Subgroups — C₃×D₉×#137 #13153507

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41×D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Q₈×D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63511	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63505	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63480	C₂	1	2	yes	maximal, complemented
7	C₃×Q₁₆×D₉	C₂	1	2	yes	maximal, complemented
8	C₃×#63477	C₂	1	2	yes	maximal, complemented
9	C₃×SD₁₆×D₉	C₂	1	2	yes	maximal, complemented
10	C₃×#63471	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×M₁₆×D₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×D₃×#137	—	3	1	no	maximal
13	D₉×#137	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₃×C₉×#41	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic, complemented
16	C₃×#3701	C₂²	1	1	yes	characteristic
17	C₃×Q₈×D₉	—	2	1	no	—
18	C₃×Q₈×D₉	—	2	1	no	—
19	C₃×Q₈×D₉	C₂²	1	2	yes	—
20	C₃×Q₈×D₉	C₂²	1	1	yes	characteristic
21	C₃×Q₈×D₉	C₂²	1	2	yes	complemented
22	C₃×#3699	—	2	1	no	—
23	C₃×#3699	C₂²	1	1	yes	characteristic
24	C₃×D₄×D₉	—	2	1	no	—
25	C₃×D₄×D₉	C₂²	1	1	yes	characteristic
26	C₃×#3697	—	2	1	no	—
27	C₃×#3697	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×D₉	—	2	1	no	—
29	C₂×C₃×C₄×D₉	—	2	1	no	—
30	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
31	C₂×C₃×Dic₁₈	—	2	1	no	—
32	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₉×Q₁₆	C₂²	1	2	yes	complemented
34	C₃×C₉×SD₁₆	C₂²	1	2	yes	complemented
35	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₃×#3675	C₂²	1	2	yes	complemented
37	C₃×#3674	C₂²	1	2	yes	complemented
38	C₃×#3674	C₂²	1	2	yes	complemented
39	C₃×#3672	C₂²	1	2	yes	complemented
40	C₃×#3667	C₂²	1	1	yes	characteristic, complemented
41	C₃×#3664	C₂²	1	2	yes	complemented
42	C₃×#3663	C₂²	1	2	yes	complemented
43	C₃×C₈×D₉	C₂²	1	2	yes	—
44	C₃×Dic₃₆	C₂²	1	2	yes	complemented
45	C₃×D₃×#41	—	3	1	no	—
46	C₂×C₃×D₃×Q₈	—	3	1	no	—
47	C₃²×#137	D₃	1	1	yes	characteristic
48	C₃×#951	—	3	1	no	—
49	C₃×#942	—	3	1	no	—
50	C₃×#918	—	3	2	no	—
51	C₃×D₃×Q₁₆	—	3	2	no	—
52	C₃×#915	—	3	2	no	—
53	C₃×D₃×SD₁₆	—	3	2	no	—
54	C₃×#909	—	3	1	no	—
55	C₃×D₃×M₁₆	—	3	1	no	—
56	#41×D₉	C₂×C₃	1	1	yes	characteristic
57	C₂×Q₈×D₉	C₂×C₃	1	1	yes	characteristic, complemented
58	C₉×#137	—	2	1	no	—
59	C₉×#137	C₂×C₃	1	1	yes	characteristic, complemented
60	#63511	C₂×C₃	1	1	yes	characteristic, complemented
61	#63505	C₂×C₃	1	1	yes	characteristic, complemented
62	#63480	C₂×C₃	1	2	yes	complemented
63	Q₁₆×D₉	C₂×C₃	1	2	yes	complemented
64	#63477	C₂×C₃	1	2	yes	complemented
65	SD₁₆×D₉	C₂×C₃	1	2	yes	complemented
66	#63471	C₂×C₃	1	1	yes	characteristic, complemented
67	M₁₆×D₉	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂²×C₃×D₉	—	2	1	no	—
