Subgroups — #13139623

Order 1728.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#88442	C₂	1	2	yes	maximal
5	#11720331	C₂	1	2	yes	maximal
6	#10591212	—	3	1	no	maximal
7	#10588608	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊Dic₉)	C₄	1	4	yes	complemented
14	C₄×C₉×Dic₃	—	2	2	no	—
15	C₃×C₄⋊Dic₉	—	2	2	no	—
16	#88442	—	2	4	no	—
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
18	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
19	C₂×C₃×C₄×Dic₃	D₃	1	1	yes	characteristic
20	C₂×#3724	—	3	2	no	—
21	#63579	—	3	2	no	—
22	C₂×C₄²×C₉	—	3	1	no	—
23	C₂×C₄⋊Dic₉	—	3	1	no	—
24	C₂×C₄⋊Dic₉	—	6	1	no	—
25	C₂×C₄⋊Dic₉	D₃	1	1	yes	characteristic
26	C₂×Dic₉⋊C₄	—	3	2	no	—
27	C₉⋊#95	—	3	2	no	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
32	C₂×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₃⋊Dic₉	—	2	2	no	—
34	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
35	C₂×C₉×Dic₃	—	2	1	no	—
36	C₂×C₉×Dic₃	—	2	1	no	—
37	C₂×C₉×Dic₃	—	2	1	no	—
38	C₂×C₉×Dic₃	—	2	1	no	—
39	C₂×C₉×Dic₃	D₄	1	2	yes	—
40	C₂×C₉×Dic₃	Q₈	1	2	yes	—
41	C₂×C₃×Dic₉	—	2	2	no	—
42	C₂×C₃×Dic₉	—	2	2	no	—
43	C₂×C₃×Dic₉	—	2	2	no	—
44	#4972	—	9	1	no	—
45	#4928	—	9	1	no	—
46	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₃⋊Dic₃	—	3	2	no	—
48	C₂²×C₃×Dic₃	—	3	2	no	—
49	C₂²×C₃×Dic₃	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×Dic₃	C₂×D₃	1	1	yes	characteristic
51	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
52	C₃×C₄⋊Dic₃	—	6	2	no	—
53	C₃×C₄×Dic₃	—	2	2	no	—
54	#3724	—	6	4	no	—
55	C₂²×C₄×C₉	—	2	1	no	—
56	C₂²×C₄×C₉	—	3	1	no	—
57	C₂²×C₄×C₉	—	3	1	no	—
58	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
59	C₂²×Dic₉	—	3	2	no	—
60	C₂²×Dic₉	—	6	2	no	—
61	C₂²×Dic₉	C₂×D₃	1	2	yes	—
62	C₄²×C₉	—	6	2	no	—
63	C₄⋊Dic₉	—	2	2	no	—
64	C₄⋊Dic₉	—	3	4	no	—
65	C₄⋊Dic₉	—	6	4	no	—
66	Dic₉⋊C₄	—	6	4	no	—
67	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₃×C₉	#41	1	1	yes	characteristic
73	C₂²×C₃×C₉	#41	1	1	yes	characteristic
74	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
75	C₃⋊Dic₉	—	2	4	no	—
76	C₉×Dic₃	—	2	2	no	—
77	C₉×Dic₃	—	2	2	no	—
78	C₃×Dic₉	—	4	2	no	—
79	C₂×C₃×C₄⋊C₄	—	9	1	no	—
80	C₂×C₃×C₄²	—	3	1	no	—
81	C₂×C₄⋊Dic₃	—	3	1	no	—
82	C₂×C₄⋊Dic₃	—	9	1	no	—
83	C₂×C₄⋊Dic₃	—	9	1	no	—
84	C₂×C₄⋊Dic₃	—	18	1	no	—
85	C₂×Dic₃⋊C₄	—	9	2	no	—
86	C₂×Dic₃⋊C₄	—	9	2	no	—
87	C₂×C₄×Dic₃	D₉	1	1	yes	characteristic
88	C₃⋊#95	—	9	2	no	—
89	C₃⋊#95	—	9	2	no	—
90	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	Dic₆	1	2	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
93	C₂×C₃²×C₄	#67	1	2	yes	—
94	C₂×C₃⋊Dic₃	—	3	4	no	—
95	C₂×C₃⋊Dic₃	—	6	2	no	—
96	C₂×C₃⋊Dic₃	—	6	2	no	—
97	C₂×C₃×Dic₃	—	2	1	no	—
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₂×C₃×Dic₃	—	2	1	no	—
