Subgroups — #13139295

Order 1728.

338 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#926	C₂	1	1	yes	maximal, characteristic
2	#11720571	C₂	1	1	yes	maximal, characteristic
3	C₃×#63389	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720258	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#10591056	—	3	1	no	maximal
9	#10588678	—	3	1	no	maximal
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₉×#217	C₄	1	1	yes	characteristic
14	C₉×#217	C₄	1	1	yes	characteristic
15	C₉×C₄⋊Dic₃	C₄	1	1	yes	characteristic
16	C₉×C₄⋊Dic₃	C₄	1	1	yes	characteristic
17	C₄×C₉×Dic₃	C₂²	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	C₃×#926	D₃	1	1	yes	characteristic
22	#63657	—	3	1	no	—
23	C₃×#830	—	3	1	no	—
24	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
25	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
26	#63554	—	3	1	no	—
27	Dic₃×C₃⋊C₈	—	3	1	no	—
28	C₉×#117	—	3	1	no	—
29	#63389	—	3	1	no	—
30	#63389	—	6	1	no	—
31	#63389	D₃	1	1	yes	characteristic
32	C₄⋊(C₉⋊C₈)	—	3	1	no	—
33	C₄⋊(C₉⋊C₈)	—	3	1	no	—
34	#63362	—	3	1	no	—
35	C₄×C₉⋊C₈	—	3	1	no	—
36	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₉×Dic₃	C₂×C₄	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	C₃⋊(C₉⋊C₈)	—	2	1	no	—
46	C₃⋊(C₉⋊C₈)	—	2	1	no	—
47	C₃×C₉⋊C₈	—	2	1	no	—
48	C₃×C₉⋊C₈	—	2	1	no	—
49	#5000	—	9	1	no	—
50	#4720	—	9	1	no	—
51	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
54	C₃×#217	Dic₃	1	1	yes	characteristic
55	C₃×#217	Dic₃	1	1	yes	characteristic
56	C₃×C₄⋊Dic₃	Dic₃	1	1	yes	characteristic
57	C₃×C₄⋊Dic₃	Dic₃	1	1	yes	characteristic
58	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₃⋊C₈	—	3	1	no	—
61	C₂×C₃×C₃⋊C₈	—	3	1	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₉×C₄⋊C₄	—	3	1	no	—
65	C₉×C₄⋊C₄	—	3	1	no	—
66	C₉×C₂²⋊C₄	—	3	1	no	—
67	C₉×C₂²⋊C₄	—	3	1	no	—
68	C₄²×C₉	—	3	1	no	—
69	C₄²×C₉	—	3	1	no	—
70	C₂×C₉⋊C₈	—	3	1	no	—
71	C₂×C₉⋊C₈	—	3	1	no	—
72	C₂×C₉⋊C₈	—	6	1	no	—
73	C₂×C₉⋊C₈	—	6	1	no	—
74	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₉	—	2	1	no	—
77	C₂²×C₃×C₉	—	2	1	no	—
78	C₂²×C₃×C₉	—	2	1	no	—
79	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×C₉	—	2	1	no	—
81	C₃×C₄×C₉	#41	1	1	yes	characteristic
82	C₃×C₄×C₉	#41	1	1	yes	characteristic
83	C₉×Dic₃	—	2	1	no	—
84	C₉×Dic₃	—	2	1	no	—
85	C₉×Dic₃	—	2	1	no	—
86	C₉×Dic₃	—	2	1	no	—
87	C₃×#117	—	3	1	no	—
88	#926	D₉	1	1	yes	characteristic
89	C₃×C₂²⋊C₈	—	9	1	no	—
90	#830	—	3	1	no	—
91	#830	—	9	1	no	—
92	#830	—	9	1	no	—
93	#830	—	18	1	no	—
94	#815	—	9	1	no	—
95	Dic₃⋊C₈	—	9	1	no	—
96	Dic₃⋊C₈	—	9	1	no	—
97	C₈×Dic₃	—	9	1	no	—
98	C₄⋊(C₃⋊C₈)	—	9	1	no	—
99	C₄⋊(C₃⋊C₈)	—	9	1	no	—
100	#803	—	9	1	no	—
101	C₄×C₃⋊C₈	—	9	1	no	—
102	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	C₂×Dic₃	1	1	yes	characteristic
111	C₂³×C₉	—	2	1	no	—
112	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
113	C₃⋊(C₃⋊C₈)	—	6	1	no	—
114	C₃⋊(C₃⋊C₈)	—	6	1	no	—
115	C₃×C₃⋊C₈	—	6	1	no	—
