Subgroups — C₃×#21559 #10640166

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3067	C₂	1	1	yes	maximal, characteristic
2	C₃×#2955	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1932	C₂	1	2	yes	maximal, complemented
8	C₃×#1906	C₂	1	2	yes	maximal, complemented
9	C₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#1757	C₂	1	2	yes	maximal, complemented
12	#21559	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#575	C₂²	1	2	yes	—
14	C₂×C₃×#135	C₂²	1	1	yes	characteristic
15	C₂×C₃×#131	C₂²	1	1	yes	characteristic
16	C₂×C₃×#131	C₂²	1	2	yes	—
17	C₂×C₃×#131	C₂²	1	1	yes	characteristic
18	C₃×C₄×#41	C₂²	1	1	yes	characteristic
19	C₃×#473	C₄	1	2	yes	complemented
20	C₃×#472	C₄	1	2	yes	complemented
21	C₃×#471	C₄	1	4	yes	complemented
22	C₃×#444	—	2	4	no	—
23	C₃×#444	—	2	1	no	—
24	C₃×#444	—	2	1	no	—
25	C₃×#444	—	2	1	no	—
26	C₃×#444	—	2	1	no	—
27	C₃×#443	—	2	4	no	—
28	C₃×#430	C₂²	1	2	yes	—
29	C₂×C₃×#108	C₂²	1	1	yes	characteristic
30	C₂×C₃×#108	C₂²	1	1	yes	characteristic
31	C₃×#427	C₂²	1	1	yes	characteristic
32	C₃×#421	C₂²	1	2	yes	—
33	C₃×#421	C₂²	1	2	yes	—
34	C₂×C₃×#104	C₂²	1	2	yes	—
35	C₃×#413	C₂²	1	2	yes	—
36	C₃×#410	C₂²	1	2	yes	—
37	C₃×#405	—	2	1	no	—
38	C₃×#405	—	2	1	no	—
39	C₃×#405	C₂²	1	1	yes	characteristic
40	C₃×#405	C₂²	1	1	yes	characteristic
41	C₃×#405	C₂²	1	2	yes	—
42	C₃×C₄×M₁₆	—	2	1	no	—
43	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
44	C₂×C₃×#97	—	2	1	no	—
45	C₂×C₃×#97	C₂²	1	1	yes	characteristic
46	C₃×#344	C₂²	1	2	yes	—
47	C₃×#344	C₂²	1	2	yes	—
48	C₃×#339	C₂²	1	2	yes	—
49	#3067	C₂×C₃	1	1	yes	characteristic
50	#2955	C₂×C₃	1	2	yes	complemented
51	C₂×#444	C₂×C₃	1	1	yes	characteristic, complemented
52	#2875	C₂×C₃	1	1	yes	characteristic, complemented
53	#1976	C₂×C₃	1	1	yes	characteristic, complemented
54	#1953	C₂×C₃	1	1	yes	characteristic, complemented
55	#1932	C₂×C₃	1	2	yes	complemented
56	#1906	C₂×C₃	1	2	yes	complemented
57	#1814	C₂×C₃	1	1	yes	characteristic, complemented
58	#1773	C₂×C₃	1	1	yes	characteristic, complemented
59	#1757	C₂×C₃	1	2	yes	complemented
60	C₂×C₃×#41	C₂³	1	2	yes	—
61	C₂×C₃×#41	C₂³	1	1	yes	characteristic
62	C₃×#137	—	2	2	no	—
63	C₃×#137	C₂×C₄	1	4	yes	—
64	C₃×AGL(1,ℤ/8)	—	2	2	no	—
65	C₃×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
66	C₃×#135	—	2	4	no	—
67	C₃×#135	—	2	1	no	—
68	C₃×#135	—	2	1	no	—
69	C₃×#135	—	2	1	no	—
70	C₃×#135	—	2	1	no	—
71	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	complemented
73	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
74	C₃×#131	—	2	1	no	—
75	C₃×#131	—	2	1	no	—
76	C₃×#131	—	2	2	no	—
77	C₃×#131	—	2	2	no	—
78	C₃×#131	—	2	2	no	—
79	C₃×#131	—	2	2	no	—
80	C₃×#131	—	2	2	no	—
81	C₃×#131	C₂×C₄	1	2	yes	—
82	C₃×#131	C₂×C₄	1	2	yes	—
