Subgroups — #13138442

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6194	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×(C₂×D₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720273	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590718	—	3	1	no	maximal
9	#611⋊C₈	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
12	C₂×#6194	C₄	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₃)⋊Dic₉	C₄	1	1	yes	characteristic
19	(C₂×D₉)⋊Dic₃	C₄	1	1	yes	characteristic
20	#88442	C₄	1	1	yes	characteristic
21	D₃×C₉⋊C₈	—	2	2	no	—
22	D₉×C₃⋊C₈	—	2	2	no	—
23	C₄×#625	—	3	1	no	—
24	C₂×D₃×C₃⋊C₈	—	3	1	no	—
25	C₂×D₃×C₃⋊C₈	—	3	1	no	—
26	C₈×C₃⋊Dic₃	—	3	1	no	—
27	C₃×(C₂×D₃)⋊C₈	—	3	1	no	—
28	C₃×(C₂×D₃)⋊C₈	D₃	1	1	yes	characteristic
29	#63572	—	3	1	no	—
30	C₄×#611	—	3	1	no	—
31	C₂²×C₉⋊C₈	—	3	1	no	—
32	C₂×C₈×D₉	—	3	1	no	—
33	C₉×C₂²⋊C₈	—	3	1	no	—
34	(C₂×D₉)⋊C₈	D₃	1	1	yes	characteristic, complemented
35	Dic₉⋊C₈	—	3	1	no	—
36	C₈×Dic₉	—	3	1	no	—
37	C₈×Dic₉	—	6	1	no	—
38	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
45	C₄×D₃×C₉	—	2	1	no	—
46	C₄×D₃×C₉	—	2	1	no	—
47	C₃×C₄×D₉	—	2	1	no	—
48	C₃×C₄×D₉	—	2	1	no	—
49	#6194	C₈	1	4	yes	complemented
50	C₃×C₈×C₉	—	2	1	no	—
51	C₃⋊(C₉⋊C₈)	D₄	1	2	yes	complemented
52	C₉×C₃⋊C₈	—	2	1	no	—
53	C₃×C₉⋊C₈	—	2	1	no	—
54	#67⋊C₈	—	9	1	no	—
55	#67⋊C₈	—	9	1	no	—
56	C₂×C₃×C₄×D₃	—	3	1	no	—
57	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
58	C₂×#625	—	3	1	no	—
59	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄×C₃⋊Dic₃	—	3	1	no	—
61	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
64	#3724	—	3	1	no	—
65	(C₂×D₃)⋊Dic₃	—	3	1	no	—
66	(C₂×D₃)⋊Dic₃	—	3	1	no	—
67	D₃×C₃⋊C₈	—	6	2	no	—
68	D₃×C₃⋊C₈	—	6	2	no	—
69	C₂²×C₄×C₉	—	3	1	no	—
70	C₂×#611	—	3	1	no	—
71	C₂×C₄×D₉	C₂×D₃	1	1	yes	characteristic
72	C₂×C₈×C₉	—	2	1	no	—
73	C₂×C₈×C₉	—	3	1	no	—
74	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
75	#3676	—	3	1	no	—
76	(C₂×D₉)⋊C₄	—	3	1	no	—
77	Dic₉⋊C₄	—	3	1	no	—
78	C₄×Dic₉	—	3	1	no	—
79	C₄×Dic₉	—	6	1	no	—
80	C₂×C₉⋊C₈	—	3	1	no	—
81	C₂×C₉⋊C₈	—	6	2	no	—
82	C₂×C₉⋊C₈	—	6	1	no	—
83	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
84	C₈×D₉	—	6	2	no	—
85	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×D₃×C₉	—	2	1	no	—
87	C₂×D₃×C₉	—	2	1	no	—
88	C₂×D₃×C₉	C₂×C₈	1	2	yes	complemented
89	C₂×C₃×D₉	—	2	1	no	—
90	C₂×C₃×D₉	—	2	1	no	—
91	C₂×C₃×D₉	C₂×C₈	1	2	yes	complemented
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₃×C₄×C₉	#41	1	1	yes	characteristic
94	C₃⋊Dic₉	—	2	1	no	—
95	C₃⋊Dic₉	C₂×C₈	1	2	yes	complemented
96	C₉×Dic₃	—	2	1	no	—
97	C₃×Dic₉	—	2	1	no	—
98	C₄×#67	—	9	1	no	—
99	C₄×#67	—	9	1	no	—
100	C₂²×C₃⋊C₈	—	9	1	no	—
101	C₂²×C₃⋊C₈	—	9	1	no	—
102	C₂×D₃×C₈	—	9	1	no	—
103	C₂×D₃×C₈	—	9	1	no	—
