Subgroups — #13137211

Order 1728.

380 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	C₉×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×#88420	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590324	—	3	1	no	maximal
9	#10588677	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₉×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
16	C₉×Dic₃⋊C₄	C₄	1	1	yes	characteristic
17	C₉×Dic₃⋊C₄	C₄	1	1	yes	characteristic
18	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	#88420	—	2	2	no	—
22	D₃×C₉⋊C₈	—	2	2	no	—
23	C₃×#872	D₃	1	1	yes	characteristic
24	C₂×#3711	—	3	1	no	—
25	C₂×D₃×C₃⋊C₈	—	3	1	no	—
26	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
27	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
28	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
29	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
30	C₉×#117	—	3	1	no	—
31	C₂×#3667	—	3	1	no	—
32	C₂²×C₉⋊C₈	—	3	1	no	—
33	C₄⋊(C₉⋊C₈)	—	3	1	no	—
34	C₄⋊(C₉⋊C₈)	—	3	1	no	—
35	C₄⋊(C₉⋊C₈)	—	3	1	no	—
36	C₄⋊(C₉⋊C₈)	—	6	1	no	—
37	C₄⋊(C₉⋊C₈)	D₃	1	1	yes	characteristic, complemented
38	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₄×D₃×C₉	D₄	1	2	yes	—
46	C₄×D₃×C₉	Q₈	1	2	yes	—
47	C₃⋊(C₉⋊C₈)	—	2	1	no	—
48	C₃⋊(C₉⋊C₈)	—	2	1	no	—
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	—	2	1	no	—
51	#4999	—	9	1	no	—
52	#4839	—	9	1	no	—
53	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
56	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
57	C₃×(C₂×D₃)⋊C₄	Dic₃	1	1	yes	characteristic
58	C₃×(C₂×D₃)⋊C₄	Dic₃	1	1	yes	characteristic
59	C₃×Dic₃⋊C₄	Dic₃	1	1	yes	characteristic
60	C₃×Dic₃⋊C₄	Dic₃	1	1	yes	characteristic
61	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	#3711	—	6	2	no	—
65	D₃×C₃⋊C₈	—	6	2	no	—
66	C₂²×C₄×C₉	—	3	1	no	—
67	C₉×C₄⋊C₄	—	3	1	no	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₂²⋊C₄	—	3	1	no	—
70	C₉×C₂²⋊C₄	—	3	1	no	—
71	C₄²×C₉	—	2	1	no	—
72	C₄²×C₉	—	3	1	no	—
73	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
74	#3667	—	6	2	no	—
75	C₂×C₉⋊C₈	—	3	1	no	—
76	C₂×C₉⋊C₈	—	3	1	no	—
77	C₂×C₉⋊C₈	—	6	2	no	—
78	C₂×C₉⋊C₈	—	6	1	no	—
79	C₂×C₉⋊C₈	—	6	1	no	—
80	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×D₃×C₉	—	2	1	no	—
84	C₂×D₃×C₉	—	2	1	no	—
85	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
86	C₃×C₄×C₉	—	2	1	no	—
87	C₃×C₄×C₉	—	2	1	no	—
88	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
90	C₉×Dic₃	—	2	1	no	—
91	C₉×Dic₃	—	2	1	no	—
92	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
93	C₃×#117	—	3	1	no	—
94	C₂×#208	—	9	1	no	—
95	C₂²×C₃⋊C₈	—	9	1	no	—
96	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
97	C₂×D₃×C₈	—	9	1	no	—
98	#872	D₉	1	1	yes	characteristic
99	C₃×C₄⋊C₈	—	9	1	no	—
100	Dic₃⋊C₈	—	9	1	no	—
101	Dic₃⋊C₈	—	9	1	no	—
102	C₄⋊(C₃⋊C₈)	—	3	1	no	—
103	C₄⋊(C₃⋊C₈)	—	9	1	no	—
104	C₄⋊(C₃⋊C₈)	—	9	1	no	—
105	C₄⋊(C₃⋊C₈)	—	9	1	no	—
106	C₄⋊(C₃⋊C₈)	—	9	1	no	—
107	C₄⋊(C₃⋊C₈)	—	18	1	no	—
108	C₂²×C₃×D₃	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
115	C₃×C₄×D₃	Dic₆	1	2	yes	—
116	C₃×C₄×D₃	D₁₂	1	2	yes	—
117	C₂³×C₉	—	3	1	no	—
118	C₃⋊(C₃⋊C₈)	—	6	1	no	—
119	C₃⋊(C₃⋊C₈)	—	6	1	no	—
120	C₃×C₃⋊C₈	—	6	1	no	—
121	C₃×C₃⋊C₈	—	6	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	3	1	no	—
126	C₂×C₄×C₉	—	3	2	no	—
127	C₂×C₄×C₉	—	3	2	no	—
