Subgroups — #72227186

Order 4096.

782 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227183	C₂	1	1	yes	maximal, characteristic
2	#72227181	C₄	1	2	yes	—
3	#72227180	C₄	1	2	yes	—
4	#72225536	C₂	1	2	yes	maximal
5	#72225532	—	2	2	no	—
6	#72225526	—	2	2	no	—
7	#72225326	C₂	1	1	yes	maximal, characteristic
8	#72225323	C₄	1	1	yes	characteristic
9	#72225322	C₄	1	1	yes	characteristic
10	#72223681	C₄	1	2	yes	—
11	#72216544	C₂	1	2	yes	maximal
12	#72216247	C₂²	1	2	yes	—
13	#72216231	C₂	1	1	yes	maximal, characteristic
14	#72216226	C₄	1	1	yes	characteristic
15	#72216226	C₄	1	1	yes	characteristic
16	C₈×#1398	C₂²	1	1	yes	characteristic
17	C₈×#1398	C₂²	1	1	yes	characteristic
18	#72216220	C₄	1	1	yes	characteristic
19	#72216215	C₂²	1	2	yes	—
20	C₁₆×M₆₄	C₄	1	1	yes	characteristic
21	#72216182	C₂²	1	1	yes	characteristic
22	#151981	—	2	2	no	—
23	C₄×#1398	C₂×C₄	1	1	yes	characteristic
24	C₄×#1398	C₂×C₄	1	1	yes	characteristic
25	C₄×#1398	D₄	1	1	yes	characteristic
26	C₄×#1398	D₄	1	1	yes	characteristic
27	#151024	—	2	2	no	—
28	C₂×#7456	C₂×C₄	1	4	yes	—
29	#149504	—	4	2	no	—
30	#149495	C₂×C₄	1	1	yes	characteristic
31	#149495	C₂×C₄	1	1	yes	characteristic
32	#149495	Q₈	1	1	yes	characteristic
33	#149495	Q₈	1	1	yes	characteristic
34	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
35	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
36	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
37	C₈×M₆₄	C₂×C₄	1	1	yes	characteristic
38	#129090	C₂×C₄	1	2	yes	—
39	#129088	C₂×C₄	1	2	yes	—
40	#129051	—	2	2	no	—
41	C₈×#346	—	2	1	no	—
42	C₈×#346	C₂³	1	1	yes	characteristic
43	C₂×#7052	—	2	1	no	—
44	C₂×#7052	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
46	#94031	—	2	2	no	—
47	#94018	—	2	2	no	—
48	#93995	—	4	2	no	—
49	#93878	—	2	4	no	—
50	#93666	—	2	2	no	—
51	#93666	C₈	1	4	yes	complemented
52	#93665	—	2	2	no	—
53	#93664	C₈	1	2	yes	complemented
54	C₁₆×C₃₂	C₈	1	2	yes	complemented
55	#93264	—	2	2	no	—
56	C₂×#1422	—	2	4	no	—
57	C₂×#1398	—	2	2	no	—
58	C₂×#1398	—	2	2	no	—
59	C₂×#1398	—	4	2	no	—
60	C₂×#1398	M₁₆	1	2	yes	—
61	C₂×#1398	M₁₆	1	2	yes	—
62	C₂×#1398	SD₁₆	1	2	yes	—
63	C₂×#1398	SD₁₆	1	2	yes	—
64	C₄×M₆₄	C₄²	1	1	yes	characteristic
65	C₄×M₆₄	C₄²	1	1	yes	characteristic
66	C₄×M₆₄	C₄²	1	1	yes	characteristic
67	C₄×M₆₄	C₄²	1	1	yes	characteristic
68	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
69	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
