Subgroups — #483147

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#500	C₂	1	1	yes	maximal, characteristic
2	#33418	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#11974	C₂	1	4	yes	maximal, complemented
5	#11971	C₂	1	4	yes	maximal, complemented
6	#11940	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄⋊Q₈	—	2	1	no	—
8	C₂×#500	—	2	1	no	—
9	C₂×#500	—	2	1	no	—
10	C₂×#500	—	2	1	no	—
11	C₂×#500	—	2	1	no	—
12	C₂×#500	—	2	1	no	—
13	C₂×#500	—	2	1	no	—
14	C₂×#500	C₄	1	2	yes	—
15	C₂×#500	C₄	1	2	yes	—
16	C₂×#500	C₄	1	2	yes	—
17	C₂×#500	C₂²	1	2	yes	—
18	C₂×#500	C₂²	1	2	yes	—
19	C₂×#500	C₂²	1	2	yes	—
20	#2920	—	2	1	no	—
21	C₂²×#107	—	2	1	no	—
22	C₂²×#107	C₄	1	2	yes	—
23	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
24	#2163	—	2	1	no	—
25	#2163	C₂²	1	2	yes	—
26	#2162	—	2	1	no	—
27	#2162	C₂²	1	2	yes	—
28	#2150	C₂²	1	2	yes	—
29	#2112	—	2	1	no	—
30	C₄×M₃₂	—	2	1	no	—
31	#1568	—	2	1	no	—
32	#1568	C₂²	1	2	yes	—
33	#1567	C₂²	1	2	yes	—
34	#1566	—	2	1	no	—
35	#1564	—	2	1	no	—
36	#1564	C₂²	1	2	yes	—
37	#1364	C₂²	1	16	yes	complemented
38	C₂³×Q₈	—	4	1	no	—
39	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Q₈	—	2	2	no	—
42	C₂×C₄⋊Q₈	—	2	2	no	—
43	C₂×C₄⋊Q₈	—	2	2	no	—
44	C₂×C₄⋊Q₈	—	2	2	no	—
45	C₂×C₄⋊Q₈	—	2	2	no	—
46	C₂×C₄⋊Q₈	—	2	2	no	—
47	C₂×C₄⋊Q₈	—	4	1	no	—
48	C₂×C₄⋊Q₈	—	4	1	no	—
49	C₂×C₄⋊Q₈	—	4	1	no	—
50	C₂×C₄⋊Q₈	—	4	1	no	—
51	C₂×C₄⋊Q₈	—	4	1	no	—
52	C₂×C₄⋊Q₈	—	4	1	no	—
53	C₂²×C₄⋊C₄	—	2	2	no	—
54	C₂²×C₄⋊C₄	—	4	1	no	—
55	C₂²×C₄²	D₄	1	1	yes	characteristic
56	C₂×M₃₂	—	2	2	no	—
57	#500	—	2	2	no	—
58	#500	—	2	2	no	—
59	#500	—	2	2	no	—
60	#500	—	2	2	no	—
61	#500	—	2	2	no	—
62	#500	—	2	2	no	—
63	#500	—	2	2	no	—
64	#500	—	2	2	no	—
65	#500	—	2	2	no	—
66	#500	—	2	2	no	—
67	#500	—	2	2	no	—
68	#500	—	2	2	no	—
69	#500	C₂×C₄	1	4	yes	—
70	#500	C₂×C₄	1	4	yes	—
71	#500	C₂×C₄	1	4	yes	—
72	#500	C₂³	1	4	yes	—
73	#432	—	2	2	no	—
74	#432	—	4	1	no	—
75	#431	—	2	2	no	—
76	#431	—	4	1	no	—
77	C₂×#107	—	2	2	no	—
78	C₂×#107	—	2	2	no	—
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	2	2	no	—
81	C₂×#107	—	2	2	no	—
82	C₂×#107	—	2	2	no	—
83	C₂×#107	—	2	2	no	—
84	C₂×#107	—	2	2	no	—
85	C₂×#107	—	2	2	no	—
86	C₂×#107	C₂×C₄	1	4	yes	—
87	C₂×#107	C₂×C₄	1	4	yes	—
88	C₂×#107	C₂×C₄	1	4	yes	—
89	#423	—	2	2	no	—
90	#407	—	4	1	no	—
91	C₄×M₁₆	—	4	1	no	—
92	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄×C₈	D₄	1	1	yes	characteristic
96	C₂×C₄×C₈	D₄	1	1	yes	characteristic
97	C₂×C₄×C₈	D₄	1	1	yes	characteristic
98	C₂×C₄×C₈	D₄	1	1	yes	characteristic
99	C₂×C₄×C₈	D₄	1	1	yes	characteristic
100	C₂×C₄×C₈	D₄	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
104	C₄⋊C₁₆	—	2	2	no	—
105	C₄⋊C₁₆	C₂³	1	4	yes	—
106	C₂²⋊C₁₆	—	2	2	no	—
107	C₂²⋊C₁₆	—	2	2	no	—
108	#346	—	2	2	no	—
109	C₄×C₁₆	—	2	2	no	—
110	#335	—	2	2	no	—
111	#335	C₂³	1	4	yes	—
112	#334	—	2	2	no	—
