Subgroups — #121595

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21177	C₂	1	1	yes	maximal, characteristic
2	#21146	C₂	1	1	yes	maximal, characteristic
3	#17814	C₂	1	1	yes	maximal, characteristic
4	C₂×#1303	C₂	1	1	yes	maximal, characteristic
5	C₂×#1303	C₂	1	1	yes	maximal, characteristic
6	#8079	C₂	1	1	yes	maximal, characteristic
7	#7882	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#430	—	2	1	no	—
11	C₂×#423	—	2	1	no	—
12	C₂×#423	C₂²	1	1	yes	characteristic
13	C₂×#423	C₂²	1	1	yes	characteristic
14	C₂×#423	C₂²	1	1	yes	characteristic
15	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	#1950	—	2	2	no	—
18	#1950	—	2	2	no	—
19	#1940	—	2	2	no	—
20	#1939	—	2	2	no	—
21	#1738	—	2	1	no	—
22	#1728	—	2	1	no	—
23	#1571	—	2	2	no	—
24	#1571	—	2	2	no	—
25	#1567	—	2	2	no	—
26	#1303	—	2	2	no	—
27	#1303	C₄	1	2	yes	—
28	#1303	C₄	1	2	yes	—
29	#1301	—	2	2	no	—
30	#1278	—	2	2	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	—	2	1	no	—
34	C₂²×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₄⋊C₄	—	2	1	no	—
37	C₂²×C₄²	—	2	1	no	—
38	C₂²×C₄²	C₂³	1	1	yes	characteristic
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	—	2	1	no	—
44	#431	C₂×C₄	1	2	yes	—
45	#431	D₄	1	2	yes	—
46	#431	Q₈	1	2	yes	—
47	#430	—	4	2	no	—
48	#430	—	4	2	no	—
49	C₂×#108	—	2	4	no	—
50	#423	—	2	1	no	—
51	#423	—	2	1	no	—
52	#423	—	2	1	no	—
53	#423	—	2	1	no	—
54	#423	—	2	1	no	—
55	#423	—	2	1	no	—
56	#423	—	2	1	no	—
57	#423	—	2	1	no	—
58	#423	—	2	1	no	—
59	#423	—	4	2	no	—
60	#423	—	4	2	no	—
61	#423	C₂×C₄	1	2	yes	—
62	#423	D₄	1	2	yes	—
63	#423	Q₈	1	2	yes	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	—	2	1	no	—
67	C₂×C₄⋊C₈	—	2	1	no	—
68	C₂×C₄⋊C₈	—	2	4	no	—
69	C₂×C₄⋊C₈	—	2	1	no	—
70	C₂×C₄⋊C₈	D₄	1	2	yes	—
71	C₂×C₄⋊C₈	D₄	1	2	yes	—
72	C₂×C₄⋊C₈	Q₈	1	2	yes	—
73	C₂×C₂²⋊C₈	—	2	1	no	—
74	C₂×C₂²⋊C₈	—	2	1	no	—
75	C₄×M₁₆	—	2	1	no	—
76	C₄×M₁₆	—	2	1	no	—
77	C₄×M₁₆	—	2	1	no	—
78	C₄×M₁₆	—	2	1	no	—
79	C₂×#97	—	2	1	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	C₂×C₄×C₈	—	2	1	no	—
83	C₄×C₄⋊C₄	—	2	1	no	—
84	C₄×C₄⋊C₄	—	2	1	no	—
85	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₄×C₄⋊C₄	D₄	1	2	yes	—
88	C₂×#95	Q₈	1	2	yes	—
89	#341	—	2	4	no	—
90	#336	—	2	4	no	—
91	#335	—	2	8	no	—
92	C₂³×C₄	—	2	2	no	—
93	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₄	#41	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂×M₁₆	—	4	1	no	—
116	C₂×M₁₆	—	4	1	no	—
117	C₂×M₁₆	—	4	1	no	—
118	C₂×M₁₆	—	4	1	no	—
119	C₂×M₁₆	—	4	1	no	—
120	C₂×M₁₆	—	4	1	no	—
121	C₂×M₁₆	—	4	1	no	—
122	C₂×M₁₆	—	4	1	no	—
123	C₂×M₁₆	—	4	1	no	—
124	C₂×M₁₆	D₈	1	2	yes	—
125	C₂×M₁₆	Q₁₆	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	1	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	4	1	no	—
132	C₂²×C₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	4	no	—
137	C₂×C₄²	—	2	1	no	—
138	C₂×C₄²	—	2	1	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	4	1	no	—
141	C₂×C₄²	—	4	1	no	—
142	C₂×C₄²	—	4	1	no	—
143	C₂×C₄²	—	4	1	no	—
144	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
151	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
152	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
153	C₂×C₄²	#41	1	1	yes	characteristic
154	C₂×C₄²	#41	1	1	yes	characteristic
155	C₂×C₄²	#41	1	1	yes	characteristic
156	#108	—	4	8	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	4	8	no	—
167	C₄⋊C₈	—	4	1	no	—
168	C₄⋊C₈	—	4	1	no	—
169	C₄⋊C₈	—	4	1	no	—
170	C₄⋊C₈	—	4	1	no	—
171	C₄⋊C₈	—	4	1	no	—
172	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
173	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	4	1	no	—
178	#97	—	4	1	no	—
179	#97	—	4	1	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	C₄×C₈	—	4	1	no	—
186	#95	—	2	2	no	—
187	#95	C₄⋊C₄	1	2	yes	—
188	#95	C₄⋊C₄	1	2	yes	—
189	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	2	no	—
198	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂²×C₄	#120	1	1	yes	characteristic
202	C₂²×C₄	#120	1	1	yes	characteristic
203	C₂²×C₄	#121	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂²×C₄	#123	1	1	yes	characteristic
209	C₂²×C₄	#123	1	1	yes	characteristic
210	C₂²×C₄	#125	1	1	yes	characteristic