100	C₂×C₃×Dic₃	—	2	1	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	Dic₆	1	2	yes	—
105	C₂×C₃×Dic₃	D₁₂	1	2	yes	—
106	C₂³×C₉	—	2	1	no	—
107	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	3	2	no	—
112	C₂×C₄×C₉	—	3	2	no	—
113	C₂×C₄×C₉	—	6	1	no	—
114	C₂×C₄×C₉	—	6	1	no	—
115	C₂×C₄×C₉	—	6	1	no	—
116	C₂×C₄×C₉	—	6	1	no	—
117	C₂×C₄×C₉	Dic₆	1	2	yes	—
118	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
119	C₂×C₄×C₉	#67	1	2	yes	—
120	C₂×Dic₉	—	2	2	no	—
121	C₂×Dic₉	—	2	2	no	—
122	C₂×Dic₉	—	2	2	no	—
123	C₂×Dic₉	—	3	4	no	—
124	C₂×Dic₉	—	6	4	no	—
125	C₂×Dic₉	—	6	2	no	—
126	C₂×Dic₉	—	6	2	no	—
127	C₂×Dic₉	—	12	2	no	—
128	C₂×Dic₉	—	12	2	no	—
129	#389	—	27	1	no	Sylow
130	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₃×C₉	#121	1	1	yes	characteristic
134	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₃×C₉	#125	1	1	yes	characteristic
136	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	—	3	1	no	—
139	C₂²×C₃×C₄	—	3	1	no	—
140	C₂²×C₃×C₄	—	9	2	no	—
141	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
143	C₂²×Dic₃	—	3	2	no	—
144	C₂²×Dic₃	—	9	2	no	—
145	C₂²×Dic₃	—	9	2	no	—
146	C₂²×Dic₃	—	18	2	no	—
147	C₂²×Dic₃	C₂×D₉	1	1	yes	characteristic
148	C₂²×Dic₃	C₂×D₉	1	1	yes	characteristic
149	C₃×C₄⋊C₄	—	18	2	no	—
150	C₃×C₄²	—	6	2	no	—
151	C₄⋊Dic₃	—	6	2	no	—
152	C₄⋊Dic₃	—	9	4	no	—
153	C₄⋊Dic₃	—	9	4	no	—
154	C₄⋊Dic₃	—	18	4	no	—
155	Dic₃⋊C₄	—	18	4	no	—
156	Dic₃⋊C₄	—	18	4	no	—
157	C₄×Dic₃	—	2	2	no	—
158	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
159	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
160	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
162	C₂²×C₃²	#235	1	1	yes	characteristic
163	C₂²×C₃²	#235	1	1	yes	characteristic
164	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
165	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
166	C₃⋊Dic₃	—	6	4	no	—
167	C₃×Dic₃	—	2	2	no	—
168	C₃×Dic₃	—	2	2	no	—
169	C₃×Dic₃	—	12	2	no	—
170	C₂²×C₉	—	2	1	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	2	1	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	1	no	—
177	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
178	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
180	C₂²×C₉	#237	1	1	yes	characteristic
181	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
182	C₂²×C₉	#239	1	1	yes	characteristic
183	C₂²×C₉	C₂×#67	1	1	yes	characteristic
184	C₄×C₉	—	2	4	no	—
185	C₄×C₉	—	6	2	no	—
186	C₄×C₉	—	6	2	no	—
187	C₄×C₉	Dic₃⋊C₄	1	4	yes	complemented
188	Dic₉	—	4	2	no	—
189	Dic₉	—	6	4	no	—
190	Dic₉	—	12	4	no	—
191	C₂×C₄⋊C₄	—	9	1	no	—
192	C₂×C₄⋊C₄	—	27	1	no	—
193	C₂×C₄⋊C₄	—	27	2	no	—
194	C₂×C₄²	—	3	1	no	—
195	#95	—	27	2	no	—