83	C₃×#131	D₄	1	4	yes	—
84	C₂×C₃×M₁₆	—	2	1	no	—
85	C₂×C₃×M₁₆	—	2	1	no	—
86	C₂×C₃×M₁₆	—	2	2	no	—
87	C₂×C₃×M₁₆	C₂³	1	2	yes	—
88	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₃×M₁₆	C₂³	1	2	yes	—
90	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
91	C₂²×C₃×C₈	—	2	1	no	—
92	C₂²×C₃×C₈	—	2	1	no	—
93	C₂²×C₃×C₈	—	2	2	no	—
94	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
96	C₃×C₄×Q₈	—	2	1	no	—
97	C₃×C₄×D₄	—	2	1	no	—
98	C₃×C₄×D₄	—	2	2	no	—
99	C₃×#117	—	2	1	no	—
100	C₃×#117	C₂³	1	2	yes	—
101	C₃×#117	C₂³	1	1	yes	characteristic
102	C₂×C₃×C₄²	—	2	1	no	—
103	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
104	C₃×#108	—	2	4	no	—
105	C₃×#108	—	2	1	no	—
106	C₃×#108	—	2	1	no	—
107	C₃×#108	C₂×C₄	1	2	yes	complemented
108	C₃×#108	C₂×C₄	1	2	yes	complemented
109	C₃×#107	—	2	1	no	—
110	C₃×#106	—	2	1	no	—
111	C₃×#104	—	2	4	no	—
112	C₃×#104	—	2	4	no	—
113	C₃×#104	—	2	2	no	—
114	C₃×#104	—	2	2	no	—
115	C₃×#101	C₂×C₄	1	4	yes	complemented
116	C₃×#100	C₂×C₄	1	4	yes	complemented
117	C₃×#99	—	2	4	no	—
118	C₃×#97	—	2	1	no	—
119	C₃×#97	—	2	1	no	—
120	C₃×#97	—	2	2	no	—
121	C₃×#97	—	2	1	no	—
122	C₃×#97	D₄	1	1	yes	characteristic
123	C₃×#97	D₄	1	1	yes	characteristic
124	C₃×#97	D₄	1	1	yes	characteristic
125	C₃×#97	D₄	1	1	yes	characteristic
126	C₃×C₄×C₈	—	2	1	no	—
127	C₃×C₄×C₈	—	2	1	no	—
128	C₃×C₄×C₈	—	2	2	no	—
129	C₃×C₄×C₈	—	2	1	no	—
130	#575	C₂²×C₃	1	2	yes	—
131	C₂×#135	C₂²×C₃	1	1	yes	characteristic
132	C₂×#131	C₂²×C₃	1	2	yes	—
133	C₂×#131	C₂²×C₃	1	1	yes	characteristic
134	C₂×#131	C₂²×C₃	1	1	yes	characteristic
135	C₄×#41	C₂²×C₃	1	1	yes	characteristic
136	#473	C₃×C₄	1	2	yes	complemented
137	#472	C₃×C₄	1	2	yes	complemented
138	#471	C₃×C₄	1	4	yes	complemented
139	#444	—	2	4	no	—
140	#444	—	2	1	no	—
141	#444	—	2	1	no	—
142	#444	—	2	1	no	—
143	#444	—	2	1	no	—
144	#443	—	2	4	no	—
145	#430	C₂²×C₃	1	2	yes	—
146	C₂×#108	C₂²×C₃	1	1	yes	characteristic
147	C₂×#108	C₂²×C₃	1	1	yes	characteristic
148	#427	C₂²×C₃	1	1	yes	characteristic
149	#421	C₂²×C₃	1	2	yes	—
150	#421	C₂²×C₃	1	2	yes	—
151	C₂×#104	C₂²×C₃	1	2	yes	—
152	#413	C₂²×C₃	1	2	yes	—
153	#410	C₂²×C₃	1	2	yes	—
154	#405	—	2	1	no	—
155	#405	—	2	1	no	—
156	#405	C₂²×C₃	1	1	yes	characteristic
157	#405	C₂²×C₃	1	1	yes	characteristic
158	#405	C₂²×C₃	1	2	yes	—
159	C₄×M₁₆	—	2	1	no	—
160	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
161	C₂×#97	—	2	1	no	—
162	C₂×#97	C₂²×C₃	1	1	yes	characteristic
163	#344	C₂²×C₃	1	2	yes	—
164	#344	C₂²×C₃	1	2	yes	—
165	#339	C₂²×C₃	1	2	yes	—
166	C₃×#41	—	2	1	no	—
167	C₃×#41	—	2	1	no	—
168	C₃×#41	—	2	2	no	—
169	C₃×#41	—	2	2	no	—
170	C₃×#41	—	2	2	no	—
171	C₃×#41	—	2	2	no	—
172	C₃×#41	C₂²×C₄	1	2	yes	—
173	C₃×#41	C₂×D₄	1	2	yes	—
174	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