104	C₃×C₂²⋊C₈	—	3	1	no	—
105	C₃×C₂²⋊C₈	—	9	1	no	—
106	(C₂×D₃)⋊C₈	—	3	1	no	—
107	(C₂×D₃)⋊C₈	D₉	1	1	yes	characteristic, complemented
108	Dic₃⋊C₈	—	9	1	no	—
109	Dic₃⋊C₈	—	9	1	no	—
110	C₈×Dic₃	—	9	1	no	—
111	C₈×Dic₃	—	9	1	no	—
112	C₈×Dic₃	—	18	1	no	—
113	C₂²×C₃×D₃	—	3	1	no	—
114	C₂²×C₃×D₃	C₄×D₃	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₃⋊Dic₃	—	3	1	no	—
117	C₂×C₃⋊Dic₃	—	3	1	no	—
118	C₂×C₃×Dic₃	—	3	1	no	—
119	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
120	C₃×C₄×D₃	—	2	1	no	—
121	C₃×C₄×D₃	—	2	1	no	—
122	C₃×C₄×D₃	—	6	1	no	—
123	C₃×C₄×D₃	—	6	1	no	—
124	#625	—	3	4	no	—
125	C₂³×C₉	—	3	1	no	—
126	C₂²×D₉	C₄×D₃	1	1	yes	characteristic
127	C₃²×C₈	—	2	1	no	—
128	C₃⋊(C₃⋊C₈)	—	3	2	no	—
129	C₃×C₃⋊C₈	—	2	1	no	—
130	C₃×C₃⋊C₈	—	6	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	3	1	no	—
133	C₂×C₄×C₉	—	6	1	no	—
134	C₂×C₄×C₉	—	6	1	no	—
135	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
136	#611	—	3	4	no	—
137	C₂×Dic₉	—	3	1	no	—
138	C₂×Dic₉	—	3	1	no	—
139	C₂×Dic₉	—	6	1	no	—
140	C₂×Dic₉	—	6	1	no	—
141	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
142	C₄×D₉	—	2	1	no	—
143	C₄×D₉	—	2	1	no	—
144	C₈×C₉	—	2	2	no	—
145	C₈×C₉	—	2	1	no	—
146	C₈×C₉	—	6	1	no	—
147	C₉⋊C₈	—	2	1	no	—
148	C₉⋊C₈	—	3	2	no	—
149	C₉⋊C₈	—	6	2	no	—
150	C₈×D₄	—	27	1	no	Sylow
151	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₃×C₉	#131	1	1	yes	characteristic
154	D₃×C₉	—	2	2	no	—
155	C₃×D₉	—	2	2	no	—
156	C₂²×C₃×C₄	—	3	1	no	—
157	C₂²×C₃×C₄	—	9	1	no	—
158	C₂×#67	—	9	1	no	—
159	C₂×#67	—	9	1	no	—
160	C₂×C₄×D₃	—	3	1	no	—
161	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
162	C₂×C₃×C₈	—	2	1	no	—
163	C₂×C₃×C₈	—	3	1	no	—
164	C₂×C₃×C₈	—	9	1	no	—
165	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
166	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
167	#217	—	9	1	no	—
168	#217	—	9	1	no	—
169	(C₂×D₃)⋊C₄	—	9	1	no	—
170	(C₂×D₃)⋊C₄	—	9	1	no	—
171	Dic₃⋊C₄	—	9	1	no	—
172	Dic₃⋊C₄	—	9	1	no	—
173	C₄×Dic₃	—	9	1	no	—
174	C₄×Dic₃	—	9	1	no	—
175	C₄×Dic₃	—	18	1	no	—
176	C₂×C₃⋊C₈	—	3	1	no	—
177	C₂×C₃⋊C₈	—	9	1	no	—
178	C₂×C₃⋊C₈	—	9	1	no	—
179	C₂×C₃⋊C₈	—	18	1	no	—
180	C₂×C₃⋊C₈	—	18	2	no	—
181	C₂×C₃⋊C₈	—	18	2	no	—
182	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
183	D₃×C₈	—	18	2	no	—
184	D₃×C₈	—	18	2	no	—
185	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₃×D₃	—	2	1	no	—
187	C₂×C₃×D₃	—	2	1	no	—
188	C₂×C₃×D₃	—	3	2	no	—
189	C₂×C₃×D₃	—	6	1	no	—
190	C₂×C₃×D₃	—	6	1	no	—
191	C₂×C₃×D₃	D₃×C₈	1	2	yes	—
192	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
193	C₃²×C₄	#237	1	1	yes	characteristic
194	C₃⋊Dic₃	—	3	2	no	—
195	C₃⋊Dic₃	—	6	1	no	—
196	C₃×Dic₃	—	2	1	no	—
197	C₃×Dic₃	—	6	1	no	—
198	C₂²×C₉	—	2	1	no	—
199	C₂²×C₉	—	3	2	no	—