113	#334	—	2	2	no	—
114	C₂²×Q₈	—	4	2	no	—
115	C₂²×Q₈	—	4	2	no	—
116	C₂²×Q₈	—	4	2	no	—
117	C₂²×Q₈	—	4	2	no	—
118	C₂²×Q₈	—	4	2	no	—
119	C₂²×Q₈	—	4	2	no	—
120	C₂²×Q₈	—	8	8	no	—
121	C₂³×C₄	—	4	2	no	—
122	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
123	C₂³×C₄	D₈	1	1	yes	characteristic
124	C₂³×C₄	SD₁₆	1	1	yes	characteristic
125	C₂×M₁₆	—	4	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	4	no	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	C₄⋊Q₈	—	4	2	no	—
146	C₄⋊Q₈	—	4	2	no	—
147	C₄⋊Q₈	—	4	2	no	—
148	C₄⋊Q₈	—	4	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	C₄⋊Q₈	—	4	2	no	—
152	C₄⋊Q₈	—	4	2	no	—
153	C₄⋊Q₈	—	4	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	4	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄⋊C₄	—	4	2	no	—
162	C₂×C₄⋊C₄	—	4	2	no	—
163	C₂×C₄⋊C₄	—	4	2	no	—
164	C₂×C₄⋊C₄	—	4	2	no	—
165	C₂×C₄⋊C₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	4	2	no	—
167	C₂×C₄⋊C₄	—	4	2	no	—
168	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄²	D₈	1	1	yes	characteristic
172	C₂×C₄²	D₈	1	1	yes	characteristic
173	C₂×C₄²	D₈	1	1	yes	characteristic
174	C₂×C₄²	SD₁₆	1	1	yes	characteristic
175	C₂×C₄²	SD₁₆	1	1	yes	characteristic
176	C₂×C₄²	SD₁₆	1	1	yes	characteristic
177	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
180	M₃₂	—	4	4	no	—
181	C₂×C₁₆	—	2	4	no	—
182	#107	—	2	4	no	—
183	#107	—	2	4	no	—
184	#107	—	2	4	no	—
185	#107	—	2	4	no	—
186	#107	—	2	4	no	—
187	#107	—	2	4	no	—
188	#107	—	2	4	no	—
189	#107	C₂²×C₄	1	8	yes	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	4	2	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	C₂²⋊C₈	—	4	2	no	—
194	#97	—	4	2	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
202	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
203	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
204	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
205	C₄×C₈	C₂×D₄	1	1	yes	characteristic
206	C₄×C₈	C₂×D₄	1	1	yes	characteristic
207	C₄×C₈	C₂×D₄	1	1	yes	characteristic
208	C₄×C₈	C₂×D₄	1	1	yes	characteristic
209	C₄×C₈	C₂×D₄	1	1	yes	characteristic
210	C₄×C₈	C₂×D₄	1	1	yes	characteristic
211	C₄×C₈	C₂⁴	1	1	yes	characteristic
212	C₂⁴	#102	1	1	yes	characteristic
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	8	8	no	—
218	C₂×Q₈	—	8	8	no	—
219	C₂×Q₈	—	8	8	no	—
220	C₂×Q₈	—	8	8	no	—
221	C₂×Q₈	—	8	8	no	—
222	C₂×Q₈	—	8	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	8	8	no	—
231	C₂²×C₄	#102	1	1	yes	characteristic
232	C₂²×C₄	#102	1	1	yes	characteristic
233	C₂²×C₄	#102	1	1	yes	characteristic
234	C₂²×C₄	#102	1	1	yes	characteristic
235	C₂²×C₄	#102	1	1	yes	characteristic
236	C₂²×C₄	#102	1	1	yes	characteristic
237	C₂²×C₄	#102	1	1	yes	characteristic
238	C₂²×C₄	#102	1	1	yes	characteristic
239	C₂²×C₄	#102	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