69	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
70	C₃×Q₈×C₉	—	2	1	no	—
71	C₃×Q₈×C₉	C₂³	1	2	yes	—
72	C₃×Q₈×C₉	C₂³	1	1	yes	characteristic
73	C₃×D₄×C₉	—	2	1	no	—
74	C₃×D₄×C₉	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₄×C₉	—	2	1	no	—
76	C₂×C₃×C₄×C₉	—	2	1	no	—
77	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
78	C₃×#611	—	2	1	no	—
79	C₃×#611	—	2	1	no	—
80	C₃×#611	—	2	1	no	—
81	C₂×C₃×Dic₉	—	2	1	no	—
82	C₂×C₃×Dic₉	—	2	1	no	—
83	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
84	C₃×D₃₆	—	2	1	no	—
85	C₃×D₃₆	C₂³	1	1	yes	characteristic
86	C₃×C₄×D₉	—	2	1	no	—
87	C₃×C₄×D₉	—	2	2	no	—
88	C₃×C₄×D₉	—	2	2	no	—
89	C₃×C₄×D₉	—	2	1	no	—
90	C₃×C₄×D₉	—	2	1	no	—
91	C₃×C₄×D₉	D₄	1	1	yes	characteristic
92	C₃×C₄×D₉	D₄	1	1	yes	characteristic
93	C₃×C₄×D₉	C₂³	1	1	yes	characteristic
94	C₃×C₄×D₉	C₂³	1	1	yes	characteristic
95	C₃×Dic₁₈	—	2	1	no	—
96	C₃×Dic₁₈	—	2	2	no	—
97	C₃×Dic₁₈	—	2	2	no	—
98	C₃×Dic₁₈	—	2	1	no	—
99	C₃×Dic₁₈	C₂³	1	2	yes	—
100	C₃×Dic₁₈	C₂³	1	1	yes	characteristic
101	C₃×C₈×C₉	C₂³	1	2	yes	—
102	C₃×C₉⋊C₈	C₂³	1	2	yes	—
103	C₂×C₃×#137	—	9	1	no	—
104	D₃×#137	—	3	1	no	—
105	C₃²×#41	C₂×D₃	1	1	yes	characteristic
106	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
107	C₃×#239	—	3	1	no	—
108	C₃×D₃×Q₈	—	3	2	no	—
109	C₃×D₃×Q₈	—	3	2	no	—
110	C₃×D₃×Q₈	—	3	1	no	—
111	C₃×D₃×Q₈	—	6	1	no	—
112	C₃×D₃×Q₈	—	6	1	no	—
113	C₃×#237	—	3	1	no	—
114	C₃×#237	—	6	1	no	—
115	C₃×D₃×D₄	—	3	1	no	—
116	C₃×D₃×D₄	—	6	1	no	—
117	C₃×#235	—	3	1	no	—
118	C₃×#235	—	6	1	no	—
119	C₂×C₃×C₄×D₃	—	3	1	no	—
120	C₂×C₃×C₄×D₃	—	6	1	no	—
121	C₂×C₃×C₄×D₃	—	6	1	no	—
122	C₂×C₃×Dic₆	—	3	1	no	—
123	C₂×C₃×Dic₆	—	6	1	no	—
124	C₃²×Q₁₆	C₂×D₃	1	2	yes	—
125	C₃²×SD₁₆	C₂×D₃	1	2	yes	—
126	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
127	C₃×#216	—	3	2	no	—
128	C₃×#216	—	3	2	no	—
129	C₃×#215	—	3	2	no	—
130	C₃×#214	—	3	2	no	—
131	C₃×#208	—	3	1	no	—
132	C₃×Dic₁₂	—	3	2	no	—
133	C₃×#204	—	3	2	no	—
134	C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
135	C₃×D₃×C₈	—	3	2	no	—
136	C₉×#41	—	2	1	no	—
137	C₉×#41	C₂²×C₃	1	1	yes	characteristic
138	C₂×Q₈×C₉	—	2	1	no	—
139	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
140	#3701	C₂²×C₃	1	1	yes	characteristic
141	Q₈×D₉	—	2	1	no	—
142	Q₈×D₉	—	2	1	no	—
143	Q₈×D₉	C₂²×C₃	1	2	yes	—
144	Q₈×D₉	C₂²×C₃	1	1	yes	characteristic
145	Q₈×D₉	C₂²×C₃	1	2	yes	complemented
146	#3699	—	2	1	no	—
147	#3699	C₂²×C₃	1	1	yes	characteristic
148	D₄×D₉	—	2	1	no	—
149	D₄×D₉	C₂²×C₃	1	1	yes	characteristic
150	#3697	—	2	1	no	—
151	#3697	C₂²×C₃	1	1	yes	characteristic
152	C₂×C₄×D₉	—	2	1	no	—
153	C₂×C₄×D₉	—	2	1	no	—
154	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