116	C₃×C₃⋊C₈	—	6	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	3	1	no	—
122	C₂×C₄×C₉	—	3	1	no	—
123	C₂×C₄×C₉	—	3	1	no	—
124	C₂×C₄×C₉	—	3	1	no	—
125	C₂×C₄×C₉	—	4	1	no	—
126	C₂×C₄×C₉	—	4	1	no	—
127	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
129	C₉⋊C₈	—	2	1	no	—
130	C₉⋊C₈	—	2	1	no	—
131	C₉⋊C₈	—	6	1	no	—
132	C₉⋊C₈	—	6	1	no	—
133	C₉⋊C₈	—	12	1	no	—
134	C₉⋊C₈	—	12	1	no	—
135	#433	—	27	1	no	Sylow
136	C₂×C₃×C₉	—	4	1	no	—
137	C₂×C₃×C₉	#117	1	1	yes	characteristic
138	C₂×C₃×C₉	#131	1	1	yes	characteristic
139	C₂×C₃×C₉	#131	1	1	yes	characteristic
140	C₂²×C₃×C₄	—	2	1	no	—
141	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
143	C₂×C₃×C₈	—	9	1	no	—
144	C₂×C₃×C₈	—	9	1	no	—
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₄⋊C₄	—	3	1	no	—
147	C₃×C₂²⋊C₄	—	3	1	no	—
148	C₃×C₂²⋊C₄	—	3	1	no	—
149	C₃×C₄²	—	3	1	no	—
150	C₃×C₄²	—	3	1	no	—
151	#217	Dic₉	1	1	yes	characteristic
152	#217	Dic₉	1	1	yes	characteristic
153	C₄⋊Dic₃	Dic₉	1	1	yes	characteristic
154	C₄⋊Dic₃	Dic₉	1	1	yes	characteristic
155	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
156	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
157	C₂×C₃⋊C₈	—	3	1	no	—
158	C₂×C₃⋊C₈	—	3	1	no	—
159	C₂×C₃⋊C₈	—	9	1	no	—
160	C₂×C₃⋊C₈	—	9	1	no	—
161	C₂×C₃⋊C₈	—	9	1	no	—
162	C₂×C₃⋊C₈	—	9	1	no	—
163	C₂×C₃⋊C₈	—	18	1	no	—
164	C₂×C₃⋊C₈	—	18	1	no	—
165	C₂²×C₃²	—	2	1	no	—
166	C₂²×C₃²	—	2	1	no	—
167	C₂²×C₃²	—	2	1	no	—
168	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
169	C₃²×C₄	—	2	1	no	—
170	C₃²×C₄	#235	1	1	yes	characteristic
171	C₃²×C₄	#235	1	1	yes	characteristic
172	C₃×Dic₃	—	2	1	no	—
173	C₃×Dic₃	—	2	1	no	—
174	C₃×Dic₃	—	2	1	no	—
175	C₃×Dic₃	—	2	1	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	2	1	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	2	1	no	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	4	1	no	—
182	C₂²×C₉	—	4	1	no	—
183	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	—	6	1	no	—
189	C₄×C₉	—	6	1	no	—
190	C₄×C₉	—	6	1	no	—
191	C₄×C₉	—	6	1	no	—
192	C₄×C₉	#237	1	1	yes	characteristic
193	C₄×C₉	#237	1	1	yes	characteristic
194	#117	—	3	1	no	—
195	C₄⋊C₈	—	27	1	no	—
196	C₄⋊C₈	—	27	1	no	—
197	C₂²⋊C₈	—	9	1	no	—
198	C₂²⋊C₈	—	27	1	no	—
199	#97	—	27	1	no	—
200	C₄×C₈	—	27	1	no	—
201	C₃×C₉	#433	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	2	1	no	—
203	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
204	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	3	1	no	—
212	C₂×C₃×C₄	—	3	1	no	—
213	C₂×C₃×C₄	—	3	1	no	—
214	C₂×C₃×C₄	—	3	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	4	1	no	—
217	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
219	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
221	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
222	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
223	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
224	C₂×Dic₃	C₂×Dic₉	1	1	yes	characteristic
