Subgroups — C₃×#10588729 #13142871

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3675	C₂	1	2	yes	maximal, complemented
4	C₃×#63389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63369	C₂	1	2	yes	maximal, complemented
6	C₃×#5046	—	3	1	no	maximal
7	#10588729	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃×D₉	—	2	1	no	—
9	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic, complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃₆	—	2	1	no	—
12	C₂×C₃×D₃₆	—	2	4	no	—
13	C₂×C₃×D₃₆	C₂²	1	2	yes	—
14	C₃×C₉×C₄⋊C₄	—	2	1	no	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
16	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₃×#3675	—	2	4	no	—
18	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	complemented
19	C₂²×C₃×D₁₂	—	3	1	no	—
20	C₃²×C₂²⋊Q₈	D₃	1	1	yes	characteristic
21	C₂×C₃×#215	—	3	2	no	—
22	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
23	C₃×#830	—	3	1	no	—
24	C₃×#809	—	3	2	no	—
25	C₉×C₂²⋊Q₈	—	2	1	no	—
26	C₉×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#3675	C₂×C₃	1	2	yes	complemented
28	#63389	C₂×C₃	1	1	yes	characteristic, complemented
29	#63369	C₂×C₃	1	2	yes	complemented
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₃×D₉	—	2	4	no	—
32	C₂²×C₃×D₉	—	2	2	no	—
33	C₂²×C₃×D₉	—	4	2	no	—
34	C₂²×C₃×D₉	—	4	2	no	—
35	C₃×Q₈×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	—	2	2	no	—
37	C₂×C₃×C₄×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₃×D₃₆	—	2	4	no	—
42	C₃×D₃₆	—	2	2	no	—
43	C₃×D₃₆	D₄	1	4	yes	—
44	C₃×C₉⋊C₈	—	2	2	no	—
45	C₃×#450	—	9	1	no	—
46	#5046	—	3	1	no	—
47	C₂³×C₃×D₃	—	6	1	no	—
48	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×D₁₂	—	3	2	no	—
51	C₂×C₃×D₁₂	—	6	4	no	—
52	C₂×C₃×D₁₂	—	6	1	no	—
53	C₂³×D₉	—	2	1	no	—
54	C₃²×C₄⋊C₄	—	2	1	no	—
55	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₂²⋊C₄	—	2	1	no	—
57	C₃×#215	—	6	4	no	—
58	C₂×C₃×C₃⋊C₈	—	3	2	no	—
59	C₂×Q₈×C₉	—	2	1	no	—
60	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₂²×C₄×C₉	—	2	1	no	—
62	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
63	C₂×D₃₆	—	2	1	no	—
64	C₂×D₃₆	—	2	4	no	—
65	C₂×D₃₆	C₂²×C₃	1	2	yes	—
66	C₉×C₄⋊C₄	—	2	2	no	—
67	C₉×C₄⋊C₄	—	2	1	no	—
68	C₉×C₄⋊C₄	—	2	1	no	—
69	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₉×C₂²⋊C₄	—	2	2	no	—
71	C₉×C₂²⋊C₄	—	2	1	no	—
72	#3675	—	2	4	no	—
73	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	complemented
74	C₂²×C₃×C₉	—	2	1	no	—
75	C₂²×C₃×C₉	—	2	1	no	—
76	C₂²×C₃×C₉	SD₁₆	1	2	yes	complemented
77	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×D₉	—	2	4	no	—
79	C₂×C₃×D₉	—	4	8	no	—
80	C₂×C₃×D₉	—	4	2	no	—
81	C₂×C₃×D₉	—	4	2	no	—
82	C₃×C₄×C₉	—	2	1	no	—
83	C₃×C₄×C₉	—	4	1	no	—
84	C₃×C₄×C₉	—	4	1	no	—
85	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
86	C₂²×C₃×D₄	—	9	1	no	—
87	C₂²×D₁₂	—	3	1	no	—
88	C₂×C₃×SD₁₆	—	9	2	no	—
89	C₃×C₂²⋊Q₈	—	2	1	no	—
90	C₃×C₂²⋊Q₈	D₉	1	1	yes	characteristic, complemented
91	C₃×C₂²⋊Q₈	C₃×D₃	1	1	yes	characteristic
92	C₂×#215	—	3	2	no	—
93	C₃×#102	—	9	2	no	—
94	C₃×C₂²⋊C₈	—	9	1	no	—
95	#830	—	3	1	no	—
96	#809	—	3	2	no	—
97	C₂³×C₃²	#67	1	1	yes	characteristic
98	C₂²×C₃×D₃	—	6	4	no	—
99	C₂²×C₃×D₃	—	6	2	no	—
100	C₂²×C₃×D₃	—	12	2	no	—
101	C₂²×C₃×D₃	—	12	2	no	—
102	C₃²×Q₈	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	#67	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₃×D₁₂	—	3	4	no	—
109	C₃×D₁₂	—	6	4	no	—
110	C₃×D₁₂	—	6	2	no	—
111	C₂³×C₉	—	2	1	no	—
112	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
113	C₂²×D₉	—	2	4	no	—
114	C₂²×D₉	—	2	2	no	—
115	C₂²×D₉	—	4	2	no	—
116	C₂²×D₉	—	4	2	no	—