196	C₃×C₉	#389	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	2	1	no	—
198	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
199	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	3	2	no	—
204	C₂×C₃×C₄	—	3	2	no	—
205	C₂×C₃×C₄	—	6	1	no	—
206	C₂×C₃×C₄	—	6	1	no	—
207	C₂×C₃×C₄	—	6	1	no	—
208	C₂×C₃×C₄	—	6	1	no	—
209	C₂×C₃×C₄	—	18	2	no	—
210	C₂×C₃×C₄	—	18	2	no	—
211	C₂×C₃×C₄	—	18	2	no	—
212	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
213	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
214	C₂×C₃×C₄	#611	1	2	yes	—
215	C₂×C₃×C₄	#624	1	2	yes	—
216	C₂×C₃×C₄	#625	1	2	yes	—
217	C₂×C₃×C₄	#627	1	2	yes	—
218	C₂×Dic₃	—	2	1	no	—
219	C₂×Dic₃	—	2	1	no	—
220	C₂×Dic₃	—	2	1	no	—
221	C₂×Dic₃	—	2	1	no	—
222	C₂×Dic₃	—	6	2	no	—
223	C₂×Dic₃	—	6	2	no	—
224	C₂×Dic₃	—	6	2	no	—
225	C₂×Dic₃	—	9	4	no	—
226	C₂×Dic₃	—	9	4	no	—
227	C₂×Dic₃	—	18	4	no	—
228	C₂×Dic₃	—	18	2	no	—
229	C₂×Dic₃	—	18	2	no	—
230	C₂×Dic₃	—	18	2	no	—
231	C₂×Dic₃	—	18	2	no	—
232	C₂×Dic₃	—	36	2	no	—
233	C₂×Dic₃	—	36	2	no	—
234	C₂×Dic₃	Dic₁₈	1	2	yes	—
235	C₂×Dic₃	D₃₆	1	2	yes	—
236	C₂×C₃²	C₄×Dic₆	1	1	yes	characteristic
237	C₂×C₃²	#869	1	1	yes	characteristic
238	C₂×C₃²	#870	1	1	yes	characteristic
239	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
240	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
241	C₂×C₃²	#924	1	1	yes	characteristic
242	C₂×C₃²	#930	1	1	yes	characteristic
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	2	1	no	—
247	C₂×C₉	—	2	1	no	—
248	C₂×C₉	—	2	1	no	—
249	C₂×C₉	—	2	1	no	—
250	C₂×C₉	#887	1	1	yes	characteristic
251	C₂×C₉	#888	1	1	yes	characteristic
252	C₂×C₉	#890	1	1	yes	characteristic
253	C₂×C₉	#893	1	1	yes	characteristic
254	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
255	C₂×C₉	#938	1	1	yes	characteristic
256	C₂×C₉	#946	1	1	yes	characteristic
257	C₂²×C₄	—	3	1	no	—
258	C₂²×C₄	—	3	1	no	—
259	C₂²×C₄	—	9	2	no	—
260	C₂²×C₄	—	27	2	no	—
261	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
262	C₄⋊C₄	—	18	2	no	—
263	C₄⋊C₄	—	27	4	no	—
264	C₄⋊C₄	—	54	4	no	—
265	C₄²	—	6	2	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
274	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
275	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
276	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
277	C₂²×C₃	#3697	1	1	yes	characteristic
278	C₂²×C₃	#3697	1	1	yes	characteristic
279	C₂²×C₃	C₂×#611	1	1	yes	characteristic
280	C₂²×C₃	D₃×Dic₆	1	1	yes	characteristic
281	C₂²×C₃	#3795	1	1	yes	characteristic
282	C₂²×C₃	#3799	1	1	yes	characteristic
283	C₂²×C₃	D₃×D₁₂	1	1	yes	characteristic
284	C₂²×C₃	C₂×#624	1	1	yes	characteristic
285	C₂²×C₃	C₂×#625	1	1	yes	characteristic
286	C₂²×C₃	C₂×#627	1	1	yes	characteristic
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	6	2	no	—
