Subgroups — #109661

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20441	C₂	1	1	yes	maximal, characteristic
2	C₂×#1571	C₂	1	1	yes	maximal, characteristic
3	#10855	C₂	1	1	yes	maximal, characteristic
4	#8095	C₂	1	1	yes	maximal, characteristic
5	#7981	C₂	1	1	yes	maximal, characteristic
6	#7932	C₂	1	1	yes	maximal, characteristic
7	#7893	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#405	—	2	1	no	—
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	#2285	C₂²	1	1	yes	characteristic
15	#2285	C₂²	1	1	yes	characteristic
16	#1771	—	2	2	no	—
17	#1770	—	2	1	no	—
18	C₈×C₄⋊C₄	—	2	1	no	—
19	C₂×#344	—	2	1	no	—
20	C₂×#344	—	2	1	no	—
21	#1729	—	2	1	no	—
22	#1571	—	2	1	no	—
23	#1571	—	2	1	no	—
24	#1571	—	2	2	no	—
25	C₂×#335	C₄	1	4	yes	complemented
26	#1303	—	2	2	no	—
27	#1289	C₄	1	4	yes	—
28	#1283	C₄	1	4	yes	complemented
29	#1279	—	2	2	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂²×M₁₆	D₄	1	1	yes	characteristic
33	C₂³×C₈	—	2	1	no	—
34	C₂×#117	—	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	—	2	1	no	—
44	#431	C₂×C₄	1	2	yes	—
45	#423	—	2	1	no	—
46	#423	—	2	1	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₈	D₄	1	2	yes	—
60	C₂×C₂²⋊C₈	—	2	1	no	—
61	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	#405	—	4	2	no	—
63	#405	—	4	2	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	C₂×C₄	1	2	yes	—
67	C₄×M₁₆	Q₈	1	2	yes	—
68	C₂×#97	—	2	1	no	—
69	C₂×#97	—	2	2	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	2	no	—
73	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
74	#384	—	2	1	no	—
75	#384	—	2	1	no	—
76	#382	—	2	1	no	—
77	#382	Q₈	1	2	yes	—
78	#380	—	2	1	no	—
79	#380	—	2	1	no	—
80	#380	C₂×C₄	1	2	yes	—
81	#380	C₂×C₄	1	2	yes	—
82	C₄×C₄⋊C₄	—	2	1	no	—
83	C₄×C₄⋊C₄	—	2	1	no	—
84	C₄×C₄⋊C₄	—	2	1	no	—
85	C₄×C₄⋊C₄	D₄	1	2	yes	—
86	#344	—	4	2	no	—
87	#344	—	4	2	no	—
88	#336	—	2	2	no	—
89	#336	—	2	2	no	—
90	#335	—	2	4	no	—
91	#335	—	2	4	no	—
92	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₄	#41	1	1	yes	characteristic
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	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
110	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
111	C₂×M₁₆	D₈	1	2	yes	—
112	C₂×M₁₆	Q₁₆	1	2	yes	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	4	1	no	—
121	C₂²×C₈	—	4	1	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	—	4	1	no	—
125	C₂²×C₈	C₄²	1	4	yes	—
126	#117	—	4	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	#117	—	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×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₂²⋊C₄	—	2	2	no	—
141	C₂×C₂²⋊C₄	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	4	1	no	—
144	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₄	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×Q₈	1	1	yes	characteristic
150	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
151	C₂×C₄²	#41	1	1	yes	characteristic
152	C₂×C₄²	#41	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	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	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	—	4	1	no	—
166	C₄⋊C₈	—	4	1	no	—
167	C₄⋊C₈	—	4	1	no	—
168	C₄⋊C₈	C₂²⋊C₄	1	2	yes	—
169	C₄⋊C₈	C₂²⋊C₄	1	2	yes	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	2	2	no	—
173	C₂²⋊C₈	—	4	1	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	#97	—	4	4	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	4	4	no	—
183	C₄×C₈	—	4	1	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	#95	—	4	1	no	—
187	#95	C₄⋊C₄	1	4	yes	—
188	C₂⁴	C₄×D₄	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	8	1	no	—
193	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	#117	1	1	yes	characteristic
197	C₂²×C₄	#117	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
202	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
203	C₂²×C₄	#120	1	1	yes	characteristic
204	C₂²×C₄	#121	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	#123	1	1	yes	characteristic
208	C₂²×C₄	#123	1	1	yes	characteristic