70	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
71	C₄×M₆₄	C₄⋊C₄	1	1	yes	characteristic
72	C₄×#364	—	2	2	no	—
73	C₄×#346	—	2	2	no	—
74	C₄×#346	C₂²×C₄	1	1	yes	characteristic
75	C₄×#346	C₂×D₄	1	1	yes	characteristic
76	C₂×#1369	—	2	2	no	—
77	#11999	—	2	2	no	—
78	#11405	—	4	2	no	—
79	C₂×#1311	—	2	2	no	—
80	C₂×#1311	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1311	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₆	—	2	1	no	—
83	C₂×C₈×C₁₆	—	2	1	no	—
84	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₆	#41	1	1	yes	characteristic
86	#8266	—	2	2	no	—
87	C₄×C₈²	#41	1	1	yes	characteristic
88	#7468	—	4	2	no	—
89	#7457	—	4	4	no	—
90	#7456	—	2	4	no	—
91	#7456	—	2	4	no	—
92	#7333	—	8	4	no	—
93	#7330	—	2	2	no	—
94	#7330	—	2	2	no	—
95	#7330	—	2	2	no	—
96	#7330	M₁₆	1	4	yes	—
97	#7329	—	2	1	no	—
98	#7329	C₂×C₈	1	2	yes	—
99	#7329	C₂×C₈	1	2	yes	—
100	#7329	M₁₆	1	2	yes	—
101	C₈×C₃₂	—	2	1	no	—
102	C₈×C₃₂	C₂×C₈	1	2	yes	—
103	C₈×C₃₂	C₂×C₈	1	2	yes	—
104	C₈×C₃₂	M₁₆	1	2	yes	—
105	#7161	—	2	2	no	—
106	#7157	—	2	2	no	—
107	#7140	—	4	2	no	—
108	#7052	—	2	4	no	—
109	#7052	—	2	1	no	—
110	#7052	C₂×C₈	1	2	yes	—
111	C₁₆²	—	2	1	no	—
112	C₁₆²	C₂×C₈	1	1	yes	characteristic
113	C₁₆²	C₂×C₈	1	1	yes	characteristic
114	C₂×M₆₄	—	2	1	no	—
115	C₂×M₆₄	—	2	1	no	—
116	C₂×M₆₄	—	2	1	no	—
117	C₂×M₆₄	—	2	1	no	—
118	C₂×M₆₄	—	4	4	no	—
119	C₂×M₆₄	#97	1	2	yes	—
120	C₂×M₆₄	#97	1	1	yes	characteristic
121	C₂×M₆₄	#97	1	1	yes	characteristic
122	C₂×M₆₄	#106	1	2	yes	—
123	C₂×M₆₄	#106	1	1	yes	characteristic
124	C₂×M₆₄	#106	1	1	yes	characteristic
125	C₂×#364	—	4	2	no	—
126	C₄×M₃₂	—	4	2	no	—
127	C₂×#346	—	4	4	no	—
128	C₂×#346	C₂×M₁₆	1	2	yes	—
129	C₂×#346	#131	1	2	yes	—
130	C₂×#346	C₂×SD₁₆	1	2	yes	—
131	C₂×#346	#135	1	2	yes	—
132	C₂×C₄×C₁₆	—	2	2	no	—
133	C₂×C₄×C₁₆	—	2	2	no	—
134	C₂×C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₆	C₄×Q₈	1	1	yes	characteristic
138	C₄²×C₈	—	2	1	no	—
139	C₄²×C₈	C₄×D₄	1	1	yes	characteristic
140	#1564	—	4	2	no	—
141	#1450	—	4	2	no	—
142	C₂×C₈²	—	2	1	no	—
143	C₂×C₈²	#117	1	1	yes	characteristic
144	C₂×C₈²	C₄×Q₈	1	1	yes	characteristic
145	#1422	—	4	4	no	—
146	#1422	—	4	4	no	—
147	#1398	—	2	4	no	—
148	#1398	—	2	4	no	—
149	#1398	—	4	2	no	—
150	#1398	—	4	2	no	—
151	#1398	—	4	4	no	—
152	#1398	—	4	4	no	—
153	#1398	—	8	2	no	—
154	#1398	—	8	2	no	—
155	#1397	—	2	1	no	—
156	#1397	—	2	1	no	—
157	#1397	—	2	1	no	—
158	#1397	—	2	1	no	—
159	#1397	#97	1	2	yes	—
160	#1397	#97	1	2	yes	—
161	#1397	#108	1	2	yes	—