155	C₂×Dic₁₈	—	2	1	no	—
156	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic, complemented
157	C₉×Q₁₆	—	2	2	no	—
158	C₉×Q₁₆	C₂²×C₃	1	2	yes	complemented
159	C₉×SD₁₆	—	2	2	no	—
160	C₉×SD₁₆	C₂²×C₃	1	2	yes	complemented
161	C₉×M₁₆	—	2	1	no	—
162	C₉×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
163	#3675	C₂²×C₃	1	2	yes	complemented
164	#3674	C₂²×C₃	1	2	yes	complemented
165	#3674	C₂²×C₃	1	2	yes	complemented
166	#3672	C₂²×C₃	1	2	yes	complemented
167	#3667	C₂²×C₃	1	1	yes	characteristic, complemented
168	#3664	C₂²×C₃	1	2	yes	complemented
169	#3663	C₂²×C₃	1	2	yes	complemented
170	C₈×D₉	C₂²×C₃	1	2	yes	—
171	Dic₃₆	C₂²×C₃	1	2	yes	complemented
172	C₂²×C₃×C₉	—	2	1	no	—
173	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×D₉	—	2	2	no	—
175	C₂×C₃×D₉	—	2	1	no	—
176	C₂×C₃×D₉	—	4	2	no	—
177	C₂×C₃×D₉	C₂×D₄	1	1	yes	characteristic
178	C₂×C₃×D₉	C₂×D₄	1	1	yes	characteristic
179	C₃×C₄×C₉	—	2	2	no	—
180	C₃×C₄×C₉	—	2	1	no	—
181	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
182	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
183	C₃×Dic₉	—	2	2	no	—
184	C₃×Dic₉	—	2	1	no	—
185	C₃×Dic₉	C₂×D₄	1	1	yes	characteristic
186	C₃×Dic₉	C₂×D₄	1	1	yes	characteristic
187	C₂×C₃×#41	—	9	1	no	—
188	C₂²×C₃×Q₈	—	9	1	no	—
189	D₃×#41	—	3	1	no	—
190	C₂×D₃×Q₈	—	3	1	no	—
191	C₃×#137	—	2	1	no	—
192	C₃×#137	—	9	1	no	—
193	C₃×#137	—	9	1	no	—
194	C₃×#137	—	9	2	no	—
195	C₃×#137	—	9	2	no	—
196	C₃×#137	—	9	1	no	—
197	C₃×#137	D₉	1	1	yes	characteristic, direct product factor
198	C₃×#137	C₃×D₃	1	1	yes	characteristic
199	C₂×C₃×Q₁₆	—	9	2	no	—
200	C₂×C₃×SD₁₆	—	9	2	no	—
201	C₂×C₃×M₁₆	—	9	1	no	—
202	#951	—	3	1	no	—
203	#942	—	3	1	no	—
204	#918	—	3	2	no	—
205	D₃×Q₁₆	—	3	2	no	—
206	#915	—	3	2	no	—
207	D₃×SD₁₆	—	3	2	no	—
208	#909	—	3	1	no	—
209	D₃×M₁₆	—	3	1	no	—
210	C₂²×C₃×D₃	—	3	1	no	—
211	C₂²×C₃×D₃	—	6	1	no	—
212	C₃²×Q₈	—	2	1	no	—
213	C₃²×Q₈	C₂²×D₃	1	2	yes	—
214	C₃²×Q₈	C₂²×D₃	1	1	yes	characteristic
215	C₃²×D₄	—	2	1	no	—
216	C₃²×D₄	C₂²×D₃	1	1	yes	characteristic
217	C₂×C₃²×C₄	—	2	1	no	—
218	C₂×C₃²×C₄	—	2	1	no	—
219	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
220	C₃×#67	—	6	1	no	—
221	C₃×#67	—	6	1	no	—
222	C₃×#67	—	6	1	no	—
223	C₂×C₃×Dic₃	—	3	1	no	—
224	C₂×C₃×Dic₃	—	6	1	no	—
225	C₂×C₃×Dic₃	—	6	1	no	—
226	C₃×D₁₂	—	3	1	no	—
227	C₃×D₁₂	—	6	1	no	—
228	C₃×C₄×D₃	—	3	1	no	—
229	C₃×C₄×D₃	—	3	1	no	—
230	C₃×C₄×D₃	—	3	1	no	—
231	C₃×C₄×D₃	—	3	1	no	—
232	C₃×C₄×D₃	—	6	1	no	—
233	C₃×C₄×D₃	—	6	1	no	—
234	C₃×C₄×D₃	—	6	2	no	—
235	C₃×C₄×D₃	—	6	2	no	—
236	C₃×C₄×D₃	—	6	1	no	—
237	C₃×Dic₆	—	3	2	no	—
238	C₃×Dic₆	—	3	1	no	—
239	C₃×Dic₆	—	6	2	no	—
240	C₃×Dic₆	—	6	2	no	—
241	C₃×Dic₆	—	6	1	no	—
242	C₃×Dic₆	—	6	1	no	—
243	C₂²×D₉	—	2	1	no	—
