no	—
107	#803	—	9	1	no	—
108	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
109	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
110	C₃²×D₄	—	2	2	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	1	no	—
113	C₂×C₃²×C₄	—	2	1	no	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
118	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂³×C₉	C₂×Dic₃	1	3	yes	—
121	C₂³×C₉	C₂×Dic₃	1	3	yes	—
122	C₃⋊(C₃⋊C₈)	—	3	2	no	—
123	C₃⋊(C₃⋊C₈)	—	6	1	no	—
124	C₃⋊(C₃⋊C₈)	—	6	1	no	—
125	C₃⋊(C₃⋊C₈)	—	6	1	no	—
126	D₄×C₉	—	2	6	no	—
127	C₂×C₄×C₉	—	2	3	no	—
128	C₂×C₄×C₉	—	2	3	no	—
129	C₂×C₄×C₉	—	2	3	no	—
130	C₂×C₄×C₉	—	2	3	no	—
131	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
132	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
133	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
134	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
135	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
136	C₉⋊C₈	—	3	6	no	—
137	C₉⋊C₈	—	6	3	no	—
138	C₉⋊C₈	—	6	3	no	—
139	C₉⋊C₈	—	6	3	no	—
140	#435	—	27	1	no	Sylow
141	C₂×C₃×C₉	—	2	2	no	—
142	C₂×C₃×C₉	—	4	1	no	—
143	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₃×C₉	#131	1	1	yes	characteristic
146	C₂×C₃×D₄	Dic₉	1	3	yes	—
147	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
149	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
150	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
152	C₃×C₄⋊C₄	Dic₉	1	3	yes	—
153	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	Dic₉	1	3	yes	—
155	C₃×C₂²⋊C₄	Dic₉	1	3	yes	—
156	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
157	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
158	C₃×C₄²	C₂×D₉	1	3	yes	—
159	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
160	#208	—	18	2	no	—
161	#208	—	18	6	no	—
162	C₂×C₃⋊C₈	—	9	3	no	—
163	C₂×C₃⋊C₈	—	9	1	no	—
164	C₂×C₃⋊C₈	—	9	3	no	—
165	C₂×C₃⋊C₈	—	9	3	no	—
166	C₂×C₃⋊C₈	—	9	3	no	—
167	C₂×C₃⋊C₈	—	9	1	no	—
168	C₂×C₃⋊C₈	—	9	1	no	—
169	C₂×C₃⋊C₈	—	9	1	no	—
170	C₂×C₃⋊C₈	—	18	2	no	—
171	C₂×C₃⋊C₈	—	18	6	no	—
172	C₂²×C₃²	—	2	1	no	—
173	C₂²×C₃²	—	2	1	no	—
174	C₂²×C₃²	—	2	1	no	—
175	C₂²×C₃²	—	2	1	no	—
176	C₂²×C₃²	—	2	1	no	—
177	C₂²×C₃²	#208	1	2	yes	—
178	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
179	C₃²×C₄	—	2	1	no	—
180	C₃²×C₄	—	2	1	no	—
181	C₃²×C₄	—	2	1	no	—
182	C₃²×C₄	—	2	1	no	—
183	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
184	C₃²×C₄	#237	1	1	yes	characteristic
185	C₂²×C₉	—	2	3	no	—
186	C₂²×C₉	—	2	3	no	—
187	C₂²×C₉	—	2	3	no	—
188	C₂²×C₉	—	2	3	no	—
189	C₂²×C₉	—	2	3	no	—
190	C₂²×C₉	#208	1	6	yes	complemented
191	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
192	C₄×C₉	—	2	3	no	—
193	C₄×C₉	—	2	3	no	—
194	C₄×C₉	—	2	3	no	—
195	C₄×C₉	—	2	3	no	—
196	C₄×C₉	D₃×D₄	1	3	yes	—
197	C₄×C₉	#237	1	3	yes	—
198	C₂×M₁₆	—	27	1	no	—
199	C₂²×C₈	—	27	1	no	—
200	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic
201	C₄⋊C₈	—	27	1	no	—
202	C₂²⋊C₈	—	27	1	no	—
203	C₂²⋊C₈	—	27	1	no	—
204	#97	—	27	1	no	—
205	C₃×C₉	#435	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	C₂×Dic₉	1	3	yes	—
207	C₂³×C₃	C₂×Dic₉	1	3	yes	—
208	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
209	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
210	C₃×D₄	—	2	6	no	—
211	C₃×D₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	3	no	—
213	C₂×C₃×C₄	—	2	3	no	—
214	C₂×C₃×C₄	—	2	3	no	—
215	C₂×C₃×C₄	—	2	3	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
221	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
222	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
