Subgroups — #72225463

Order 2048.

624 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225456	C₂	1	2	yes	maximal, complemented
2	#72225454	C₂	1	2	yes	maximal
3	#72225400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₁₆	C₂	1	1	yes	maximal, characteristic
5	#72216239	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×D₈	—	2	1	no	—
8	#151042	C₂²	1	2	yes	—
9	#150138	—	2	2	no	—
10	#135796	—	2	1	no	—
11	#135796	C₂²	1	2	yes	—
12	#135781	—	2	1	no	—
13	C₁₆×D₁₆	—	2	1	no	—
14	C₁₆×D₁₆	—	2	1	no	—
15	C₁₆×D₁₆	C₄	1	2	yes	—
16	C₁₆×D₁₆	C₄	1	2	yes	—
17	C₂×#7100	C₂²	1	1	yes	characteristic
18	C₂×#7068	—	2	1	no	—
19	C₂×C₁₆²	C₂²	1	1	yes	characteristic
20	#94403	—	2	2	no	—
21	#93883	—	2	1	no	—
22	#93879	—	2	1	no	—
23	C₂×C₄×D₁₆	D₄	1	1	yes	characteristic
24	C₂×D₄×C₁₆	—	4	1	no	—
25	C₂×C₈×D₈	—	2	1	no	—
26	#13640	—	2	2	no	—
27	#12909	—	4	2	no	—
28	C₈×D₁₆	—	2	1	no	—
29	C₈×D₁₆	—	2	1	no	—
30	C₈×D₁₆	C₂×C₄	1	2	yes	—
31	C₈×D₁₆	Q₈	1	2	yes	—
32	C₁₆×D₈	—	2	2	no	—
33	C₁₆×D₈	—	2	2	no	—
34	C₁₆×D₈	—	4	1	no	—
35	C₁₆×D₈	—	4	1	no	—
36	#12023	—	2	2	no	—
37	#12023	—	2	2	no	—
38	#12023	—	4	1	no	—
39	#12023	—	4	1	no	—
40	C₂×#1371	D₄	1	1	yes	characteristic
41	C₂×#1370	—	2	1	no	—
42	C₂×#1331	—	2	1	no	—
43	C₂×D₄⋊C₁₆	—	4	1	no	—
44	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
47	#8823	—	2	2	no	—
48	#7458	—	2	2	no	—
49	#7455	—	2	2	no	—
50	#7385	—	4	4	no	—
51	#7100	—	2	1	no	—
52	#7100	C₂×C₄	1	2	yes	—
53	#7068	—	2	2	no	—
54	#7068	—	2	2	no	—
55	C₁₆²	C₂×C₄	1	1	yes	characteristic
56	C₁₆²	C₂×C₄	1	1	yes	characteristic
57	C₁₆²	D₄	1	1	yes	characteristic
58	C₁₆²	D₄	1	1	yes	characteristic
59	C₂²×D₁₆	Q₁₆	1	1	yes	characteristic
60	C₂³×C₁₆	—	8	1	no	—
61	C₂×C₄×D₈	—	2	1	no	—
62	C₂×C₈×D₄	—	4	1	no	—
63	C₄×D₁₆	—	2	1	no	—
64	C₄×D₁₆	—	2	1	no	—
65	C₄×D₁₆	C₄⋊C₄	1	2	yes	—
66	C₄×D₁₆	D₈	1	2	yes	—
67	D₄×C₁₆	—	4	2	no	—
68	D₄×C₁₆	—	4	2	no	—
69	D₄×C₁₆	—	8	1	no	—
70	D₄×C₁₆	—	8	1	no	—
71	C₂×#368	—	8	4	no	—
72	C₂×#366	Q₁₆	1	1	yes	characteristic
73	C₂×C₄⋊C₁₆	—	4	1	no	—
74	C₂×#357	—	2	1	no	—
75	C₂×C₂²⋊C₁₆	—	8	1	no	—
76	C₂×C₄×C₁₆	—	2	1	no	—
77	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₆	#41	1	1	yes	characteristic
81	#1848	—	4	2	no	—
82	C₈×D₈	—	2	2	no	—
83	C₈×D₈	—	2	2	no	—
84	C₈×D₈	—	4	1	no	—
85	C₈×D₈	—	4	1	no	—
86	#1567	—	4	2	no	—
87	#1567	—	4	2	no	—
88	C₂×#335	—	2	1	no	—
89	C₂×D₄⋊C₈	—	4	1	no	—
90	C₂×C₈²	C₂×D₄	1	1	yes	characteristic
91	C₄⋊C₃₂	—	4	2	no	—
92	C₂²⋊C₃₂	—	4	2	no	—
93	#1397	—	4	2	no	—
94	#1372	—	4	2	no	—
95	#1372	—	4	2	no	—
96	#1372	—	8	2	no	—
97	#1372	—	8	1	no	—
98	#1372	—	8	1	no	—
99	#1372	—	8	2	no	—
100	#1371	—	2	1	no	—
101	#1371	C₄⋊C₄	1	2	yes	—