244	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
245	C₃²×C₈	C₂²×D₃	1	2	yes	—
246	C₃×C₃⋊C₈	—	3	2	no	—
247	Q₈×C₉	—	2	1	no	—
248	Q₈×C₉	—	2	2	no	—
249	Q₈×C₉	—	2	1	no	—
250	Q₈×C₉	—	4	1	no	—
251	Q₈×C₉	C₂³×C₃	1	1	yes	characteristic
252	Q₈×C₉	C₂³×C₃	1	2	yes	—
253	D₄×C₉	—	2	1	no	—
254	D₄×C₉	—	2	1	no	—
255	D₄×C₉	—	4	1	no	—
256	D₄×C₉	C₂³×C₃	1	1	yes	characteristic
257	C₂×C₄×C₉	—	2	1	no	—
258	C₂×C₄×C₉	—	2	1	no	—
259	C₂×C₄×C₉	—	2	1	no	—
260	C₂×C₄×C₉	—	4	1	no	—
261	C₂×C₄×C₉	—	4	1	no	—
262	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
263	#611	—	2	1	no	—
264	#611	—	2	1	no	—
265	#611	—	2	1	no	—
266	C₂×Dic₉	—	2	1	no	—
267	C₂×Dic₉	—	2	1	no	—
268	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
269	D₃₆	—	2	1	no	—
270	D₃₆	C₂³×C₃	1	1	yes	characteristic
271	C₄×D₉	—	2	1	no	—
272	C₄×D₉	—	2	2	no	—
273	C₄×D₉	—	2	2	no	—
274	C₄×D₉	—	2	1	no	—
275	C₄×D₉	—	2	1	no	—
276	C₄×D₉	C₃×D₄	1	1	yes	characteristic
277	C₄×D₉	C₃×D₄	1	1	yes	characteristic
278	C₄×D₉	C₂³×C₃	1	1	yes	characteristic
279	C₄×D₉	C₂³×C₃	1	1	yes	characteristic
280	Dic₁₈	—	2	1	no	—
281	Dic₁₈	—	2	2	no	—
282	Dic₁₈	—	2	2	no	—
283	Dic₁₈	—	2	1	no	—
284	Dic₁₈	C₂³×C₃	1	1	yes	characteristic
285	Dic₁₈	C₂³×C₃	1	2	yes	—
286	C₈×C₉	—	2	2	no	—
287	C₈×C₉	C₂³×C₃	1	2	yes	—
288	C₉⋊C₈	C₂³×C₃	1	2	yes	—
289	C₂×#137	—	9	1	no	Sylow
290	C₂×C₃×C₉	—	2	1	no	—
291	C₂×C₃×C₉	—	4	1	no	—
292	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
293	C₃×D₉	—	2	1	no	—
294	C₃×D₉	—	4	1	no	—
295	C₃×D₉	#137	1	2	yes	direct product factor
296	C₃×#41	—	2	1	no	—
297	C₃×#41	—	9	1	no	—
298	C₃×#41	—	9	1	no	—
299	C₃×#41	—	9	1	no	—
300	C₃×#41	—	18	1	no	—
301	C₃×#41	—	18	1	no	—
302	C₃×#41	C₂×D₉	1	1	yes	characteristic
303	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
304	C₂×C₃×Q₈	—	2	1	no	—
305	C₂×C₃×Q₈	—	9	2	no	—
306	C₂×C₃×Q₈	—	9	1	no	—
307	C₂×C₃×Q₈	—	9	2	no	—
308	C₂×C₃×Q₈	—	9	1	no	—
309	C₂×C₃×Q₈	—	18	1	no	—
310	C₂×C₃×Q₈	—	18	1	no	—
311	C₂×C₃×Q₈	—	18	1	no	—
312	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic, complemented
313	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
314	C₂×C₃×D₄	—	9	1	no	—
315	C₂×C₃×D₄	—	18	1	no	—
316	C₂²×C₃×C₄	—	9	1	no	—
317	C₂²×C₃×C₄	—	18	1	no	—
318	C₂²×C₃×C₄	—	18	1	no	—
319	#239	—	3	1	no	—
320	D₃×Q₈	—	3	2	no	—
321	D₃×Q₈	—	3	1	no	—
322	D₃×Q₈	—	3	2	no	—
323	D₃×Q₈	—	6	1	no	—
324	D₃×Q₈	—	6	1	no	—
325	#237	—	3	1	no	—
326	#237	—	6	1	no	—
327	D₃×D₄	—	3	1	no	—
328	D₃×D₄	—	6	1	no	—
329	#235	—	3	1	no	—
330	#235	—	6	1	no	—
331	C₂×C₄×D₃	—	3	1	no	—
332	C₂×C₄×D₃	—	6	1	no	—
333	C₂×C₄×D₃	—	6	1	no	—
334	C₂×Dic₆	—	3	1	no	—
335	C₂×Dic₆	—	6	1	no	—
336	C₃×Q₁₆	—	2	2	no	—
337	C₃×Q₁₆	—	9	2	no	—
338	C₃×Q₁₆	—	9	2	no	—
339	C₃×Q₁₆	—	9	2	no	—