162	#1397	#108	1	2	yes	—
163	C₄×C₃₂	—	2	1	no	—
164	C₄×C₃₂	—	2	1	no	—
165	C₄×C₃₂	C₄×C₈	1	2	yes	—
166	C₄×C₃₂	C₄×C₈	1	2	yes	—
167	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
168	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
169	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
170	C₄×C₃₂	C₄⋊C₈	1	2	yes	—
171	#1376	—	4	2	no	—
172	#1369	—	4	2	no	—
173	#1369	—	4	2	no	—
174	#1368	—	4	2	no	—
175	#1368	—	4	2	no	—
176	#1313	—	8	2	no	—
177	#1313	—	8	2	no	—
178	#1311	—	2	1	no	—
179	#1311	—	2	1	no	—
180	#1311	—	4	4	no	—
181	#1311	C₂×M₁₆	1	2	yes	—
182	#1311	#131	1	2	yes	—
183	C₈×C₁₆	—	2	2	no	—
184	C₈×C₁₆	—	2	2	no	—
185	C₈×C₁₆	—	2	2	no	—
186	C₈×C₁₆	—	2	1	no	—
187	C₈×C₁₆	—	2	1	no	—
188	C₈×C₁₆	—	2	2	no	—
189	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
190	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
191	C₈×C₁₆	C₂×M₁₆	1	1	yes	characteristic
192	C₈×C₁₆	C₂×M₁₆	1	1	yes	characteristic
193	#1276	—	4	2	no	—
194	C₂×M₃₂	—	8	2	no	—
195	C₂²×C₁₆	—	2	4	no	—
196	C₂²×C₁₆	—	4	2	no	—
197	C₂²×C₁₆	C₂×#97	1	1	yes	characteristic
198	C₂²×C₁₆	#405	1	1	yes	characteristic
199	C₂²×C₁₆	C₂×#106	1	1	yes	characteristic
200	C₂²×C₁₆	#427	1	1	yes	characteristic
201	C₂²×C₁₆	C₄×SD₁₆	1	2	yes	—
202	C₂²×C₁₆	#446	1	2	yes	—
203	#431	—	8	2	no	—
204	C₂×C₄×C₈	—	2	1	no	—
205	C₂×C₄×C₈	—	2	1	no	—
206	C₂×C₄×C₈	—	4	2	no	—
207	C₂×C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₄×C₈	#436	1	2	yes	—
209	C₂×C₄×C₈	C₄×SD₁₆	1	2	yes	—
210	C₂×C₄×C₈	#443	1	1	yes	characteristic
211	C₄³	#443	1	1	yes	characteristic
212	M₆₄	—	2	4	no	—
213	M₆₄	—	2	4	no	—
214	M₆₄	—	2	2	no	—
215	M₆₄	—	2	2	no	—
216	M₆₄	—	2	2	no	—
217	M₆₄	—	2	2	no	—
218	M₆₄	—	2	2	no	—
219	M₆₄	—	2	2	no	—
220	M₆₄	—	4	2	no	—
221	M₆₄	—	4	2	no	—
222	M₆₄	—	8	4	no	—
223	M₆₄	—	8	4	no	—
224	M₆₄	#347	1	4	yes	—
225	M₆₄	#365	1	4	yes	—
226	C₂×C₃₂	—	2	2	no	—
227	C₂×C₃₂	—	2	2	no	—
228	C₂×C₃₂	—	2	2	no	—
229	C₂×C₃₂	—	2	2	no	—
230	C₂×C₃₂	—	2	2	no	—
231	C₂×C₃₂	—	8	4	no	—
232	C₂×C₃₂	#322	1	2	yes	—
233	C₂×C₃₂	#322	1	2	yes	—
234	C₂×C₃₂	#334	1	4	yes	—
235	C₂×C₃₂	#334	1	2	yes	—
236	C₂×C₃₂	#334	1	2	yes	—
237	#364	—	8	2	no	—
238	#364	—	8	2	no	—
239	#346	—	2	2	no	—
240	#346	—	2	2	no	—
241	#346	—	2	2	no	—
242	#346	—	2	2	no	—
243	#346	—	4	2	no	—
244	#346	—	4	2	no	—
245	#346	—	8	2	no	—
246	#346	—	8	8	no	—
247	C₄×C₁₆	—	2	2	no	—
248	C₄×C₁₆	—	2	2	no	—
249	C₄×C₁₆	—	2	4	no	—
250	C₄×C₁₆	—	2	4	no	—
251	C₄×C₁₆	—	2	1	no	—
252	C₄×C₁₆	—	2	1	no	—
253	C₄×C₁₆	—	4	2	no	—