102	#1370	—	2	2	no	—
103	#1370	—	4	1	no	—
104	#1331	—	2	2	no	—
105	#1331	—	2	2	no	—
106	D₄⋊C₁₆	—	4	2	no	—
107	D₄⋊C₁₆	—	4	2	no	—
108	C₈×C₁₆	—	2	1	no	—
109	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₁₆	SD₁₆	1	1	yes	characteristic
114	C₈×C₁₆	SD₁₆	1	1	yes	characteristic
115	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
117	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
118	C₈×C₁₆	#41	1	1	yes	characteristic
119	C₂²×D₈	—	2	1	no	—
120	C₂³×C₈	—	8	1	no	—
121	C₂×C₄×D₄	—	4	1	no	—
122	C₂×D₁₆	—	2	4	no	—
123	C₂×D₁₆	—	2	1	no	—
124	C₂×D₁₆	#107	1	2	yes	—
125	C₂²×C₁₆	—	2	2	no	—
126	C₂²×C₁₆	—	2	1	no	—
127	C₂²×C₁₆	—	2	1	no	—
128	C₂²×C₁₆	—	8	1	no	—
129	C₂²×C₁₆	—	8	2	no	—
130	C₂²×C₁₆	—	8	2	no	—
131	C₂²×C₁₆	—	16	4	no	—
132	C₂²×C₁₆	SD₃₂	1	1	yes	characteristic
133	C₂²×C₁₆	SD₃₂	1	1	yes	characteristic
134	C₂²×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
135	C₂²×C₁₆	#135	1	1	yes	characteristic
136	C₄×D₈	—	2	2	no	—
137	C₄×D₈	—	2	2	no	—
138	C₄×D₈	—	4	1	no	—
139	C₄×D₈	—	4	1	no	—
140	C₈×D₄	—	4	2	no	—
141	C₈×D₄	—	4	2	no	—
142	C₈×D₄	—	8	1	no	—
143	C₈×D₄	—	8	1	no	—
144	C₂×#108	—	8	4	no	—
145	C₂×#107	—	2	1	no	—
146	#423	—	8	2	no	—
147	C₂×C₄⋊C₈	—	4	1	no	—
148	C₂×#102	—	4	1	no	—
149	C₂×C₂²⋊C₈	—	8	1	no	—
150	C₂×C₄×C₈	#121	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₃₂	—	8	2	no	—
154	#368	—	16	8	no	—
155	#366	—	2	1	no	—
156	#366	#107	1	2	yes	—
157	C₄⋊C₁₆	—	4	2	no	—
158	C₄⋊C₁₆	—	8	1	no	—
159	#357	—	2	2	no	—
160	#357	—	2	2	no	—
161	C₂²⋊C₁₆	—	8	2	no	—
162	C₂²⋊C₁₆	—	8	2	no	—
163	C₄×C₁₆	—	2	2	no	—
164	C₄×C₁₆	—	2	1	no	—
165	C₄×C₁₆	—	2	1	no	—
166	C₄×C₁₆	—	2	1	no	—
167	C₄×C₁₆	—	2	1	no	—
168	C₄×C₁₆	#102	1	1	yes	characteristic
169	C₄×C₁₆	#102	1	1	yes	characteristic
170	C₄×C₁₆	#102	1	1	yes	characteristic
171	C₄×C₁₆	#102	1	1	yes	characteristic
172	C₄×C₁₆	D₁₆	1	1	yes	characteristic
173	C₄×C₁₆	D₁₆	1	1	yes	characteristic
174	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₄×C₁₆	#117	1	1	yes	characteristic
176	C₄×C₁₆	#121	1	1	yes	characteristic
177	C₄×C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
178	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
179	C₄×C₁₆	#135	1	1	yes	characteristic
180	#335	—	2	2	no	—
181	#335	—	4	1	no	—
182	#335	—	8	2	no	—
183	#335	—	8	2	no	—
184	#335	—	8	2	no	—
185	#335	—	8	2	no	—
186	D₄⋊C₈	—	4	2	no	—
187	D₄⋊C₈	—	4	2	no	—
188	C₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₈²	C₂²⋊Q₈	1	1	yes	characteristic
190	C₈²	C₂×SD₁₆	1	1	yes	characteristic
191	C₈²	#137	1	1	yes	characteristic
192	C₂²×D₄	—	4	1	no	—
193	C₂³×C₄	—	8	1	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	4	2	no	—