340	C₃×Q₁₆	C₂×D₉	1	2	yes	complemented
341	C₃×Q₁₆	C₂×C₃×D₃	1	2	yes	—
342	C₃×SD₁₆	—	2	2	no	—
343	C₃×SD₁₆	—	9	2	no	—
344	C₃×SD₁₆	—	9	2	no	—
345	C₃×SD₁₆	—	9	2	no	—
346	C₃×SD₁₆	C₂×D₉	1	2	yes	complemented
347	C₃×SD₁₆	C₂×C₃×D₃	1	2	yes	—
348	C₃×M₁₆	—	2	1	no	—
349	C₃×M₁₆	—	9	1	no	—
350	C₃×M₁₆	—	9	2	no	—
351	C₃×M₁₆	C₂×D₉	1	1	yes	characteristic, complemented
352	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
353	C₂×C₃×C₈	—	9	2	no	—
354	#216	—	3	2	no	—
355	#216	—	3	2	no	—
356	#215	—	3	2	no	—
357	#214	—	3	2	no	—
358	#208	—	3	1	no	—
359	Dic₁₂	—	3	2	no	—
360	#204	—	3	2	no	—
361	ΓL(1,𝔽₂₅)	—	3	2	no	—
362	D₃×C₈	—	3	2	no	—
363	C₂²×C₃²	—	2	1	no	—
364	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
365	C₂×C₃×D₃	—	3	1	no	—
366	C₂×C₃×D₃	—	3	1	no	—
367	C₂×C₃×D₃	—	6	2	no	—
368	C₂×C₃×D₃	—	6	1	no	—
369	C₂×C₃×D₃	—	12	2	no	—
370	C₃²×C₄	—	2	2	no	—
371	C₃²×C₄	—	2	1	no	—
372	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
373	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
374	C₃×Dic₃	—	3	1	no	—
375	C₃×Dic₃	—	3	1	no	—
376	C₃×Dic₃	—	6	2	no	—
377	C₃×Dic₃	—	6	1	no	—
378	C₂²×C₉	—	2	1	no	—
379	C₂²×C₉	—	2	1	no	—
380	C₂²×C₉	—	4	1	no	—
381	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
382	C₂×D₉	—	2	2	no	—
383	C₂×D₉	—	2	1	no	—
384	C₂×D₉	—	4	2	no	—
385	C₂×D₉	C₂×C₃×D₄	1	1	yes	characteristic
386	C₂×D₉	C₂×C₃×D₄	1	1	yes	characteristic
387	C₄×C₉	—	2	2	no	—
388	C₄×C₉	—	2	1	no	—
389	C₄×C₉	—	2	1	no	—
390	C₄×C₉	—	2	1	no	—
391	C₄×C₉	—	4	1	no	—
392	C₄×C₉	—	4	2	no	—
393	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
394	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
395	Dic₉	—	2	2	no	—
396	Dic₉	—	2	1	no	—
397	Dic₉	C₂×C₃×D₄	1	1	yes	characteristic
398	Dic₉	C₂×C₃×D₄	1	1	yes	characteristic
399	C₂×#41	—	9	1	no	—
400	C₂²×Q₈	—	9	1	no	—
401	#137	—	9	1	no	—
402	#137	—	9	1	no	—
403	#137	—	9	2	no	—
404	#137	—	9	2	no	—
405	#137	—	9	1	no	—
406	#137	C₃×D₉	1	1	yes	characteristic, direct product factor
407	C₂×Q₁₆	—	9	2	no	—
408	C₂×SD₁₆	—	9	2	no	—
409	C₂×M₁₆	—	9	1	no	—
410	C₃×C₉	C₂×#137	1	1	yes	characteristic, complemented, Sylow
411	C₂³×C₃	—	9	1	no	—
412	C₂³×C₃	—	18	1	no	—
413	C₂²×D₃	—	3	1	no	—
414	C₂²×D₃	—	6	1	no	—
415	C₃×Q₈	—	2	1	no	—
416	C₃×Q₈	—	2	1	no	—
417	C₃×Q₈	—	2	2	no	—
418	C₃×Q₈	—	2	1	no	—
419	C₃×Q₈	—	4	1	no	—
420	C₃×Q₈	—	9	2	no	—
421	C₃×Q₈	—	9	1	no	—
422	C₃×Q₈	—	18	2	no	—
423	C₃×Q₈	—	18	2	no	—
424	C₃×Q₈	—	18	1	no	—
425	C₃×Q₈	—	18	1	no	—
426	C₃×Q₈	C₂²×D₉	1	2	yes	—
427	C₃×Q₈	C₂²×D₉	1	1	yes	characteristic
428	C₃×Q₈	C₂²×C₃×D₃	1	2	yes	—
429	C₃×Q₈	C₂²×C₃×D₃	1	1	yes	characteristic
430	C₃×D₄	—	2	1	no	—
431	C₃×D₄	—	2	1	no	—
432	C₃×D₄	—	2	1	no	—
433	C₃×D₄	—	4	1	no	—
434	C₃×D₄	—	9	1	no	—
435	C₃×D₄	—	18	1	no	—