175	C₂×C₃×Q₈	#41	1	1	yes	characteristic
176	C₂×C₃×D₄	—	2	1	no	—
177	C₂×C₃×D₄	—	2	2	no	—
178	C₂×C₃×D₄	C₂²×C₄	1	2	yes	—
179	C₂×C₃×D₄	#41	1	1	yes	characteristic
180	C₂²×C₃×C₄	—	2	1	no	—
181	C₂²×C₃×C₄	—	2	2	no	—
182	C₂²×C₃×C₄	—	2	1	no	—
183	C₂²×C₃×C₄	—	2	2	no	—
184	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
185	C₃×Q₁₆	—	2	4	no	—
186	C₃×Q₁₆	—	2	1	no	—
187	C₃×Q₁₆	—	2	1	no	—
188	C₃×SD₁₆	—	2	4	no	—
189	C₃×SD₁₆	—	2	4	no	—
190	C₃×SD₁₆	—	2	2	no	—
191	C₃×SD₁₆	—	2	2	no	—
192	C₃×D₈	—	2	4	no	—
193	C₃×D₈	—	2	1	no	—
194	C₃×D₈	—	2	1	no	—
195	C₃×M₁₆	—	2	1	no	—
196	C₃×M₁₆	—	2	1	no	—
197	C₃×M₁₆	—	2	2	no	—
198	C₃×M₁₆	—	2	2	no	—
199	C₃×M₁₆	—	2	2	no	—
200	C₃×M₁₆	—	2	4	no	—
201	C₃×M₁₆	—	2	2	no	—
202	C₃×M₁₆	—	2	2	no	—
203	C₃×M₁₆	—	4	2	no	—
204	C₃×M₁₆	—	4	2	no	—
205	C₃×M₁₆	C₂²×C₄	1	2	yes	—
206	C₃×M₁₆	C₂²×C₄	1	4	yes	—
207	C₃×M₁₆	C₂×D₄	1	2	yes	—
208	C₂×C₃×C₈	—	2	1	no	—
209	C₂×C₃×C₈	—	2	1	no	—
210	C₂×C₃×C₈	—	2	2	no	—
211	C₂×C₃×C₈	—	2	2	no	—
212	C₂×C₃×C₈	—	2	2	no	—
213	C₂×C₃×C₈	—	2	4	no	—
214	C₂×C₃×C₈	—	2	2	no	—
215	C₂×C₃×C₈	—	4	2	no	—
216	C₂×C₃×C₈	—	4	2	no	—
217	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
218	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
219	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
220	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
221	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
222	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
223	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
224	C₂×C₃×C₈	#41	1	1	yes	characteristic
225	C₂×C₃×C₈	#41	1	1	yes	characteristic
226	C₂×C₃×C₈	#41	1	2	yes	—
227	C₃×C₄⋊C₄	—	2	2	no	—
228	C₃×C₄⋊C₄	—	2	1	no	—
229	C₃×C₄⋊C₄	—	2	2	no	—
230	C₃×C₂²⋊C₄	—	2	2	no	—
231	C₃×C₂²⋊C₄	—	2	2	no	—
232	C₃×C₂²⋊C₄	#41	1	2	yes	—
233	C₃×C₄²	—	2	2	no	—
234	C₃×C₄²	—	2	1	no	—
235	C₃×C₄²	—	2	1	no	—
236	C₃×C₄²	—	2	2	no	—
237	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
238	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
239	C₂×#41	C₂³×C₃	1	2	yes	—
240	C₂×#41	C₂³×C₃	1	1	yes	characteristic
241	#137	—	2	2	no	—
242	#137	C₂×C₃×C₄	1	4	yes	—
243	AGL(1,ℤ/8)	—	2	2	no	—
244	AGL(1,ℤ/8)	C₂×C₃×C₄	1	4	yes	—
245	#135	—	2	4	no	—
246	#135	—	2	1	no	—
247	#135	—	2	1	no	—
248	#135	—	2	1	no	—
249	#135	—	2	1	no	—
250	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic, complemented
251	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	complemented
252	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
253	#131	—	2	1	no	—
254	#131	—	2	1	no	—
255	#131	—	2	2	no	—
256	#131	—	2	2	no	—
257	#131	—	2	2	no	—
258	#131	—	2	2	no	—
259	#131	—	2	2	no	—
260	#131	C₂×C₃×C₄	1	2	yes	—
261	#131	C₂×C₃×C₄	1	2	yes	—
262	#131	C₃×D₄	1	4	yes	—