223	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
224	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
225	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
230	C₃⋊C₈	—	9	6	no	—
231	C₃⋊C₈	—	9	2	no	—
232	C₃⋊C₈	—	18	3	no	—
233	C₃⋊C₈	—	18	3	no	—
234	C₃⋊C₈	—	18	3	no	—
235	C₃⋊C₈	—	18	1	no	—
236	C₃⋊C₈	—	18	1	no	—
237	C₃⋊C₈	—	18	1	no	—
238	C₂×C₃²	—	2	2	no	—
239	C₂×C₃²	—	4	1	no	—
240	C₂×C₃²	C₂×#208	1	1	yes	characteristic
241	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
242	C₂×C₃²	#948	1	1	yes	characteristic
243	C₂×C₉	—	2	6	no	—
244	C₂×C₉	—	4	3	no	—
245	C₂×C₉	C₂×#208	1	3	yes	—
246	C₂×C₉	D₄×Dic₃	1	3	yes	—
247	C₂×C₉	#948	1	3	yes	—
248	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
251	M₁₆	—	54	2	no	—
252	C₂×C₈	—	27	1	no	—
253	C₂×C₈	—	27	1	no	—
254	C₂×C₈	—	27	1	no	—
255	C₂×C₈	—	27	1	no	—
256	C₂×C₈	—	54	2	no	—
257	C₄⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
258	C₂²⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
259	C₂²⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
260	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
261	C₂²×C₃	—	2	3	no	—
262	C₂²×C₃	—	2	3	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	3	no	—
266	C₂²×C₃	—	2	3	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	3	no	—
271	C₂²×C₃	#3667	1	6	yes	complemented
272	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
273	C₂²×C₃	#3748	1	2	yes	—
274	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
275	C₃×C₄	—	2	3	no	—
276	C₃×C₄	—	2	3	no	—
277	C₃×C₄	—	2	3	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	3	no	—
283	C₃×C₄	D₄×D₉	1	3	yes	—
284	C₃×C₄	#3699	1	3	yes	—
285	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
286	C₃×C₄	#3830	1	1	yes	characteristic
287	C₃²	#5011	1	1	yes	characteristic
288	C₉	#5011	1	3	yes	complemented
289	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
290	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
291	D₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
298	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
301	C₈	—	27	2	no	—
302	C₈	—	54	1	no	—
303	C₈	—	54	1	no	—
304	C₈	—	54	1	no	—
305	C₂×C₃	—	2	6	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	4	3	no	—
308	C₂×C₃	—	4	1	no	—
309	C₂×C₃	C₂×#3667	1	3	yes	—
310	C₂×C₃	D₄×Dic₉	1	3	yes	—
311	C₂×C₃	#63510	1	3	yes	—
312	C₂×C₃	C₂×#3748	1	1	yes	characteristic
313	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
314	C₂×C₃	#64157	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	#88532	1	2	yes	complemented
321	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
327	C₄	#88742	1	1	yes	characteristic
328	C₃	#10588688	1	3	yes	complemented
329	C₃	#10592710	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	4	1	no	—
332	C₂	C₂×#88532	1	1	yes	characteristic
333	C₂	D₄×C₃⋊Dic₉	1	1	yes	characteristic
334	C₂	#11721865	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	#435
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88532	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. #11720571	C₂	semidirect
5. #11720571	C₂	semidirect
6. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
7. #11720516	C₂	semidirect
52. C₃⋊(C₉⋊C₈)	D₄	semidirect
86. C₂²×C₃×C₉	M₁₆	semidirect
190. C₂²×C₉	#208	semidirect
205. C₃×C₉	#435	semidirect
271. C₂²×C₃	#3667	semidirect
288. C₉	#5011	semidirect
320. C₂²	#88532	semidirect
328. C₃	#10588688	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₂×#88532	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720571	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#11720516	2
#10592710	3
#10588688	3