128	C₂×C₄×C₉	—	3	1	no	—
129	C₂×C₄×C₉	—	3	1	no	—
130	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
131	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
132	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
133	C₉⋊C₈	—	2	1	no	—
134	C₉⋊C₈	—	2	1	no	—
135	C₉⋊C₈	—	6	1	no	—
136	C₉⋊C₈	—	6	1	no	—
137	C₉⋊C₈	—	12	1	no	—
138	C₉⋊C₈	—	12	1	no	—
139	#424	—	27	1	no	Sylow
140	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₉	#131	1	1	yes	characteristic
142	C₂×C₃×C₉	#131	1	1	yes	characteristic
143	D₃×C₉	—	2	2	no	—
144	C₂²×C₃×C₄	—	3	1	no	—
145	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
146	C₂×C₃×C₈	—	9	1	no	—
147	C₂×C₃×C₈	—	9	1	no	—
148	C₃×C₄⋊C₄	—	3	1	no	—
149	C₃×C₄⋊C₄	—	3	1	no	—
150	C₃×C₂²⋊C₄	—	3	1	no	—
151	C₃×C₂²⋊C₄	—	3	1	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	3	1	no	—
154	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
155	C₃×C₄²	D₃²	1	1	yes	characteristic
156	(C₂×D₃)⋊C₄	Dic₉	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	Dic₉	1	1	yes	characteristic
158	Dic₃⋊C₄	Dic₉	1	1	yes	characteristic
159	Dic₃⋊C₄	Dic₉	1	1	yes	characteristic
160	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
161	#208	—	18	2	no	—
162	C₂×C₃⋊C₈	—	3	1	no	—
163	C₂×C₃⋊C₈	—	3	1	no	—
164	C₂×C₃⋊C₈	—	9	1	no	—
165	C₂×C₃⋊C₈	—	9	1	no	—
166	C₂×C₃⋊C₈	—	9	1	no	—
167	C₂×C₃⋊C₈	—	9	1	no	—
168	C₂×C₃⋊C₈	—	18	1	no	—
169	C₂×C₃⋊C₈	—	18	1	no	—
170	C₂×C₃⋊C₈	—	18	2	no	—
171	ΓL(1,𝔽₂₅)	—	18	2	no	—
172	D₃×C₈	—	18	2	no	—
173	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
174	C₂×C₃×D₃	—	2	1	no	—
175	C₂×C₃×D₃	—	2	1	no	—
176	C₂×C₃×D₃	C₄⋊Dic₃	1	2	yes	—
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	—	2	1	no	—
179	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
181	C₃×Dic₃	—	2	1	no	—
182	C₃×Dic₃	—	2	1	no	—
183	C₃×Dic₃	C₄⋊Dic₃	1	2	yes	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	3	2	no	—
186	C₂²×C₉	—	6	1	no	—
187	C₂²×C₉	—	6	1	no	—
188	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
189	C₄×C₉	—	2	2	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	3	2	no	—
195	C₄×C₉	—	4	1	no	—
196	C₄×C₉	—	6	1	no	—
197	C₄×C₉	—	6	1	no	—
198	C₄×C₉	D₃×D₄	1	1	yes	characteristic
199	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
200	C₂×M₁₆	—	27	1	no	—
201	C₂²×C₈	—	27	1	no	—
202	#117	—	3	1	no	—
203	C₄⋊C₈	—	9	1	no	—
204	C₄⋊C₈	—	27	1	no	—
205	C₄⋊C₈	—	27	1	no	—
206	C₄⋊C₈	—	27	1	no	—
207	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	3	1	no	—
209	C₂²×D₃	C₂×Dic₉	1	1	yes	characteristic
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	3	1	no	—
214	C₂×C₃×C₄	—	3	2	no	—
215	C₂×C₃×C₄	—	3	2	no	—
216	C₂×C₃×C₄	—	3	1	no	—
217	C₂×C₃×C₄	—	3	1	no	—
218	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
221	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
222	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
224	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
225	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
226	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
227	C₄×D₃	Dic₁₈	1	2	yes	—
228	C₄×D₃	D₃₆	1	2	yes	—
229	C₃×C₈	—	18	1	no	—
230	C₃×C₈	—	18	1	no	—
231	C₃⋊C₈	—	6	1	no	—
232	C₃⋊C₈	—	6	1	no	—
233	C₃⋊C₈	—	18	1	no	—
234	C₃⋊C₈	—	18	1	no	—
235	C₃⋊C₈	—	18	1	no	—
236	C₃⋊C₈	—	18	1	no	—
237	C₃⋊C₈	—	36	1	no	—