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	C₂³×C₃	1	2	yes	—
267	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
268	C₂×M₁₆	C₂³×C₃	1	2	yes	—
269	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	2	no	—
273	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
275	C₄×Q₈	—	2	1	no	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	—	2	2	no	—
278	#117	—	2	1	no	—
279	#117	C₂³×C₃	1	2	yes	—
280	#117	C₂³×C₃	1	1	yes	characteristic
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
283	#108	—	2	4	no	—
284	#108	—	2	1	no	—
285	#108	—	2	1	no	—
286	#108	C₂×C₃×C₄	1	2	yes	complemented
287	#108	C₂×C₃×C₄	1	2	yes	complemented
288	#107	—	2	1	no	—
289	#106	—	2	1	no	—
290	#104	—	2	4	no	—
291	#104	—	2	4	no	—
292	#104	—	2	2	no	—
293	#104	—	2	2	no	—
294	#101	C₂×C₃×C₄	1	4	yes	complemented
295	#100	C₂×C₃×C₄	1	4	yes	complemented
296	#99	—	2	4	no	—
297	#97	—	2	1	no	—
298	#97	—	2	1	no	—
299	#97	—	2	2	no	—
300	#97	—	2	1	no	—
301	#97	C₃×D₄	1	1	yes	characteristic
302	#97	C₃×D₄	1	1	yes	characteristic
303	#97	C₃×D₄	1	1	yes	characteristic
304	#97	C₃×D₄	1	1	yes	characteristic
305	C₄×C₈	—	2	1	no	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	1	no	—
309	C₂³×C₃	—	2	1	no	—
310	C₂³×C₃	—	2	2	no	—
311	C₂³×C₃	C₂×#41	1	1	yes	characteristic
312	C₃×Q₈	—	2	1	no	—
313	C₃×Q₈	—	2	2	no	—
314	C₃×Q₈	—	2	2	no	—
315	C₃×Q₈	C₄×D₄	1	2	yes	—
316	C₃×D₄	—	2	2	no	—
317	C₃×D₄	—	2	2	no	—
318	C₃×D₄	—	2	2	no	—
319	C₃×D₄	—	2	2	no	—
320	C₃×D₄	—	2	1	no	—
321	C₃×D₄	—	4	2	no	—
322	C₃×D₄	—	4	2	no	—
323	C₃×D₄	C₄×D₄	1	2	yes	—
324	C₂×C₃×C₄	—	2	1	no	—
325	C₂×C₃×C₄	—	2	2	no	—
326	C₂×C₃×C₄	—	2	2	no	—
327	C₂×C₃×C₄	—	2	2	no	—
328	C₂×C₃×C₄	—	2	1	no	—
329	C₂×C₃×C₄	—	2	2	no	—
330	C₂×C₃×C₄	—	2	1	no	—
331	C₂×C₃×C₄	—	4	2	no	—
332	C₂×C₃×C₄	—	4	2	no	—
333	C₂×C₃×C₄	—	4	2	no	—
334	C₂×C₃×C₄	—	8	1	no	—
335	C₂×C₃×C₄	—	8	2	no	—
336	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
337	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
339	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
340	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
341	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
342	C₃×C₈	—	2	2	no	—
343	C₃×C₈	—	2	4	no	—
344	C₃×C₈	—	2	2	no	—
345	C₃×C₈	—	4	2	no	—
346	C₃×C₈	C₄×D₄	1	1	yes	characteristic
347	C₃×C₈	C₄×D₄	1	1	yes	characteristic
348	C₃×C₈	C₄×D₄	1	1	yes	characteristic
349	C₃×C₈	C₄×D₄	1	1	yes	characteristic
350	C₃×C₈	#121	1	2	yes	—
351	C₃×C₈	#121	1	2	yes	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	#41	C₂²×C₃×C₄	1	2	yes	—
359	#41	C₂×C₃×D₄	1	2	yes	—
360	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
361	C₂×Q₈	C₃×#41	1	1	yes	characteristic
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	C₂²×C₃×C₄	1	2	yes	—
365	C₂×D₄	C₃×#41	1	1	yes	characteristic