254	C₄×C₁₆	—	8	2	no	—
255	C₄×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
256	C₄×C₁₆	C₂×#97	1	1	yes	characteristic
257	C₄×C₁₆	C₄×M₁₆	1	1	yes	characteristic
258	C₄×C₁₆	#405	1	1	yes	characteristic
259	C₄×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
260	C₄×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
261	C₄×C₁₆	C₂×#108	1	1	yes	characteristic
262	C₄×C₁₆	C₂×#108	1	1	yes	characteristic
263	C₄×C₁₆	#436	1	2	yes	—
264	C₄×C₁₆	#443	1	2	yes	—
265	#335	—	8	2	no	—
266	#334	—	8	2	no	—
267	#322	—	8	2	no	—
268	C₈²	—	2	1	no	—
269	C₈²	—	2	1	no	—
270	C₈²	—	4	2	no	—
271	C₈²	—	8	2	no	—
272	C₈²	#431	1	1	yes	characteristic
273	C₈²	#431	1	1	yes	characteristic
274	C₈²	#446	1	2	yes	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	4	1	no	—
277	C₂²×C₈	#1590	1	2	yes	—
278	C₂²×C₈	#1770	1	1	yes	characteristic
279	C₂²×C₈	C₄×#106	1	1	yes	characteristic
280	C₂×C₄²	—	4	1	no	—
281	C₂×C₄²	#1590	1	2	yes	—
282	C₂×C₄²	#1772	1	1	yes	characteristic
283	M₃₂	—	16	2	no	—
284	M₃₂	—	16	2	no	—
285	C₂×C₁₆	—	2	4	no	—
286	C₂×C₁₆	—	2	2	no	—
287	C₂×C₁₆	—	2	2	no	—
288	C₂×C₁₆	—	2	2	no	—
289	C₂×C₁₆	—	2	2	no	—
290	C₂×C₁₆	—	2	4	no	—
291	C₂×C₁₆	—	2	2	no	—
292	C₂×C₁₆	—	2	2	no	—
293	C₂×C₁₆	—	2	2	no	—
294	C₂×C₁₆	—	4	8	no	—
295	C₂×C₁₆	—	4	4	no	—
296	C₂×C₁₆	—	8	4	no	—
297	C₂×C₁₆	—	16	2	no	—
298	C₂×C₁₆	C₂×#322	1	1	yes	characteristic
299	C₂×C₁₆	#1450	1	1	yes	characteristic
300	C₂×C₁₆	#1451	1	1	yes	characteristic
301	C₂×C₁₆	#1451	1	1	yes	characteristic
302	C₂×C₁₆	C₂×#334	1	1	yes	characteristic
303	C₂×C₁₆	C₂×#334	1	1	yes	characteristic
304	C₂×C₁₆	#1564	1	1	yes	characteristic
305	C₂×C₁₆	#1564	1	1	yes	characteristic
306	C₂×C₁₆	#1590	1	4	yes	—
307	C₂×C₁₆	C₂×#347	1	2	yes	—
308	C₂×C₁₆	#2110	1	2	yes	—
309	C₂×C₁₆	C₂×#365	1	2	yes	—
310	C₂×C₁₆	#2155	1	2	yes	—
311	C₄⋊C₈	—	16	2	no	—
312	C₂²⋊C₈	—	16	2	no	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	2	2	no	—
317	C₄×C₈	—	2	2	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	2	1	no	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	4	1	no	—
322	C₄×C₈	—	4	1	no	—
323	C₄×C₈	—	8	4	no	—
324	C₄×C₈	—	16	2	no	—
325	C₄×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
326	C₄×C₈	#1770	1	1	yes	characteristic
327	C₄×C₈	#1772	1	1	yes	characteristic
328	C₄×C₈	C₄×#108	1	1	yes	characteristic
329	C₃₂	—	2	4	no	—
330	C₃₂	—	2	4	no	—
331	C₃₂	—	2	2	no	—
332	C₃₂	—	2	2	no	—
333	C₃₂	—	2	4	no	—
334	C₃₂	—	4	2	no	—
335	C₃₂	—	4	2	no	—
336	C₃₂	—	4	2	no	—