289	C₃×C₄	—	6	2	no	—
290	C₃×C₄	—	36	2	no	—
291	C₃×C₄	Dic₉⋊C₄	1	4	yes	complemented
292	C₃×C₄	#3724	1	4	yes	—
293	Dic₃	—	2	2	no	—
294	Dic₃	—	2	2	no	—
295	Dic₃	—	12	2	no	—
296	Dic₃	—	18	4	no	—
297	Dic₃	—	18	4	no	—
298	Dic₃	—	36	4	no	—
299	C₃²	#4928	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	#4972	1	1	yes	characteristic, complemented
302	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
303	C₂×C₄	—	3	2	no	—
304	C₂×C₄	—	3	2	no	—
305	C₂×C₄	—	6	1	no	—
306	C₂×C₄	—	6	1	no	—
307	C₂×C₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	18	2	no	—
310	C₂×C₄	—	18	2	no	—
311	C₂×C₄	—	18	2	no	—
312	C₂×C₄	—	27	4	no	—
313	C₂×C₄	—	54	2	no	—
314	C₂×C₄	—	54	2	no	—
315	C₂×C₄	#6189	1	2	yes	—
316	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
317	C₂×C₄	#6194	1	2	yes	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	C₄×Dic₁₈	1	1	yes	characteristic
326	C₂×C₃	#63431	1	1	yes	characteristic
327	C₂×C₃	#63432	1	1	yes	characteristic
328	C₂×C₃	C₄×D₃₆	1	1	yes	characteristic
329	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
330	C₂×C₃	#63486	1	1	yes	characteristic
331	C₂×C₃	#63492	1	1	yes	characteristic
332	C₂×C₃	#63844	1	1	yes	characteristic
333	C₂×C₃	#63856	1	1	yes	characteristic
334	C₂×C₃	#63869	1	1	yes	characteristic
335	C₂×C₃	#63894	1	1	yes	characteristic
336	C₂×C₃	#63908	1	1	yes	characteristic
337	C₂×C₃	#63918	1	1	yes	characteristic
338	C₂×C₃	C₂×#3724	1	1	yes	characteristic
339	C₂²	D₃×Dic₁₈	1	1	yes	characteristic
340	C₂²	#88641	1	1	yes	characteristic
341	C₂²	#88642	1	1	yes	characteristic
342	C₂²	D₃×D₃₆	1	1	yes	characteristic
343	C₂²	C₂×#6189	1	1	yes	characteristic
344	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
345	C₂²	C₂×#6194	1	1	yes	characteristic
346	C₄	—	6	2	no	—
347	C₄	—	6	2	no	—
348	C₄	—	36	2	no	—
349	C₄	—	54	4	no	—
350	C₄	#88442	1	4	yes	complemented
351	C₃	—	2	1	no	—
352	C₃	#10588608	1	1	yes	characteristic, complemented
353	C₃	#10591212	1	1	yes	characteristic
354	C₂	#11720912	1	1	yes	characteristic
355	C₂	#11720927	1	1	yes	characteristic
356	C₂	#11720960	1	1	yes	characteristic
357	C₂	#11720970	1	1	yes	characteristic
358	C₂	#11720993	1	1	yes	characteristic
359	C₂	#11721037	1	1	yes	characteristic
360	C₂	C₂×#88442	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#389
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
13. C₄⋊(C₃⋊Dic₉)	C₄	semidirect
74. C₃×C₄×C₉	C₄⋊C₄	semidirect
187. C₄×C₉	Dic₃⋊C₄	semidirect
196. C₃×C₉	#389	semidirect
291. C₃×C₄	Dic₉⋊C₄	semidirect
301. C₉	#4972	semidirect
350. C₄	#88442	semidirect
352. C₃	#10588608	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂×C₄×C₉×Dic₃	2
C₂×C₃×C₄⋊Dic₉	2
C₂×#88442	2
#11720331	2
#10591212	3
#10588608	3