247	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
248	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
249	M₁₆	—	8	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	4	no	—
261	C₂×C₈	#101	1	4	yes	—
262	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₈	#137	1	4	yes	—
269	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
271	C₄⋊C₄	—	2	8	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄²	#102	1	1	yes	characteristic
280	C₄²	#102	1	1	yes	characteristic
281	C₄²	#102	1	1	yes	characteristic
282	C₄²	#102	1	1	yes	characteristic
283	C₄²	#102	1	1	yes	characteristic
284	C₄²	#102	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
288	C₄²	C₂×D₈	1	1	yes	characteristic
289	C₄²	C₂×D₈	1	1	yes	characteristic
290	C₄²	C₂×D₈	1	1	yes	characteristic
291	C₄²	C₂×SD₁₆	1	1	yes	characteristic
292	C₄²	C₂×SD₁₆	1	1	yes	characteristic
293	C₄²	C₂×SD₁₆	1	1	yes	characteristic
294	C₄²	C₂²×D₄	1	1	yes	characteristic
295	C₁₆	—	4	4	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	C₂×#102	1	1	yes	characteristic
300	C₂³	C₂×#102	1	1	yes	characteristic
301	C₂³	C₂×#102	1	1	yes	characteristic
302	Q₈	—	8	8	no	—
303	Q₈	—	8	8	no	—
304	Q₈	—	8	8	no	—
305	Q₈	—	8	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	8	no	—
324	C₂×C₄	—	8	8	no	—
325	C₂×C₄	—	8	8	no	—
326	C₂×C₄	—	8	8	no	—
327	C₂×C₄	#362	1	4	yes	—
328	C₂×C₄	C₂×#101	1	2	yes	—
329	C₂×C₄	C₂×#102	1	1	yes	characteristic
330	C₂×C₄	C₂×#102	1	1	yes	characteristic
331	C₂×C₄	C₂×#102	1	1	yes	characteristic
332	C₂×C₄	C₂×#102	1	1	yes	characteristic
333	C₂×C₄	C₂×#102	1	1	yes	characteristic
334	C₂×C₄	C₂×#102	1	1	yes	characteristic
335	C₂×C₄	C₂×#102	1	1	yes	characteristic
336	C₂×C₄	C₂×#102	1	1	yes	characteristic
337	C₂×C₄	C₂×#102	1	1	yes	characteristic
338	C₂×C₄	#419	1	2	yes	—
339	C₂×C₄	#509	1	4	yes	—
340	C₂×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
342	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	8	4	no	—
348	C₈	C₂×#101	1	2	yes	—
349	C₈	C₂×#101	1	2	yes	—
350	C₈	#419	1	2	yes	—
351	C₈	C₂×#137	1	2	yes	—
352	C₂²	—	2	8	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	#1546	1	2	yes	—
360	C₂²	#2140	1	2	yes	—
361	C₂²	C₂×#362	1	2	yes	—
362	C₂²	C₂²×#102	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	8	8	no	—
370	C₄	#1546	1	2	yes	—
371	C₄	#2140	1	2	yes	—
372	C₄	C₂×#362	1	2	yes	—
373	C₄	C₂×#362	1	2	yes	—
374	C₄	C₂²×#101	1	1	yes	characteristic
375	C₄	C₂×#419	1	1	yes	characteristic
376	C₄	C₂×#509	1	2	yes	—
377	C₂	—	2	2	no	—
378	C₂	—	2	2	no	—
379	C₂	—	2	2	no	—
380	C₂	C₂×#1546	1	1	yes	characteristic
381	C₂	C₂×#2140	1	1	yes	characteristic
382	C₂	C₂²×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #11974	C₂	semidirect
5. #11971	C₂	semidirect
6. #11940	C₂	semidirect
37. #1364	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#500	2
#33418	2
#17793	2
#11974	2
#11971	2
#11940	2