Subgroups — #121496

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 387 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	#21145	C₂	1	1	yes	maximal, characteristic
3	C₂×#1570	C₂	1	1	yes	maximal, characteristic
4	#8095	C₂	1	1	yes	maximal, characteristic
5	#7957	C₂	1	1	yes	maximal, characteristic
6	#7882	C₂	1	1	yes	maximal, characteristic
7	#7877	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×#424	—	2	1	no	—
12	C₂×#423	C₂²	1	1	yes	characteristic
13	C₂×#423	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
16	#2285	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	#1729	—	2	1	no	—
23	#1570	—	2	2	no	—
24	#1570	—	2	2	no	—
25	C₂×#334	—	2	2	no	—
26	#1303	—	2	2	no	—
27	#1286	C₄	1	4	yes	—
28	#1278	—	2	2	no	—
29	#1277	—	2	2	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂³×C₈	—	2	1	no	—
35	C₂×#117	—	2	1	no	—
36	C₂²×C₄²	C₂³	1	1	yes	characteristic
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	C₂×C₄	1	2	yes	—
44	#431	Q₈	1	2	yes	—
45	#430	—	4	2	no	—
46	#430	—	4	2	no	—
47	C₂×#108	—	2	4	no	—
48	#424	—	4	2	no	—
49	#424	—	4	2	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	C₂×C₄	1	2	yes	—
56	#423	D₄	1	2	yes	—
57	#423	Q₈	1	2	yes	—
58	C₂×C₄⋊C₈	—	2	1	no	—
59	C₂×C₄⋊C₈	—	2	1	no	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₄⋊C₈	—	2	1	no	—
62	C₂×C₄⋊C₈	—	2	1	no	—
63	C₂×C₄⋊C₈	—	2	1	no	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	—	2	4	no	—
67	C₂×C₄⋊C₈	—	2	1	no	—
68	C₂×C₄⋊C₈	D₄	1	2	yes	—
69	C₂×C₄⋊C₈	D₄	1	2	yes	—
70	C₂×C₄⋊C₈	Q₈	1	2	yes	—
71	C₂×C₂²⋊C₈	—	2	1	no	—
72	C₂×C₂²⋊C₈	—	2	1	no	—
73	C₄×M₁₆	—	2	1	no	—
74	C₄×M₁₆	—	2	1	no	—
75	C₄×M₁₆	—	2	1	no	—
76	C₄×M₁₆	—	2	1	no	—
77	C₂×#97	—	2	1	no	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	C₂×C₄×C₈	D₄	1	2	yes	—
81	#384	—	2	1	no	—
82	#382	C₂×C₄	1	2	yes	—
83	#380	—	2	1	no	—
84	#380	Q₈	1	2	yes	—
85	C₄×C₄⋊C₄	—	2	1	no	—
86	C₄×C₄⋊C₄	D₄	1	2	yes	—
87	#341	—	2	4	no	—
88	#336	—	2	4	no	—
89	#334	—	2	8	no	—
90	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
92	C₂³×C₄	#41	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	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	2	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	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	SD₁₆	1	2	yes	—
116	C₂×M₁₆	SD₁₆	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	—	4	1	no	—
125	C₂²×C₈	—	4	1	no	—
126	C₂²×C₈	—	4	1	no	—
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	—	4	1	no	—
129	C₂²×C₈	—	4	1	no	—
130	#117	—	4	1	no	—
131	#117	—	4	1	no	—
132	#117	—	4	1	no	—
133	#117	—	4	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₂²⋊C₄	—	2	2	no	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
146	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
147	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
148	C₂×C₄²	#41	1	1	yes	characteristic
149	C₂×C₄²	#41	1	1	yes	characteristic
150	C₂×C₄²	#41	1	1	yes	characteristic
151	#108	—	4	8	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	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	—	4	8	no	—
164	C₄⋊C₈	—	4	1	no	—
165	C₄⋊C₈	—	4	1	no	—
166	C₄⋊C₈	—	4	1	no	—
167	C₄⋊C₈	—	4	1	no	—
168	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	4	1	no	—
172	C₂²⋊C₈	—	4	1	no	—
173	#97	—	4	1	no	—
174	#97	—	4	1	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	C₄⋊C₄	1	4	yes	—
181	#95	—	2	2	no	—
182	#95	C₄⋊C₄	1	4	yes	—
183	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	1	no	—
187	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
190	C₂²×C₄	#120	1	1	yes	characteristic
191	C₂²×C₄	#120	1	1	yes	characteristic
192	C₂²×C₄	#121	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	#123	1	1	yes	characteristic
198	C₂²×C₄	#123	1	1	yes	characteristic
199	C₂²×C₄	#125	1	1	yes	characteristic
200	C₂²×C₄	#125	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
206	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
207	C₂²×C₄	#135	1	1	yes	characteristic