200	C₂²×C₉	—	6	1	no	—
201	C₂²×C₉	—	6	1	no	—
202	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×D₉	—	2	1	no	—
204	C₂×D₉	—	2	1	no	—
205	C₂×D₉	D₃×C₈	1	2	yes	complemented
206	C₄×C₉	—	2	1	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	6	1	no	—
209	C₄×C₉	D₃×D₄	1	1	yes	characteristic
210	C₄×C₉	#237	1	1	yes	characteristic
211	Dic₉	—	2	1	no	—
212	Dic₉	—	3	2	no	—
213	Dic₉	—	6	2	no	—
214	Dic₉	—	6	2	no	—
215	Dic₉	—	6	1	no	—
216	C₂²×C₈	—	27	1	no	—
217	C₂²×C₈	—	27	1	no	—
218	C₄×D₄	—	27	1	no	—
219	C₄⋊C₈	—	27	1	no	—
220	C₂²⋊C₈	—	3	1	no	—
221	C₂²⋊C₈	—	9	1	no	—
222	C₄×C₈	—	27	1	no	—
223	C₃×C₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
224	C₂³×C₃	—	3	1	no	—
225	C₂³×C₃	—	9	1	no	—
226	C₂²×D₃	—	3	1	no	—
227	C₂²×D₃	C₄×D₉	1	1	yes	characteristic
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	3	1	no	—
230	C₂×C₃×C₄	—	6	1	no	—
231	C₂×C₃×C₄	—	6	1	no	—
232	C₂×C₃×C₄	—	9	1	no	—
233	C₂×C₃×C₄	—	18	1	no	—
234	C₂×C₃×C₄	—	18	1	no	—
235	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
237	#67	—	9	4	no	—
238	#67	—	9	4	no	—
239	C₂×Dic₃	—	3	1	no	—
240	C₂×Dic₃	—	9	1	no	—
241	C₂×Dic₃	—	9	1	no	—
242	C₂×Dic₃	—	9	1	no	—
243	C₂×Dic₃	—	9	1	no	—
244	C₂×Dic₃	—	18	1	no	—
245	C₂×Dic₃	—	18	1	no	—
246	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
247	C₄×D₃	—	2	1	no	—
248	C₄×D₃	—	2	1	no	—
249	C₄×D₃	—	6	1	no	—
250	C₄×D₃	—	6	1	no	—
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	—	2	1	no	—
254	C₃×C₈	—	6	1	no	—
255	C₃×C₈	—	18	1	no	—
256	C₃⋊C₈	—	2	1	no	—
257	C₃⋊C₈	—	6	1	no	—
258	C₃⋊C₈	—	9	2	no	—
259	C₃⋊C₈	—	9	2	no	—
260	C₃⋊C₈	—	18	2	no	—
261	C₂×C₃²	#67⋊C₄	1	1	yes	characteristic
262	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
263	C₂×C₃²	#907	1	1	yes	characteristic
264	C₃×D₃	—	2	2	no	—
265	C₃×D₃	—	6	2	no	—
266	C₂×C₉	—	2	1	no	—
267	C₂×C₉	—	2	1	no	—
268	C₂×C₉	—	2	1	no	—
269	C₂×C₉	—	6	2	no	—
270	C₂×C₉	#67⋊C₄	1	1	yes	characteristic
271	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
272	C₂×C₉	#907	1	1	yes	characteristic
273	D₉	—	2	2	no	—
274	C₂×D₄	—	27	1	no	—
275	C₂²×C₄	—	3	1	no	—
276	C₂²×C₄	—	9	1	no	—
277	C₂×C₈	—	3	1	no	—
278	C₂×C₈	—	9	1	no	—
279	C₂×C₈	—	27	1	no	—
280	C₂×C₈	—	54	2	no	—
281	C₂×C₈	—	54	2	no	—
282	C₂×C₈	D₃×D₉	1	1	yes	characteristic
283	C₄⋊C₄	—	27	1	no	—
284	C₂²⋊C₄	—	27	1	no	—
285	C₂²⋊C₄	—	27	1	no	—
286	C₄²	—	27	1	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	3	2	no	—
289	C₂²×C₃	—	6	1	no	—
290	C₂²×C₃	—	6	1	no	—
291	C₂²×C₃	—	9	2	no	—
292	C₂²×C₃	—	18	1	no	—
293	C₂²×C₃	—	18	1	no	—
294	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
295	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
296	C₂×D₃	—	2	1	no	—
297	C₂×D₃	—	2	1	no	—
298	C₂×D₃	—	3	2	no	—
299	C₂×D₃	—	6	1	no	—
300	C₂×D₃	—	6	1	no	—