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
371	Q₁₆	—	2	4	no	—
372	Q₁₆	—	2	1	no	—
373	Q₁₆	—	2	1	no	—
374	SD₁₆	—	2	4	no	—
375	SD₁₆	—	2	4	no	—
376	SD₁₆	—	2	2	no	—
377	SD₁₆	—	2	2	no	—
378	D₈	—	2	4	no	—
379	D₈	—	2	1	no	—
380	D₈	—	2	1	no	—
381	M₁₆	—	2	1	no	—
382	M₁₆	—	2	1	no	—
383	M₁₆	—	2	2	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	2	4	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	4	2	no	—
390	M₁₆	—	4	2	no	—
391	M₁₆	C₂²×C₃×C₄	1	2	yes	—
392	M₁₆	C₂²×C₃×C₄	1	4	yes	—
393	M₁₆	C₂×C₃×D₄	1	2	yes	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	2	no	—
402	C₂×C₈	—	4	2	no	—
403	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
404	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
405	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
406	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
407	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
408	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
409	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
410	C₂×C₈	C₃×#41	1	1	yes	characteristic
411	C₂×C₈	C₃×#41	1	1	yes	characteristic
412	C₂×C₈	C₃×#41	1	2	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	C₃×#41	1	2	yes	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	2	no	—
423	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
424	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
425	C₂²×C₃	—	2	2	no	—
426	C₂²×C₃	—	4	1	no	—
427	C₂²×C₃	—	4	2	no	—
428	C₂²×C₃	—	4	2	no	—
429	C₂²×C₃	—	8	2	no	—
430	C₂²×C₃	C₄×#41	1	1	yes	characteristic
431	C₂²×C₃	#525	1	1	yes	characteristic
432	C₂²×C₃	#532	1	1	yes	characteristic
433	C₃×C₄	—	2	2	no	—
434	C₃×C₄	—	4	2	no	—
435	C₃×C₄	—	4	1	no	—
436	C₃×C₄	—	4	1	no	—
437	C₃×C₄	—	4	2	no	—
438	C₃×C₄	—	8	2	no	—
439	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
440	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
441	C₃×C₄	C₂×#121	1	1	yes	characteristic
442	C₃×C₄	#529	1	1	yes	characteristic
443	C₂³	—	2	1	no	—
444	C₂³	—	2	2	no	—
445	C₂³	C₂×C₃×#41	1	1	yes	characteristic
446	Q₈	—	2	1	no	—
447	Q₈	—	2	2	no	—
448	Q₈	—	2	2	no	—
449	Q₈	C₃×C₄×D₄	1	2	yes	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	2	no	—
454	D₄	—	2	1	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	C₃×C₄×D₄	1	2	yes	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	8	1	no	—
469	C₂×C₄	—	8	2	no	—
470	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
471	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
472	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
473	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
474	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
475	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