225	C₃×C₈	—	18	1	no	—
226	C₃×C₈	—	18	1	no	—
227	C₃⋊C₈	—	6	1	no	—
228	C₃⋊C₈	—	6	1	no	—
229	C₃⋊C₈	—	18	1	no	—
230	C₃⋊C₈	—	18	1	no	—
231	C₃⋊C₈	—	18	1	no	—
232	C₃⋊C₈	—	18	1	no	—
233	C₃⋊C₈	—	36	1	no	—
234	C₃⋊C₈	—	36	1	no	—
235	C₂×C₃²	—	4	1	no	—
236	C₂×C₃²	#926	1	1	yes	characteristic
237	C₂×C₃²	#948	1	1	yes	characteristic
238	C₂×C₃²	#948	1	1	yes	characteristic
239	C₂×C₉	—	2	1	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	4	2	no	—
243	C₂×C₉	—	4	1	no	—
244	C₂×C₉	#877	1	1	yes	characteristic
245	C₂×C₉	#901	1	1	yes	characteristic
246	C₂×C₉	#907	1	1	yes	characteristic
247	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
248	C₂×C₈	—	9	1	no	—
249	C₂×C₈	—	9	1	no	—
250	C₂×C₈	—	27	1	no	—
251	C₂×C₈	—	27	1	no	—
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₂²⋊C₄	—	3	1	no	—
255	C₂²⋊C₄	—	3	1	no	—
256	C₄²	—	3	1	no	—
257	C₄²	—	3	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
269	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	1	no	—
275	C₃×C₄	—	6	1	no	—
276	C₃×C₄	—	6	1	no	—
277	C₃×C₄	—	6	1	no	—
278	C₃×C₄	—	6	1	no	—
279	C₃×C₄	#3697	1	1	yes	characteristic
280	C₃×C₄	#3697	1	1	yes	characteristic
281	C₃×C₄	#3804	1	1	yes	characteristic
282	C₃×C₄	#3804	1	1	yes	characteristic
283	Dic₃	—	2	1	no	—
284	Dic₃	—	2	1	no	—
285	Dic₃	—	2	1	no	—
286	Dic₃	—	2	1	no	—
287	C₃²	#5000	1	1	yes	characteristic
288	C₉	—	2	1	no	—
289	C₉	#4720	1	1	yes	characteristic, complemented
290	C₂³	D₃×Dic₉	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
299	C₈	—	18	1	no	—
300	C₈	—	18	1	no	—
301	C₈	—	54	1	no	—
302	C₈	—	54	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	4	2	no	—
307	C₂×C₃	—	4	1	no	—
308	C₂×C₃	—	4	1	no	—
309	C₂×C₃	#63488	1	1	yes	characteristic
310	C₂×C₃	#63510	1	1	yes	characteristic
311	C₂×C₃	#63510	1	1	yes	characteristic
312	C₂×C₃	#63814	1	1	yes	characteristic
313	C₂×C₃	#63815	1	1	yes	characteristic
314	C₂×C₃	#63955	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	6	1	no	—
321	C₄	—	6	1	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	#88658	1	1	yes	characteristic
325	C₄	#88658	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	#10588678	1	1	yes	characteristic, complemented
328	C₃	#10591056	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	#11720877	1	1	yes	characteristic
331	C₂	#11720878	1	1	yes	characteristic
332	C₂	#11721244	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	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#433
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
5. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
6. #11720258	C₂	semidirect
7. Dic₃×C₉⋊C₈	C₂	semidirect
201. C₃×C₉	#433	semidirect
289. C₉	#4720	semidirect
327. C₃	#10588678	semidirect

Maximal subgroups

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

subgroup	index
C₉×#926	2
#11720571	2
C₃×#63389	2
Dic₃⋊(C₉⋊C₈)	2
#11720258	2
Dic₃×C₉⋊C₈	2
#10591056	3
#10588678	3