436	C₃×D₄	—	18	1	no	—
437	C₃×D₄	—	18	1	no	—
438	C₃×D₄	—	18	1	no	—
439	C₃×D₄	C₂²×D₉	1	1	yes	characteristic
440	C₃×D₄	C₂²×C₃×D₃	1	1	yes	characteristic
441	C₂×C₃×C₄	—	2	1	no	—
442	C₂×C₃×C₄	—	2	1	no	—
443	C₂×C₃×C₄	—	2	1	no	—
444	C₂×C₃×C₄	—	2	1	no	—
445	C₂×C₃×C₄	—	2	1	no	—
446	C₂×C₃×C₄	—	4	1	no	—
447	C₂×C₃×C₄	—	4	1	no	—
448	C₂×C₃×C₄	—	9	1	no	—
449	C₂×C₃×C₄	—	9	1	no	—
450	C₂×C₃×C₄	—	9	1	no	—
451	C₂×C₃×C₄	—	9	1	no	—
452	C₂×C₃×C₄	—	9	1	no	—
453	C₂×C₃×C₄	—	18	1	no	—
454	C₂×C₃×C₄	—	18	1	no	—
455	C₂×C₃×C₄	—	18	2	no	—
456	C₂×C₃×C₄	—	18	2	no	—
457	C₂×C₃×C₄	—	18	1	no	—
458	C₂×C₃×C₄	—	18	1	no	—
459	C₂×C₃×C₄	—	18	1	no	—
460	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
461	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
462	#67	—	6	1	no	—
463	#67	—	6	1	no	—
464	#67	—	6	1	no	—
465	C₂×Dic₃	—	3	1	no	—
466	C₂×Dic₃	—	6	1	no	—
467	C₂×Dic₃	—	6	1	no	—
468	D₁₂	—	3	1	no	—
469	D₁₂	—	6	1	no	—
470	C₄×D₃	—	3	1	no	—
471	C₄×D₃	—	3	1	no	—
472	C₄×D₃	—	3	1	no	—
473	C₄×D₃	—	3	1	no	—
474	C₄×D₃	—	6	1	no	—
475	C₄×D₃	—	6	1	no	—
476	C₄×D₃	—	6	2	no	—
477	C₄×D₃	—	6	2	no	—
478	C₄×D₃	—	6	1	no	—
479	Dic₆	—	3	2	no	—
480	Dic₆	—	3	1	no	—
481	Dic₆	—	6	2	no	—
482	Dic₆	—	6	2	no	—
483	Dic₆	—	6	1	no	—
484	Dic₆	—	6	1	no	—
485	C₃×C₈	—	2	2	no	—
486	C₃×C₈	—	9	2	no	—
487	C₃×C₈	C₂²×D₉	1	2	yes	—
488	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
489	C₃⋊C₈	—	3	2	no	—
490	C₂×C₃²	—	2	1	no	—
491	C₂×C₃²	—	4	1	no	—
492	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
493	C₃×D₃	—	3	2	no	—
494	C₃×D₃	—	6	1	no	—
495	C₃×D₃	—	12	1	no	—
496	C₂×C₉	—	2	1	no	—
497	C₂×C₉	—	2	1	no	—
498	C₂×C₉	—	4	1	no	—
499	C₂×C₉	—	4	1	no	—
500	C₂×C₉	—	8	1	no	—
501	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
502	D₉	—	2	1	no	—
503	D₉	—	4	1	no	—
504	D₉	C₃×#137	1	2	yes	direct product factor
505	#41	—	9	1	no	—
506	#41	—	9	1	no	—
507	#41	—	9	1	no	—
508	#41	—	18	1	no	—
509	#41	—	18	1	no	—
510	#41	C₂×C₃×D₉	1	1	yes	characteristic
511	C₂×Q₈	—	9	2	no	—
512	C₂×Q₈	—	9	1	no	—
513	C₂×Q₈	—	9	2	no	—
514	C₂×Q₈	—	9	1	no	—
515	C₂×Q₈	—	18	1	no	—
516	C₂×Q₈	—	18	1	no	—
517	C₂×Q₈	—	18	1	no	—
518	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic, complemented
519	C₂×D₄	—	9	1	no	—
520	C₂×D₄	—	18	1	no	—
521	C₂²×C₄	—	9	1	no	—
522	C₂²×C₄	—	18	1	no	—
523	C₂²×C₄	—	18	1	no	—
524	Q₁₆	—	9	2	no	—
525	Q₁₆	—	9	2	no	—
526	Q₁₆	—	9	2	no	—
527	Q₁₆	C₂×C₃×D₉	1	2	yes	complemented
528	SD₁₆	—	9	2	no	—
529	SD₁₆	—	9	2	no	—
530	SD₁₆	—	9	2	no	—
531	SD₁₆	C₂×C₃×D₉	1	2	yes	complemented
532	M₁₆	—	9	1	no	—
533	M₁₆	—	9	2	no	—
534	M₁₆	C₂×C₃×D₉	1	1	yes	characteristic, complemented
535	C₂×C₈	—	9	2	no	—
536	C₂²×C₃	—	2	1	no	—
537	C₂²×C₃	—	2	1	no	—
538	C₂²×C₃	—	2	1	no	—
539	C₂²×C₃	—	4	1	no	—