301	C₂×D₃	C₈×D₉	1	2	yes	complemented
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	6	1	no	—
305	C₃×C₄	—	18	1	no	—
306	C₃×C₄	D₄×D₉	1	1	yes	characteristic
307	C₃×C₄	#3699	1	1	yes	characteristic
308	C₃×C₄	#3802	1	1	yes	characteristic
309	C₃×C₄	#3805	1	1	yes	characteristic
310	Dic₃	—	2	1	no	—
311	Dic₃	—	6	1	no	—
312	Dic₃	—	9	2	no	—
313	Dic₃	—	9	2	no	—
314	Dic₃	—	18	2	no	—
315	Dic₃	—	18	2	no	—
316	Dic₃	—	18	1	no	—
317	Dic₃	—	18	1	no	—
318	C₃²	#67⋊C₈	1	1	yes	characteristic
319	C₉	—	2	1	no	—
320	C₉	#67⋊C₈	1	1	yes	characteristic, complemented
321	C₂³	—	3	1	no	—
322	C₂³	—	9	1	no	—
323	D₄	—	27	4	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	6	1	no	—
327	C₂×C₄	—	9	1	no	—
328	C₂×C₄	—	18	1	no	—
329	C₂×C₄	—	18	1	no	—
330	C₂×C₄	—	27	1	no	—
331	C₂×C₄	—	27	1	no	—
332	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	6	1	no	—
335	C₈	—	18	1	no	—
336	C₈	—	27	2	no	—
337	C₂×C₃	—	2	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	6	2	no	—
341	C₂×C₃	—	18	2	no	—
342	C₂×C₃	#611⋊C₄	1	1	yes	characteristic
343	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
344	C₂×C₃	#63469	1	1	yes	characteristic
345	C₂×C₃	D₃²×C₈	1	1	yes	characteristic
346	C₂×C₃	#63805	1	1	yes	characteristic
347	C₂×C₃	#63902	1	1	yes	characteristic
348	D₃	—	2	2	no	—
349	D₃	—	6	2	no	—
350	C₂²	—	3	2	no	—
351	C₂²	—	6	1	no	—
352	C₂²	—	6	1	no	—
353	C₂²	—	9	2	no	—
354	C₂²	—	18	1	no	—
355	C₂²	—	18	1	no	—
356	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
357	C₄	—	6	1	no	—
358	C₄	—	18	1	no	—
359	C₄	—	27	2	no	—
360	C₄	—	54	1	no	—
361	C₄	#88646	1	1	yes	characteristic
362	C₄	#88663	1	1	yes	characteristic
363	C₃	—	2	1	no	—
364	C₃	#611⋊C₈	1	1	yes	characteristic, complemented
365	C₃	#10590718	1	1	yes	characteristic
366	C₂	—	6	2	no	—
367	C₂	—	18	2	no	—
368	C₂	D₃×C₈×D₉	1	1	yes	characteristic
369	C₂	#11720809	1	1	yes	characteristic
370	C₂	#11721028	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂×#6194	C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	C₈×D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×D₃×C₉⋊C₈	C₂	semidirect
3. C₂×D₉×C₃⋊C₈	C₂	semidirect
4. C₈×C₃⋊Dic₉	C₂	semidirect
5. C₉×(C₂×D₃)⋊C₈	C₂	semidirect
6. C₃×(C₂×D₉)⋊C₈	C₂	semidirect
7. #11720273	C₂	semidirect
34. (C₂×D₉)⋊C₈	D₃	semidirect
49. #6194	C₈	semidirect
51. C₃⋊(C₉⋊C₈)	D₄	semidirect
88. C₂×D₃×C₉	C₂×C₈	semidirect
91. C₂×C₃×D₉	C₂×C₈	semidirect
95. C₃⋊Dic₉	C₂×C₈	semidirect
107. (C₂×D₃)⋊C₈	D₉	semidirect
205. C₂×D₉	D₃×C₈	semidirect
223. C₃×C₉	C₈×D₄	semidirect
301. C₂×D₃	C₈×D₉	semidirect
320. C₉	#67⋊C₈	semidirect
364. C₃	#611⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6194	2
C₂×D₃×C₉⋊C₈	2
C₂×D₉×C₃⋊C₈	2
C₈×C₃⋊Dic₉	2
C₉×(C₂×D₃)⋊C₈	2
C₃×(C₂×D₉)⋊C₈	2
#11720273	2
#10590718	3
#611⋊C₈	3