208	C₂²×C₄	#135	1	1	yes	characteristic
209	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂²×C₄	#137	1	1	yes	characteristic
212	C₂²×C₄	#137	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	8	1	no	—
228	M₁₆	—	8	1	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	8	1	no	—
247	C₂×C₈	—	8	1	no	—
248	C₂×C₈	—	8	1	no	—
249	C₂×C₈	—	8	1	no	—
250	C₂×C₈	—	8	1	no	—
251	C₂×C₈	—	8	1	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
261	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
262	C₄²	C₄×D₄	1	1	yes	characteristic
263	C₄²	C₄×Q₈	1	1	yes	characteristic
264	C₄²	#120	1	1	yes	characteristic
265	C₄²	#121	1	1	yes	characteristic
266	C₄²	#121	1	1	yes	characteristic
267	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
268	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
269	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
270	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
271	C₄²	#123	1	1	yes	characteristic
272	C₄²	#125	1	1	yes	characteristic
273	C₄²	#127	1	1	yes	characteristic
274	C₄²	C₄⋊Q₈	1	1	yes	characteristic
275	C₄²	C₄⋊Q₈	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	—	4	1	no	—
278	C₂³	#385	1	1	yes	characteristic
279	C₂³	#395	1	1	yes	characteristic
280	C₂³	#398	1	1	yes	characteristic
281	C₂³	#475	1	1	yes	characteristic
282	C₂³	#476	1	1	yes	characteristic
283	C₂³	#478	1	1	yes	characteristic
284	C₂³	#479	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#384	1	1	yes	characteristic
301	C₂×C₄	#385	1	1	yes	characteristic
302	C₂×C₄	#385	1	1	yes	characteristic
303	C₂×C₄	#389	1	1	yes	characteristic
304	C₂×C₄	#389	1	1	yes	characteristic
305	C₂×C₄	#389	1	1	yes	characteristic
306	C₂×C₄	#392	1	1	yes	characteristic
307	C₂×C₄	#395	1	1	yes	characteristic
308	C₂×C₄	#395	1	1	yes	characteristic
309	C₂×C₄	#398	1	1	yes	characteristic
310	C₂×C₄	#398	1	1	yes	characteristic
311	C₂×C₄	#399	1	1	yes	characteristic
312	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
313	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
314	C₂×C₄	#441	1	1	yes	characteristic
315	C₂×C₄	#442	1	1	yes	characteristic
316	C₂×C₄	#448	1	1	yes	characteristic
317	C₂×C₄	#448	1	1	yes	characteristic
318	C₂×C₄	#449	1	1	yes	characteristic
319	C₂×C₄	#449	1	1	yes	characteristic
320	C₂×C₄	#450	1	1	yes	characteristic
321	C₂×C₄	#450	1	1	yes	characteristic
322	C₂×C₄	#452	1	1	yes	characteristic
323	C₂×C₄	#452	1	1	yes	characteristic
324	C₂×C₄	#465	1	1	yes	characteristic
325	C₂×C₄	#465	1	1	yes	characteristic
326	C₂×C₄	#468	1	1	yes	characteristic
327	C₂×C₄	#468	1	1	yes	characteristic
328	C₂×C₄	#475	1	1	yes	characteristic
329	C₂×C₄	#476	1	1	yes	characteristic
330	C₂×C₄	#478	1	1	yes	characteristic
331	C₂×C₄	#479	1	1	yes	characteristic
332	C₂×C₄	#498	1	1	yes	characteristic
333	C₂×C₄	#499	1	1	yes	characteristic
334	C₂×C₄	#501	1	1	yes	characteristic
335	C₂×C₄	#501	1	1	yes	characteristic
336	C₈	—	4	4	no	—
337	C₈	—	4	4	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	8	2	no	—
341	C₈	—	8	2	no	—
342	C₈	—	8	2	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	#1444	1	1	yes	characteristic
354	C₂²	#1864	1	1	yes	characteristic
355	C₂²	#1865	1	1	yes	characteristic
356	C₂²	#2058	1	1	yes	characteristic
357	C₂²	#2059	1	1	yes	characteristic
358	C₂²	#2088	1	1	yes	characteristic
359	C₂²	#2088	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	4	no	—
368	C₄	#1864	1	1	yes	characteristic
369	C₄	#1865	1	1	yes	characteristic
370	C₄	#1941	1	1	yes	characteristic
371	C₄	#1943	1	1	yes	characteristic
372	C₄	#2000	1	1	yes	characteristic
373	C₄	#2000	1	1	yes	characteristic
374	C₄	#2058	1	1	yes	characteristic
375	C₄	#2059	1	1	yes	characteristic
376	C₄	#2098	1	2	yes	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	#11327	1	1	yes	characteristic
384	C₂	#11334	1	1	yes	characteristic
385	C₂	#11336	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
#21145	2
C₂×#1570	2
#8095	2
#7957	2
#7882	2
#7877	2