337	C₃₂	—	16	4	no	—
338	C₃₂	#1313	1	4	yes	—
339	C₃₂	#1368	1	4	yes	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	#8283	1	1	yes	characteristic
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	4	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	8	2	no	—
351	C₂×C₈	—	32	2	no	—
352	C₂×C₈	#8260	1	1	yes	characteristic
353	C₂×C₈	C₄×#334	1	1	yes	characteristic
354	C₂×C₈	#8283	1	1	yes	characteristic
355	C₂×C₈	#8293	1	1	yes	characteristic
356	C₂×C₈	#12666	1	2	yes	—
357	C₂×C₈	#12667	1	2	yes	—
358	C₂×C₈	#12668	1	2	yes	—
359	C₂×C₈	C₄×#365	1	2	yes	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	8	2	no	—
364	C₄²	#8266	1	1	yes	characteristic
365	C₄²	#8293	1	1	yes	characteristic
366	C₁₆	—	2	8	no	—
367	C₁₆	—	2	1	no	—
368	C₁₆	—	2	1	no	—
369	C₁₆	—	2	1	no	—
370	C₁₆	—	2	1	no	—
371	C₁₆	—	2	4	no	—
372	C₁₆	—	4	4	no	—
373	C₁₆	—	4	2	no	—
374	C₁₆	—	8	4	no	—
375	C₁₆	—	32	2	no	—
376	C₁₆	C₂×#1313	1	1	yes	characteristic
377	C₁₆	#11405	1	1	yes	characteristic
378	C₁₆	#11414	1	2	yes	—
379	C₁₆	C₂×#1368	1	1	yes	characteristic
380	C₁₆	#11999	1	1	yes	characteristic
381	C₁₆	#12000	1	2	yes	—
382	C₂³	#129159	1	1	yes	characteristic
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	2	no	—
388	C₂×C₄	#96889	1	1	yes	characteristic
389	C₂×C₄	#129158	1	2	yes	—
390	C₂×C₄	#129159	1	1	yes	characteristic
391	C₂×C₄	#129160	1	2	yes	—
392	C₈	—	2	4	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	2	no	—
395	C₈	—	8	2	no	—
396	C₈	—	64	2	no	—
397	C₈	#129049	1	1	yes	characteristic
398	C₈	#129051	1	1	yes	characteristic
399	C₈	#129088	1	1	yes	characteristic
400	C₈	C₄×#1368	1	1	yes	characteristic
401	C₈	#129160	1	2	yes	—
402	C₈	#129195	1	2	yes	—
403	C₂²	—	4	1	no	—
404	C₂²	#72216215	1	1	yes	characteristic
405	C₂²	#72227184	1	2	yes	—
406	C₄	—	2	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	8	2	no	—
409	C₄	#72225325	1	1	yes	characteristic
410	C₄	#72225535	1	1	yes	characteristic
411	C₄	#72227184	1	2	yes	—
412	C₂	—	2	1	no	—
413	C₂	—	2	2	no	—
414	C₂	#72227185	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#129088
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72227185
prime radical	C₂³	#129159
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₆₄ ◁ C₄×#1398 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
51. #93666	C₈	semidirect
53. #93664	C₈	semidirect
54. C₁₆×C₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72227183	2
#72225536	2
#72225326	2
#72216544	2
#72216231	2