540	C₂²×C₃	—	9	1	no	—
541	C₂²×C₃	—	9	1	no	—
542	C₂²×C₃	—	18	2	no	—
543	C₂²×C₃	—	18	1	no	—
544	C₂²×C₃	—	36	2	no	—
545	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
546	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
547	C₂×D₃	—	3	1	no	—
548	C₂×D₃	—	3	1	no	—
549	C₂×D₃	—	6	2	no	—
550	C₂×D₃	—	6	1	no	—
551	C₂×D₃	—	12	2	no	—
552	C₃×C₄	—	2	2	no	—
553	C₃×C₄	—	2	1	no	—
554	C₃×C₄	—	2	1	no	—
555	C₃×C₄	—	2	2	no	—
556	C₃×C₄	—	2	1	no	—
557	C₃×C₄	—	2	1	no	—
558	C₃×C₄	—	4	1	no	—
559	C₃×C₄	—	4	2	no	—
560	C₃×C₄	—	9	1	no	—
561	C₃×C₄	—	9	1	no	—
562	C₃×C₄	—	18	1	no	—
563	C₃×C₄	—	18	2	no	—
564	C₃×C₄	D₄×D₉	1	1	yes	characteristic
565	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
566	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
567	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
568	Dic₃	—	3	1	no	—
569	Dic₃	—	3	1	no	—
570	Dic₃	—	6	2	no	—
571	Dic₃	—	6	1	no	—
572	C₃²	D₃×#137	1	1	yes	characteristic
573	C₉	—	2	1	no	—
574	C₉	C₂×C₃×#137	1	1	yes	characteristic, complemented
575	C₂³	—	9	1	no	—
576	C₂³	—	18	1	no	—
577	Q₈	—	2	1	no	—
578	Q₈	—	9	2	no	—
579	Q₈	—	9	1	no	—
580	Q₈	—	18	2	no	—
581	Q₈	—	18	2	no	—
582	Q₈	—	18	1	no	—
583	Q₈	—	18	1	no	—
584	Q₈	C₂²×C₃×D₉	1	2	yes	—
585	Q₈	C₂²×C₃×D₉	1	1	yes	characteristic
586	D₄	—	2	1	no	—
587	D₄	—	9	1	no	—
588	D₄	—	18	1	no	—
589	D₄	—	18	1	no	—
590	D₄	—	18	1	no	—
591	D₄	—	18	1	no	—
592	D₄	C₂²×C₃×D₉	1	1	yes	characteristic
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	9	1	no	—
596	C₂×C₄	—	9	1	no	—
597	C₂×C₄	—	9	1	no	—
598	C₂×C₄	—	9	1	no	—
599	C₂×C₄	—	9	1	no	—
600	C₂×C₄	—	18	1	no	—
601	C₂×C₄	—	18	1	no	—
602	C₂×C₄	—	18	2	no	—
603	C₂×C₄	—	18	2	no	—
604	C₂×C₄	—	18	1	no	—
605	C₂×C₄	—	18	1	no	—
606	C₂×C₄	—	18	1	no	—
607	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
608	C₈	—	9	2	no	—
609	C₈	C₂²×C₃×D₉	1	2	yes	—
610	C₂×C₃	—	2	1	no	—
611	C₂×C₃	—	2	1	no	—
612	C₂×C₃	—	2	1	no	—
613	C₂×C₃	—	4	1	no	—
614	C₂×C₃	—	4	1	no	—
615	C₂×C₃	—	4	1	no	—
616	C₂×C₃	—	8	1	no	—
617	C₂×C₃	—	9	2	no	—
618	C₂×C₃	—	18	1	no	—
619	C₂×C₃	—	36	1	no	—
620	C₂×C₃	C₂×D₄×D₉	1	1	yes	characteristic
621	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
622	D₃	—	3	2	no	—
623	D₃	—	6	1	no	—
624	D₃	—	12	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	9	1	no	—
627	C₂²	—	9	1	no	—
628	C₂²	—	18	2	no	—
629	C₂²	—	18	1	no	—
630	C₂²	—	36	2	no	—
631	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
632	C₄	—	2	1	no	—
633	C₄	—	2	2	no	—
634	C₄	—	9	1	no	—
635	C₄	—	9	1	no	—
636	C₄	—	18	1	no	—
637	C₄	—	18	2	no	—
638	C₄	C₃×D₄×D₉	1	1	yes	characteristic
639	C₄	C₂³×C₃×D₉	1	1	yes	characteristic
640	C₃	—	2	1	no	—
641	C₃	D₉×#137	1	1	yes	characteristic, direct product factor