211	C₂²×C₄	#125	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
218	C₂²×C₄	#135	1	1	yes	characteristic
219	C₂²×C₄	#135	1	1	yes	characteristic
220	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₄	#137	1	1	yes	characteristic
223	C₂²×C₄	#137	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	1	no	—
231	M₁₆	—	4	1	no	—
232	M₁₆	—	4	1	no	—
233	M₁₆	—	4	1	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	8	1	no	—
239	M₁₆	—	8	1	no	—
240	M₁₆	—	8	1	no	—
241	M₁₆	—	8	1	no	—
242	M₁₆	—	8	1	no	—
243	M₁₆	—	8	1	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	8	2	no	—
260	C₂×C₈	—	8	2	no	—
261	C₂×C₈	—	8	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
273	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
274	C₄²	C₄×D₄	1	1	yes	characteristic
275	C₄²	C₄×Q₈	1	1	yes	characteristic
276	C₄²	#120	1	1	yes	characteristic
277	C₄²	#121	1	1	yes	characteristic
278	C₄²	#121	1	1	yes	characteristic
279	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
280	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
281	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
282	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
283	C₄²	#123	1	1	yes	characteristic
284	C₄²	#125	1	1	yes	characteristic
285	C₄²	#127	1	1	yes	characteristic
286	C₄²	C₄⋊Q₈	1	1	yes	characteristic
287	C₄²	C₄⋊Q₈	1	1	yes	characteristic
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	#385	1	1	yes	characteristic
291	C₂³	#395	1	1	yes	characteristic
292	C₂³	#398	1	1	yes	characteristic
293	C₂³	#474	1	1	yes	characteristic
294	C₂³	#477	1	1	yes	characteristic
295	C₂³	#478	1	1	yes	characteristic
296	C₂³	#479	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	#384	1	1	yes	characteristic
318	C₂×C₄	#385	1	1	yes	characteristic
319	C₂×C₄	#385	1	1	yes	characteristic
320	C₂×C₄	#389	1	1	yes	characteristic
321	C₂×C₄	#389	1	1	yes	characteristic
322	C₂×C₄	#389	1	1	yes	characteristic
323	C₂×C₄	#392	1	1	yes	characteristic
324	C₂×C₄	#395	1	1	yes	characteristic
325	C₂×C₄	#395	1	1	yes	characteristic
326	C₂×C₄	#398	1	1	yes	characteristic
327	C₂×C₄	#398	1	1	yes	characteristic
328	C₂×C₄	#399	1	1	yes	characteristic
329	C₂×C₄	C₄×D₈	1	1	yes	characteristic
330	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
331	C₂×C₄	#440	1	1	yes	characteristic
332	C₂×C₄	#440	1	1	yes	characteristic
333	C₂×C₄	#447	1	1	yes	characteristic
334	C₂×C₄	#447	1	1	yes	characteristic
335	C₂×C₄	#449	1	1	yes	characteristic
336	C₂×C₄	#449	1	1	yes	characteristic
337	C₂×C₄	#451	1	1	yes	characteristic
338	C₂×C₄	#451	1	1	yes	characteristic
339	C₂×C₄	#452	1	1	yes	characteristic
340	C₂×C₄	#452	1	1	yes	characteristic
341	C₂×C₄	#466	1	1	yes	characteristic
342	C₂×C₄	#467	1	1	yes	characteristic
343	C₂×C₄	#469	1	1	yes	characteristic
344	C₂×C₄	#470	1	1	yes	characteristic
345	C₂×C₄	#474	1	1	yes	characteristic
346	C₂×C₄	#477	1	1	yes	characteristic
347	C₂×C₄	#478	1	1	yes	characteristic
348	C₂×C₄	#479	1	1	yes	characteristic
349	C₂×C₄	#499	1	1	yes	characteristic
350	C₂×C₄	#500	1	1	yes	characteristic
351	C₂×C₄	#501	1	1	yes	characteristic
352	C₂×C₄	#501	1	1	yes	characteristic
353	C₈	—	4	4	no	—
354	C₈	—	4	4	no	—
355	C₈	—	4	2	no	—
356	C₈	—	4	2	no	—
357	C₈	—	8	2	no	—
358	C₈	—	8	2	no	—
359	C₈	—	8	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	#1444	1	1	yes	characteristic
371	C₂²	#1862	1	1	yes	characteristic
372	C₂²	#1863	1	1	yes	characteristic
373	C₂²	#2057	1	1	yes	characteristic
374	C₂²	#2059	1	1	yes	characteristic
375	C₂²	#2086	1	1	yes	characteristic
376	C₂²	#2087	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	4	no	—
385	C₄	—	8	2	no	—
386	C₄	#1861	1	2	yes	—
387	C₄	#1862	1	1	yes	characteristic
388	C₄	#1863	1	1	yes	characteristic
389	C₄	#1942	1	1	yes	characteristic
390	C₄	#1944	1	1	yes	characteristic
391	C₄	#1999	1	1	yes	characteristic
392	C₄	#2001	1	1	yes	characteristic
393	C₄	#2057	1	1	yes	characteristic
394	C₄	#2059	1	1	yes	characteristic
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	—	2	1	no	—
401	C₂	#11328	1	1	yes	characteristic
402	C₂	#11334	1	1	yes	characteristic
403	C₂	#11341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#21177	2
#21146	2
#17814	2
C₂×#1303	2
#8079	2
#7882	2