117	C₃×C₃⋊C₈	—	6	2	no	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	4	2	no	—
128	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
130	D₃₆	—	2	4	no	—
131	D₃₆	—	2	2	no	—
132	D₃₆	C₃×D₄	1	4	yes	—
133	C₉⋊C₈	—	2	2	no	—
134	#450	—	9	1	no	Sylow
135	C₂×C₃×C₉	—	2	2	no	—
136	C₂×C₃×C₉	#120	1	1	yes	characteristic
137	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₃×D₉	—	4	4	no	—
140	C₂⁴×C₃	—	18	1	no	—
141	C₂³×D₃	—	6	1	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic, complemented
144	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
145	C₂×C₃×D₄	—	9	2	no	—
146	C₂×C₃×D₄	—	18	4	no	—
147	C₂×C₃×D₄	—	18	1	no	—
148	C₂²×C₃×C₄	—	2	1	no	—
149	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂×D₁₂	—	3	2	no	—
152	C₂×D₁₂	—	6	4	no	—
153	C₂×D₁₂	—	6	1	no	—
154	C₃×SD₁₆	—	18	4	no	—
155	C₂×C₃×C₈	—	9	2	no	—
156	C₃×C₄⋊C₄	—	2	2	no	—
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic, complemented
161	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₃×C₂²⋊C₄	—	2	2	no	—
163	C₃×C₂²⋊C₄	—	2	1	no	—
164	C₃×C₂²⋊C₄	—	2	1	no	—
165	#215	—	6	4	no	—
166	C₂×C₃⋊C₈	—	3	2	no	—
167	C₂²×C₃²	—	2	1	no	—
168	C₂²×C₃²	—	2	1	no	—
169	C₂²×C₃²	#215	1	2	yes	—
170	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
171	C₂×C₃×D₃	—	6	4	no	—
172	C₂×C₃×D₃	—	12	8	no	—
173	C₂×C₃×D₃	—	12	2	no	—
174	C₂×C₃×D₃	—	12	2	no	—
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	—	4	1	no	—
177	C₃²×C₄	—	4	1	no	—
178	C₃²×C₄	D₃×D₄	1	2	yes	—
179	C₂²×C₉	—	2	1	no	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	2	2	no	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	—	4	1	no	—
184	C₂²×C₉	—	4	1	no	—
185	C₂²×C₉	C₃×SD₁₆	1	2	yes	complemented
186	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×D₉	—	2	4	no	—
188	C₂×D₉	—	4	8	no	—
189	C₂×D₉	—	4	2	no	—
190	C₂×D₉	—	4	2	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	—	4	1	no	—
194	C₄×C₉	—	4	2	no	—
195	C₄×C₉	—	4	2	no	—
196	C₄×C₉	—	4	1	no	—
197	C₄×C₉	—	4	1	no	—
198	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
199	C₂²×D₄	—	9	1	no	—
200	C₂×SD₁₆	—	9	2	no	—
201	C₂²⋊Q₈	C₃×D₉	1	1	yes	characteristic, complemented
202	#102	—	9	2	no	—
203	C₂²⋊C₈	—	9	1	no	—
204	C₃×C₉	#450	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₃	—	2	1	no	—
206	C₂³×C₃	—	18	4	no	—
207	C₂³×C₃	—	18	2	no	—
208	C₂³×C₃	—	36	2	no	—
209	C₂³×C₃	—	36	2	no	—
210	C₂³×C₃	#611	1	1	yes	characteristic
211	C₂³×C₃	C₃×#67	1	1	yes	characteristic
212	C₂²×D₃	—	6	4	no	—
213	C₂²×D₃	—	6	2	no	—
214	C₂²×D₃	—	12	2	no	—
215	C₂²×D₃	—	12	2	no	—
216	C₃×Q₈	—	2	2	no	—
217	C₃×Q₈	—	2	2	no	—
218	C₃×Q₈	—	4	2	no	—
219	C₃×D₄	—	9	4	no	—
220	C₃×D₄	—	18	4	no	—
221	C₃×D₄	—	18	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	4	2	no	—
233	C₂×C₃×C₄	#611	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
235	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
237	D₁₂	—	3	4	no	—
238	D₁₂	—	6	4	no	—
239	D₁₂	—	6	2	no	—
240	C₃×C₈	—	18	2	no	—
241	C₃⋊C₈	—	6	2	no	—
242	C₂×C₃²	—	2	2	no	—
243	C₂×C₃²	#937	1	1	yes	characteristic
244	C₂×C₃²	C₂×#215	1	1	yes	characteristic
245	C₂×C₃²	#949	1	1	yes	characteristic
246	C₃×D₃	—	12	4	no	—
247	C₂×C₉	—	2	2	no	—
248	C₂×C₉	—	2	1	no	—
249	C₂×C₉	—	2	1	no	—
250	C₂×C₉	—	2	1	no	—
251	C₂×C₉	—	4	2	no	—
252	C₂×C₉	C₃×#120	1	1	yes	characteristic
253	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
254	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
255	D₉	—	4	4	no	—
256	C₂⁴	—	18	1	no	—
257	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic, complemented
258	C₂×D₄	—	9	2	no	—
259	C₂×D₄	—	18	4	no	—