197	C₂×D₈	—	4	1	no	—
198	C₂×M₁₆	—	16	2	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	8	1	no	—
201	C₂²×C₈	—	8	2	no	—
202	C₂²×C₈	—	8	2	no	—
203	C₂²×C₈	—	16	2	no	—
204	C₂²×C₈	—	16	2	no	—
205	C₂²×C₈	#467	1	1	yes	characteristic
206	C₂²×C₈	#473	1	1	yes	characteristic
207	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
208	C₂²×C₈	#510	1	1	yes	characteristic
209	C₄×D₄	—	4	2	no	—
210	C₄×D₄	—	4	2	no	—
211	C₄×D₄	—	8	1	no	—
212	C₄×D₄	—	8	1	no	—
213	C₂×C₄⋊C₄	—	4	1	no	—
214	C₂×C₂²⋊C₄	—	8	1	no	—
215	C₂×C₄²	#459	1	1	yes	characteristic
216	D₁₆	—	2	4	no	—
217	D₁₆	—	2	2	no	—
218	D₁₆	#368	1	4	yes	complemented
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	8	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	2	2	no	—
225	C₂×C₁₆	—	2	2	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	4	1	no	—
229	C₂×C₁₆	—	4	1	no	—
230	C₂×C₁₆	—	4	2	no	—
231	C₂×C₁₆	—	4	2	no	—
232	C₂×C₁₆	—	8	2	no	—
233	C₂×C₁₆	—	16	4	no	—
234	C₂×C₁₆	—	16	2	no	—
235	C₂×C₁₆	—	16	4	no	—
236	C₂×C₁₆	#357	1	1	yes	characteristic
237	C₂×C₁₆	#357	1	1	yes	characteristic
238	C₂×C₁₆	#357	1	1	yes	characteristic
239	C₂×C₁₆	#357	1	1	yes	characteristic
240	C₂×C₁₆	C₂×#107	1	1	yes	characteristic
241	C₂×C₁₆	#427	1	1	yes	characteristic
242	C₂×C₁₆	#466	1	1	yes	characteristic
243	C₂×C₁₆	#467	1	1	yes	characteristic
244	#108	—	16	8	no	—
245	#107	—	2	2	no	—
246	#107	—	4	1	no	—
247	C₄⋊C₈	—	4	2	no	—
248	C₄⋊C₈	—	8	1	no	—
249	C₄⋊C₈	—	16	2	no	—
250	C₄⋊C₈	—	16	2	no	—
251	#102	—	4	2	no	—
252	#102	—	4	2	no	—
253	C₂²⋊C₈	—	8	2	no	—
254	C₂²⋊C₈	—	8	2	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	#380	1	1	yes	characteristic
257	C₄×C₈	C₂×#102	1	1	yes	characteristic
258	C₄×C₈	#417	1	1	yes	characteristic
259	C₄×C₈	#461	1	1	yes	characteristic
260	C₄×C₈	#466	1	1	yes	characteristic
261	C₄×C₈	#472	1	1	yes	characteristic
262	C₄×C₈	#474	1	1	yes	characteristic
263	C₄×C₈	#476	1	1	yes	characteristic
264	C₄×C₈	C₂×D₁₆	1	1	yes	characteristic
265	C₄×C₈	#509	1	1	yes	characteristic
266	C₃₂	—	16	2	no	—
267	C₂⁴	—	8	1	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	8	2	no	—
271	C₂×D₄	—	8	1	no	—
272	C₂²×C₄	—	8	1	no	—
273	C₂²×C₄	—	8	2	no	—
274	C₂²×C₄	—	8	2	no	—
275	C₂²×C₄	—	16	2	no	—
276	C₂²×C₄	—	16	2	no	—
277	C₂²×C₄	#1669	1	1	yes	characteristic
278	C₂²×C₄	#2208	1	1	yes	characteristic
279	C₂²×C₄	#2213	1	1	yes	characteristic
280	D₈	—	2	4	no	—
281	D₈	—	4	4	no	—
282	D₈	—	4	2	no	—
283	M₁₆	—	32	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	8	2	no	—
294	C₂×C₈	—	16	4	no	—
295	C₂×C₈	—	16	2	no	—
296	C₂×C₈	—	16	2	no	—
297	C₂×C₈	—	16	2	no	—
298	C₂×C₈	—	32	2	no	—