476	C₈	—	2	2	no	—
477	C₈	—	2	4	no	—
478	C₈	—	2	2	no	—
479	C₈	—	4	2	no	—
480	C₈	C₃×C₄×D₄	1	1	yes	characteristic
481	C₈	C₃×C₄×D₄	1	1	yes	characteristic
482	C₈	C₃×C₄×D₄	1	1	yes	characteristic
483	C₈	C₃×C₄×D₄	1	1	yes	characteristic
484	C₈	C₃×#121	1	2	yes	—
485	C₈	C₃×#121	1	2	yes	—
486	C₂×C₃	—	2	1	no	—
487	C₂×C₃	—	2	1	no	—
488	C₂×C₃	—	2	1	no	—
489	C₂×C₃	—	4	2	no	—
490	C₂×C₃	—	8	2	no	—
491	C₂×C₃	C₄×#121	1	1	yes	characteristic
492	C₂²	—	2	2	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	C₃×C₄×#41	1	1	yes	characteristic
498	C₂²	C₃×#525	1	1	yes	characteristic
499	C₂²	C₃×#532	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	2	no	—
505	C₄	—	8	2	no	—
506	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
507	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
508	C₄	C₂×C₃×#121	1	1	yes	characteristic
509	C₄	C₃×#529	1	1	yes	characteristic
510	C₃	#21559	1	1	yes	characteristic, direct product factor, Sylow
511	C₂	—	2	1	no	—
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	4	2	no	—
515	C₂	—	8	2	no	—
516	C₂	C₃×C₄×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×#121
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	C₄×#121
prime radical	C₃×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2955	C₂	semidirect
3. C₂×C₃×#444	C₂	semidirect
4. C₃×#2875	C₂	semidirect
5. C₃×#1976	C₂	semidirect
6. C₃×#1953	C₂	semidirect
7. C₃×#1932	C₂	semidirect
8. C₃×#1906	C₂	semidirect
9. C₃×#1814	C₂	semidirect
10. C₃×#1773	C₂	semidirect
11. C₃×#1757	C₂	semidirect
12. #21559	C₃	direct (N × H)
19. C₃×#473	C₄	semidirect
20. C₃×#472	C₄	semidirect
21. C₃×#471	C₄	semidirect
50. #2955	C₂×C₃	semidirect
51. C₂×#444	C₂×C₃	semidirect
52. #2875	C₂×C₃	semidirect
53. #1976	C₂×C₃	semidirect
54. #1953	C₂×C₃	semidirect
55. #1932	C₂×C₃	semidirect
56. #1906	C₂×C₃	semidirect
57. #1814	C₂×C₃	semidirect
58. #1773	C₂×C₃	semidirect
59. #1757	C₂×C₃	semidirect
71. C₂×C₃×Q₁₆	C₂×C₄	semidirect
72. C₂×C₃×SD₁₆	C₂×C₄	semidirect
73. C₂×C₃×D₈	C₂×C₄	semidirect
107. C₃×#108	C₂×C₄	semidirect
108. C₃×#108	C₂×C₄	semidirect
115. C₃×#101	C₂×C₄	semidirect
116. C₃×#100	C₂×C₄	semidirect
136. #473	C₃×C₄	semidirect
137. #472	C₃×C₄	semidirect
138. #471	C₃×C₄	semidirect
250. C₂×Q₁₆	C₂×C₃×C₄	semidirect
251. C₂×SD₁₆	C₂×C₃×C₄	semidirect
252. C₂×D₈	C₂×C₃×C₄	semidirect
286. #108	C₂×C₃×C₄	semidirect
287. #108	C₂×C₃×C₄	semidirect
294. #101	C₂×C₃×C₄	semidirect
295. #100	C₂×C₃×C₄	semidirect
510. C₃	#21559	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×#3067	2
C₃×#2955	2
C₂×C₃×#444	2
C₃×#2875	2
C₃×#1976	2
C₃×#1953	2
C₃×#1932	2
C₃×#1906	2
C₃×#1814	2
C₃×#1773	2
C₃×#1757	2
#21559	3

Subgroups — C3×#21559 #10640166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21559 #10640166