209	C₂²×C₄	#124	1	1	yes	characteristic
210	C₂²×C₄	#126	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
213	C₂²×C₄	#135	1	1	yes	characteristic
214	C₂²×C₄	#135	1	1	yes	characteristic
215	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₄	#137	1	1	yes	characteristic
218	C₂²×C₄	#137	1	1	yes	characteristic
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	8	1	no	—
227	M₁₆	—	8	1	no	—
228	M₁₆	—	8	1	no	—
229	M₁₆	—	8	1	no	—
230	M₁₆	#107	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	8	2	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄⋊C₄	—	8	1	no	—
261	C₄⋊C₄	—	8	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	8	1	no	—
269	C₄²	—	8	1	no	—
270	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₄²	#117	1	1	yes	characteristic
272	C₄²	C₄×D₄	1	1	yes	characteristic
273	C₄²	C₄×D₄	1	1	yes	characteristic
274	C₄²	C₄×D₄	1	1	yes	characteristic
275	C₄²	C₄×D₄	1	1	yes	characteristic
276	C₄²	C₄×Q₈	1	1	yes	characteristic
277	C₄²	C₄×Q₈	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	#123	1	1	yes	characteristic
282	C₄²	#125	1	1	yes	characteristic
283	C₄²	#125	1	1	yes	characteristic
284	C₄²	#126	1	1	yes	characteristic
285	C₄²	C₄⋊Q₈	1	1	yes	characteristic
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
289	C₂³	#385	1	1	yes	characteristic
290	C₂³	#388	1	1	yes	characteristic
291	C₂³	C₄×D₈	1	1	yes	characteristic
292	C₂³	C₄×Q₁₆	1	1	yes	characteristic
293	C₂³	#440	1	1	yes	characteristic
294	C₂³	#440	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	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	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
315	C₂×C₄	#385	1	1	yes	characteristic
316	C₂×C₄	#386	1	1	yes	characteristic
317	C₂×C₄	#387	1	1	yes	characteristic
318	C₂×C₄	#387	1	1	yes	characteristic
319	C₂×C₄	#387	1	1	yes	characteristic
320	C₂×C₄	#387	1	1	yes	characteristic
321	C₂×C₄	#388	1	1	yes	characteristic
322	C₂×C₄	#389	1	1	yes	characteristic
323	C₂×C₄	#389	1	1	yes	characteristic
324	C₂×C₄	#391	1	1	yes	characteristic
325	C₂×C₄	C₂×#107	1	1	yes	characteristic
326	C₂×C₄	#427	1	1	yes	characteristic
327	C₂×C₄	#428	1	1	yes	characteristic
328	C₂×C₄	#428	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	#441	1	1	yes	characteristic
332	C₂×C₄	#442	1	1	yes	characteristic
333	C₂×C₄	#447	1	1	yes	characteristic
334	C₂×C₄	#449	1	1	yes	characteristic
335	C₂×C₄	#451	1	1	yes	characteristic
336	C₂×C₄	#452	1	1	yes	characteristic
337	C₂×C₄	#459	1	1	yes	characteristic
338	C₂×C₄	#462	1	1	yes	characteristic
339	C₂×C₄	#463	1	1	yes	characteristic
340	C₂×C₄	#464	1	1	yes	characteristic
341	C₂×C₄	#474	1	1	yes	characteristic
342	C₂×C₄	#477	1	1	yes	characteristic
343	C₂×C₄	#478	1	1	yes	characteristic
344	C₂×C₄	#479	1	1	yes	characteristic
345	C₂×C₄	#480	1	1	yes	characteristic
346	C₂×C₄	#482	1	1	yes	characteristic
347	C₂×C₄	#484	1	1	yes	characteristic
348	C₂×C₄	#485	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	4	no	—
354	C₈	—	8	2	no	—
355	C₈	—	8	2	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	#1436	1	1	yes	characteristic
367	C₂²	C₄×#107	1	1	yes	characteristic
368	C₂²	#1776	1	1	yes	characteristic
369	C₂²	#1862	1	1	yes	characteristic
370	C₂²	#1863	1	1	yes	characteristic
371	C₂²	#1925	1	1	yes	characteristic
372	C₂²	#1928	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	8	4	no	—
381	C₄	—	16	2	no	—
382	C₄	#1757	1	2	yes	—
383	C₄	#1852	1	1	yes	characteristic
384	C₄	#1854	1	1	yes	characteristic
385	C₄	#1878	1	1	yes	characteristic
386	C₄	#1879	1	1	yes	characteristic
387	C₄	#1918	1	1	yes	characteristic
388	C₄	#1921	1	1	yes	characteristic
389	C₄	#1947	1	1	yes	characteristic
390	C₄	#1949	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	#10905	1	1	yes	characteristic
398	C₂	#10907	1	1	yes	characteristic
399	C₂	#10914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
25. C₂×#335	C₄	semidirect
28. #1283	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20441	2
C₂×#1571	2
#10855	2
#8095	2
#7981	2
#7932	2
#7893	2