238	C₃⋊C₈	—	36	1	no	—
239	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
240	C₂×C₃²	#948	1	1	yes	characteristic
241	C₂×C₃²	#948	1	1	yes	characteristic
242	C₃×D₃	—	2	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	6	2	no	—
247	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₉	#901	1	1	yes	characteristic
249	C₂×C₉	#907	1	1	yes	characteristic
250	C₂²×C₄	—	3	1	no	—
251	M₁₆	—	54	2	no	—
252	C₂×C₈	—	9	1	no	—
253	C₂×C₈	—	9	1	no	—
254	C₂×C₈	—	27	1	no	—
255	C₂×C₈	—	27	1	no	—
256	C₂×C₈	—	54	2	no	—
257	C₄⋊C₄	—	3	1	no	—
258	C₄⋊C₄	—	3	1	no	—
259	C₂²⋊C₄	—	3	1	no	—
260	C₂²⋊C₄	—	3	1	no	—
261	C₄²	—	3	1	no	—
262	C₄²	D₃×D₉	1	1	yes	characteristic
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	3	2	no	—
265	C₂²×C₃	—	6	1	no	—
266	C₂²×C₃	—	6	1	no	—
267	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
268	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
269	C₂×D₃	—	2	1	no	—
270	C₂×D₃	—	2	1	no	—
271	C₂×D₃	C₄⋊Dic₉	1	2	yes	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	3	2	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	6	1	no	—
282	C₃×C₄	—	6	1	no	—
283	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
284	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
285	C₃×C₄	D₃×Dic₆	1	1	yes	characteristic
286	C₃×C₄	D₃×D₁₂	1	1	yes	characteristic
287	Dic₃	—	2	1	no	—
288	Dic₃	—	2	1	no	—
289	Dic₃	C₄⋊Dic₉	1	2	yes	—
290	C₃²	#4999	1	1	yes	characteristic
291	C₉	—	2	1	no	—
292	C₉	#4839	1	1	yes	characteristic, complemented
293	C₂³	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	—	3	2	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	3	1	no	—
299	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
300	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
302	C₈	—	18	1	no	—
303	C₈	—	18	1	no	—
304	C₈	—	54	1	no	—
305	C₈	—	54	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	6	2	no	—
310	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
311	C₂×C₃	#63510	1	1	yes	characteristic
312	C₂×C₃	#63510	1	1	yes	characteristic
313	C₂×C₃	#63814	1	1	yes	characteristic
314	C₂×C₃	#63815	1	1	yes	characteristic
315	C₂×C₃	D₃×C₄⋊Dic₃	1	1	yes	characteristic
316	D₃	—	2	2	no	—
317	C₂²	—	3	2	no	—
318	C₂²	—	6	1	no	—
319	C₂²	—	6	1	no	—
320	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	3	2	no	—
324	C₄	—	6	1	no	—
325	C₄	—	6	1	no	—
326	C₄	D₃×Dic₁₈	1	1	yes	characteristic
327	C₄	D₃×D₃₆	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	#10588677	1	1	yes	characteristic, complemented
330	C₃	#10590324	1	1	yes	characteristic
331	C₂	—	6	2	no	—
332	C₂	#11720877	1	1	yes	characteristic
333	C₂	#11720878	1	1	yes	characteristic
334	C₂	D₃×C₄⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂²×C₃×D₃	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#424
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
6. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
7. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
37. C₄⋊(C₉⋊C₈)	D₃	semidirect
207. C₃×C₉	#424	semidirect
292. C₉	#4839	semidirect
329. C₃	#10588677	semidirect

Maximal subgroups

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

subgroup	index
C₉×#872	2
C₂×#88420	2
C₂×D₃×C₉⋊C₈	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₃×C₄⋊(C₉⋊C₈)	2
Dic₃⋊(C₉⋊C₈)	2
#10590324	3
#10588677	3