642	C₃	C₃×D₃×#137	1	1	yes	characteristic
643	C₂	—	2	1	no	—
644	C₂	—	4	1	no	—
645	C₂	—	9	2	no	—
646	C₂	—	18	1	no	—
647	C₂	—	36	1	no	—
648	C₂	C₂×C₃×D₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×D₉
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₃×C₉×#137	C₂
socle	C₂×C₃²	C₂×D₃×D₄
prime radical	C₃×#41×D₉	C₂
hypercenter	C₃×#137	D₉
hypocenter	C₉	C₂×C₃×#137
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#137

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Q₈×D₉	C₂	semidirect
3. C₃×C₉×#137	C₂	semidirect
4. C₃×#63511	C₂	semidirect
5. C₃×#63505	C₂	semidirect
6. C₃×#63480	C₂	semidirect
7. C₃×Q₁₆×D₉	C₂	semidirect
8. C₃×#63477	C₂	semidirect
9. C₃×SD₁₆×D₉	C₂	semidirect
10. C₃×#63471	C₂	semidirect
11. C₃×M₁₆×D₉	C₂	semidirect
13. D₉×#137	C₃	direct (N × H)
15. C₂×C₃×Q₈×C₉	C₂²	semidirect
21. C₃×Q₈×D₉	C₂²	semidirect
32. C₂×C₃×Dic₁₈	C₂²	semidirect
33. C₃×C₉×Q₁₆	C₂²	semidirect
34. C₃×C₉×SD₁₆	C₂²	semidirect
35. C₃×C₉×M₁₆	C₂²	semidirect
36. C₃×#3675	C₂²	semidirect
37. C₃×#3674	C₂²	semidirect
38. C₃×#3674	C₂²	semidirect
39. C₃×#3672	C₂²	semidirect
40. C₃×#3667	C₂²	semidirect
41. C₃×#3664	C₂²	semidirect
42. C₃×#3663	C₂²	semidirect
44. C₃×Dic₃₆	C₂²	semidirect
57. C₂×Q₈×D₉	C₂×C₃	semidirect
59. C₉×#137	C₂×C₃	semidirect
60. #63511	C₂×C₃	semidirect
61. #63505	C₂×C₃	semidirect
62. #63480	C₂×C₃	semidirect
63. Q₁₆×D₉	C₂×C₃	semidirect
64. #63477	C₂×C₃	semidirect
65. SD₁₆×D₉	C₂×C₃	semidirect
66. #63471	C₂×C₃	semidirect
67. M₁₆×D₉	C₂×C₃	semidirect
139. C₂×Q₈×C₉	C₂²×C₃	semidirect
145. Q₈×D₉	C₂²×C₃	semidirect
156. C₂×Dic₁₈	C₂²×C₃	semidirect
158. C₉×Q₁₆	C₂²×C₃	semidirect
160. C₉×SD₁₆	C₂²×C₃	semidirect
162. C₉×M₁₆	C₂²×C₃	semidirect
163. #3675	C₂²×C₃	semidirect
164. #3674	C₂²×C₃	semidirect
165. #3674	C₂²×C₃	semidirect
166. #3672	C₂²×C₃	semidirect
167. #3667	C₂²×C₃	semidirect
168. #3664	C₂²×C₃	semidirect
169. #3663	C₂²×C₃	semidirect
171. Dic₃₆	C₂²×C₃	semidirect
197. C₃×#137	D₉	direct (N × H)
295. C₃×D₉	#137	direct (N × H)
312. C₂×C₃×Q₈	C₂×D₉	semidirect
340. C₃×Q₁₆	C₂×D₉	semidirect
346. C₃×SD₁₆	C₂×D₉	semidirect
351. C₃×M₁₆	C₂×D₉	semidirect
406. #137	C₃×D₉	direct (N × H)
410. C₃×C₉	C₂×#137	semidirect
504. D₉	C₃×#137	direct (N × H)
518. C₂×Q₈	C₂×C₃×D₉	semidirect
527. Q₁₆	C₂×C₃×D₉	semidirect
531. SD₁₆	C₂×C₃×D₉	semidirect
534. M₁₆	C₂×C₃×D₉	semidirect
574. C₉	C₂×C₃×#137	semidirect
641. C₃	D₉×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#41×D₉	2
C₂×C₃×Q₈×D₉	2
C₃×C₉×#137	2
C₃×#63511	2
C₃×#63505	2
C₃×#63480	2
C₃×Q₁₆×D₉	2
C₃×#63477	2
C₃×SD₁₆×D₉	2
C₃×#63471	2
C₃×M₁₆×D₉	2
C₃×D₃×#137	3
D₉×#137	3

Subgroups — C3×D9×#137 #13153507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₉×#137 #13153507