299	C₂×C₈	#1673	1	1	yes	characteristic
300	C₂×C₈	#1675	1	1	yes	characteristic
301	C₂×C₈	#1808	1	1	yes	characteristic
302	C₂×C₈	#1816	1	1	yes	characteristic
303	C₂×C₈	C₂×#357	1	1	yes	characteristic
304	C₂×C₈	#2139	1	1	yes	characteristic
305	C₂×C₈	#2224	1	1	yes	characteristic
306	C₂×C₈	#2226	1	1	yes	characteristic
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	8	1	no	—
309	C₂²⋊C₄	—	8	2	no	—
310	C₂²⋊C₄	—	8	2	no	—
311	C₄²	#1667	1	1	yes	characteristic
312	C₄²	#1802	1	1	yes	characteristic
313	C₄²	#2206	1	1	yes	characteristic
314	C₄²	#2214	1	1	yes	characteristic
315	C₁₆	—	2	4	no	—
316	C₁₆	—	2	4	no	—
317	C₁₆	—	2	2	no	—
318	C₁₆	—	2	2	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	2	1	no	—
324	C₁₆	—	2	1	no	—
325	C₁₆	—	4	1	no	—
326	C₁₆	—	4	1	no	—
327	C₁₆	—	16	2	no	—
328	C₁₆	#1419	1	2	yes	—
329	C₁₆	C₂×#368	1	2	yes	—
330	C₂³	—	8	4	no	—
331	C₂³	—	8	2	no	—
332	C₂³	—	16	2	no	—
333	C₂³	—	16	2	no	—
334	C₂³	#12447	1	1	yes	characteristic
335	D₄	—	4	4	no	—
336	D₄	—	8	4	no	—
337	D₄	—	8	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	—	16	4	no	—
345	C₂×C₄	—	16	2	no	—
346	C₂×C₄	—	16	2	no	—
347	C₂×C₄	—	16	2	no	—
348	C₂×C₄	#8644	1	1	yes	characteristic
349	C₂×C₄	#12424	1	1	yes	characteristic
350	C₂×C₄	#12426	1	1	yes	characteristic
351	C₂×C₄	#12445	1	1	yes	characteristic
352	C₂×C₄	#12811	1	1	yes	characteristic
353	C₂×C₄	#12821	1	1	yes	characteristic
354	C₈	—	2	2	no	—
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	16	2	no	—
363	C₈	—	64	2	no	—
364	C₈	#12837	1	1	yes	characteristic
365	C₈	#12841	1	1	yes	characteristic
366	C₈	C₂×#1419	1	1	yes	characteristic
367	C₈	#13625	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	8	4	no	—
371	C₂²	—	16	8	no	—
372	C₂²	—	16	2	no	—
373	C₂²	—	16	2	no	—
374	C₂²	#134950	1	1	yes	characteristic
375	C₂²	#151659	1	2	yes	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	16	2	no	—
381	C₄	#134963	1	1	yes	characteristic
382	C₄	#134971	1	1	yes	characteristic
383	C₄	#152276	1	1	yes	characteristic
384	C₄	#152283	1	1	yes	characteristic
385	C₂	—	2	2	no	—
386	C₂	—	16	4	no	—
387	C₂	#72216267	1	1	yes	characteristic
388	C₂	#72225459	1	1	yes	characteristic
389	C₂	#72225461	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#134950
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂²×D₁₆	Q₁₆
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72225456	C₂	semidirect
3. #72225400	C₂	semidirect
5. #72216239	C₂	semidirect
218. D₁₆	#368	semidirect

Maximal subgroups

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

subgroup	index
#72225456	2
#72225454	2
#72225400	2
C₂×C₁₆×D₁₆	2
#72216239	2