260	C₂×D₄	—	18	1	no	—
261	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
262	SD₁₆	—	18	4	no	—
263	C₂×C₈	—	9	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic, complemented
266	C₂²⋊C₄	—	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	2	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	—	18	4	no	—
276	C₂²×C₃	—	36	8	no	—
277	C₂²×C₃	—	36	2	no	—
278	C₂²×C₃	—	36	2	no	—
279	C₂²×C₃	#3675	1	2	yes	complemented
280	C₂²×C₃	C₂×#611	1	1	yes	characteristic
281	C₂²×C₃	C₃×#215	1	2	yes	—
282	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
283	C₂×D₃	—	6	4	no	—
284	C₂×D₃	—	12	8	no	—
285	C₂×D₃	—	12	2	no	—
286	C₂×D₃	—	12	2	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	4	2	no	—
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	—	4	1	no	—
295	C₃×C₄	—	4	1	no	—
296	C₃×C₄	—	4	1	no	—
297	C₃×C₄	D₄×D₉	1	2	yes	—
298	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
299	C₃²	#5046	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	C₃×#450	1	1	yes	characteristic, complemented
302	C₂³	—	18	4	no	—
303	C₂³	—	18	2	no	—
304	C₂³	—	36	2	no	—
305	C₂³	—	36	2	no	—
306	C₂³	C₃×#611	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	9	4	no	—
309	D₄	—	18	4	no	—
310	D₄	—	18	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	C₃×#611	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
316	C₈	—	18	2	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	4	2	no	—
323	C₂×C₃	—	36	4	no	—
324	C₂×C₃	#63499	1	1	yes	characteristic
325	C₂×C₃	C₂×#3675	1	1	yes	characteristic
326	C₂×C₃	#63512	1	1	yes	characteristic
327	C₂×C₃	C₃×#937	1	1	yes	characteristic
328	C₂×C₃	C₂×C₃×#215	1	1	yes	characteristic
329	C₂×C₃	C₃×#949	1	1	yes	characteristic
330	D₃	—	12	4	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	18	4	no	—
334	C₂²	—	36	8	no	—
335	C₂²	—	36	2	no	—
336	C₂²	—	36	2	no	—
337	C₂²	C₃×#3675	1	2	yes	complemented
338	C₂²	C₂×C₃×#611	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	C₃×D₄×D₉	1	2	yes	—
343	C₃	—	2	1	no	—
344	C₃	#10588729	1	1	yes	characteristic, direct product factor
345	C₃	C₃×#5046	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	36	4	no	—
348	C₂	C₃×#63499	1	1	yes	characteristic
349	C₂	C₂×C₃×#3675	1	1	yes	characteristic
350	C₂	C₃×#63512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₂²⋊Q₈	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂²×C₃×D₃₆	C₂
hypercenter	C₃×C₂²⋊Q₈	D₉
hypocenter	C₉	C₃×#450
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₂²⋊Q₈	C₂	semidirect
3. C₂×C₃×#3675	C₂	semidirect
4. C₃×#63389	C₂	semidirect
5. C₃×#63369	C₂	semidirect
7. #10588729	C₃	direct (N × H)
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
15. C₃×C₉×C₄⋊C₄	C₂²	semidirect
18. C₂×C₃×C₉⋊C₈	C₂²	semidirect
26. C₉×C₂²⋊Q₈	C₂×C₃	semidirect
27. C₂×#3675	C₂×C₃	semidirect
28. #63389	C₂×C₃	semidirect
29. #63369	C₂×C₃	semidirect
60. C₂×Q₈×C₉	C₂²×C₃	semidirect
69. C₉×C₄⋊C₄	C₂²×C₃	semidirect
73. C₂×C₉⋊C₈	C₂²×C₃	semidirect
76. C₂²×C₃×C₉	SD₁₆	semidirect
90. C₃×C₂²⋊Q₈	D₉	semidirect
143. C₂×C₃×Q₈	C₂×D₉	semidirect
160. C₃×C₄⋊C₄	C₂×D₉	semidirect
185. C₂²×C₉	C₃×SD₁₆	semidirect
201. C₂²⋊Q₈	C₃×D₉	semidirect
204. C₃×C₉	#450	semidirect
257. C₂×Q₈	C₂×C₃×D₉	semidirect
265. C₄⋊C₄	C₂×C₃×D₉	semidirect
279. C₂²×C₃	#3675	semidirect
301. C₉	C₃×#450	semidirect
337. C₂²	C₃×#3675	semidirect
344. C₃	#10588729	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₃₆	2
C₃×C₉×C₂²⋊Q₈	2
C₂×C₃×#3675	2
C₃×#63389	2
C₃×#63369	2
C₃×#5046	3
#10588729	3

Subgroups — C3×#10588729 #13142871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588729 #13142871