Subgroups — #72225360

Order 2048.

863 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225356	C₂	1	4	yes	maximal, complemented
2	#72225355	C₂	1	4	yes	maximal, complemented
3	#72225353	C₂	1	2	yes	maximal, complemented
4	#72225352	C₂	1	1	yes	maximal, characteristic, complemented
5	#72225308	C₂	1	1	yes	maximal, characteristic
6	C₂×#93751	C₂	1	2	yes	maximal, complemented
7	C₁₆×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#512773	C₄	1	1	yes	characteristic
9	#512751	—	2	2	no	—
10	C₂×#12046	C₂²	1	2	yes	—
11	#151082	C₄	1	4	yes	—
12	#151081	C₂²	1	4	yes	—
13	#151073	—	2	4	no	—
14	#151072	C₂²	1	8	yes	complemented
15	#151035	C₄	1	1	yes	characteristic
16	C₂×#7459	C₂²	1	2	yes	—
17	#150067	C₄	1	2	yes	—
18	#150051	—	2	2	no	—
19	C₈×M₆₄	C₂²	1	1	yes	characteristic
20	C₈×M₆₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₆²	C₂²	1	1	yes	characteristic
22	#93882	C₂²	1	4	yes	complemented
23	#93751	C₂²	1	4	yes	complemented
24	#93751	C₂²	1	4	yes	complemented
25	#93664	C₂²	1	2	yes	complemented
26	C₁₆×C₃₂	C₂²	1	2	yes	complemented
27	#33981	C₈	1	1	yes	characteristic
28	#33974	—	4	2	no	—
29	C₂×#2178	C₂×C₄	1	2	yes	—
30	C₂×#2170	—	2	2	no	—
31	#13656	C₈	1	4	yes	—
32	#13655	C₂×C₄	1	4	yes	—
33	#13648	—	4	4	no	—
34	#13647	—	2	8	no	—
35	#13636	C₈	1	1	yes	characteristic
36	C₂×#1422	—	2	2	no	—
37	#13620	C₈	1	2	yes	—
38	#13558	—	4	2	no	—
39	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
40	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
41	C₄×M₆₄	D₄	1	1	yes	characteristic
42	C₄×M₆₄	D₄	1	1	yes	characteristic
43	#12046	—	2	2	no	—
44	#12046	—	2	2	no	—
45	#12046	C₂×C₄	1	4	yes	—
46	#12046	C₂³	1	4	yes	—
47	C₂×#1373	C₂×C₄	1	2	yes	—
48	C₂×#1335	—	2	2	no	—
49	C₂×C₈×C₁₆	—	2	1	no	—
50	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
51	#7462	C₂×C₄	1	4	yes	—
52	#7459	—	2	2	no	—
53	#7459	C₂³	1	4	yes	—
54	#7387	C₂×C₄	1	8	yes	—
55	#7381	—	2	8	no	—
56	#7329	C₂×C₄	1	2	yes	—
57	#7329	D₄	1	2	yes	—
58	C₈×C₃₂	C₂³	1	2	yes	—
59	C₈×C₃₂	C₂³	1	2	yes	—
60	C₁₆²	—	2	1	no	—
61	C₁₆²	C₂³	1	1	yes	characteristic
62	C₁₆²	C₂³	1	1	yes	characteristic
63	C₂×#508	C₂×C₈	1	2	yes	—
64	C₂×#504	—	4	2	no	—
65	C₂×#443	—	2	2	no	—
66	#2264	C₁₆	1	4	yes	complemented
67	#2263	C₁₆	1	4	yes	complemented
68	#2262	C₂×C₈	1	4	yes	—
69	#2258	—	4	8	no	—
70	#2258	—	8	4	no	—
71	C₂×M₆₄	—	2	1	no	—
72	C₂×M₆₄	—	2	1	no	—
73	C₂×M₆₄	—	4	2	no	—
74	C₂×M₆₄	—	8	4	no	—
75	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
76	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
77	C₂×M₆₄	D₈	1	1	yes	characteristic
78	C₂×M₆₄	D₈	1	1	yes	characteristic
79	#2178	—	2	2	no	—
80	#2178	—	2	2	no	—
81	#2178	C₂×C₈	1	4	yes	—
82	#2178	C₂²×C₄	1	4	yes	—
83	#2170	—	2	4	no	—
84	#2170	—	2	4	no	—
85	#2170	—	4	2	no	—
86	#2170	—	4	2	no	—
87	C₂×#368	C₂×C₈	1	2	yes	—
88	C₂×#364	—	2	2	no	—
89	C₂×#359	—	2	2	no	—
90	C₂×#350	—	4	2	no	—
91	C₂×C₄×C₁₆	—	2	2	no	—
92	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈²	#41	1	1	yes	characteristic
95	#1425	C₂×C₈	1	4	yes	—
96	#1422	—	2	4	no	—
97	#1422	—	4	2	no	—
98	#1417	C₂×C₈	1	8	yes	—
99	#1401	—	4	8	no	—
100	#1397	—	2	1	no	—
101	#1397	—	2	1	no	—
102	#1397	C₂×C₈	1	2	yes	—
103	#1397	D₈	1	2	yes	—
104	C₄×C₃₂	C₂²×C₄	1	2	yes	—
105	C₄×C₃₂	C₂²×C₄	1	2	yes	—
106	C₄×C₃₂	C₂×D₄	1	2	yes	—
107	C₄×C₃₂	C₂×D₄	1	2	yes	—
108	#1373	—	2	2	no	—
109	#1373	C₂²×C₄	1	4	yes	—
110	#1335	—	2	4	no	—
111	#1335	—	2	4	no	—
112	C₈×C₁₆	—	2	2	no	—
113	C₈×C₁₆	—	2	1	no	—
114	C₈×C₁₆	—	2	1	no	—
115	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
117	C₈×C₁₆	#41	1	1	yes	characteristic
118	C₈×C₁₆	C₂⁴	1	1	yes	characteristic
119	C₂×#135	—	2	2	no	—
120	C₂×#131	—	4	2	no	—
121	#508	—	2	2	no	—
122	#508	—	2	2	no	—
123	#508	C₂×C₁₆	1	4	yes	—
124	#508	C₂²×C₈	1	4	yes	—
125	C₂×Q₃₂	C₂×C₁₆	1	2	yes	—
126	C₂×SD₃₂	—	2	2	no	—
127	C₂×D₁₆	C₂×C₁₆	1	2	yes	—
128	#504	—	4	4	no	—
129	#504	—	4	4	no	—
130	#504	—	8	2	no	—
131	#504	—	8	2	no	—
132	C₂×M₃₂	—	4	2	no	—
133	C₂×M₃₂	—	8	2	no	—
134	C₂²×C₁₆	—	2	4	no	—
135	C₂²×C₁₆	—	8	2	no	—
136	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₁₆	#131	1	1	yes	characteristic
138	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
139	C₂²×C₁₆	#135	1	1	yes	characteristic
140	#443	—	2	4	no	—
141	#443	—	2	4	no	—
142	#443	—	4	2	no	—
143	#443	—	4	2	no	—
144	C₂×#108	—	2	2	no	—
145	C₂×#104	—	4	2	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
148	Q₆₄	C₂×C₁₆	1	4	yes	complemented
149	SD₆₄	C₂×C₁₆	1	8	yes	complemented
150	D₃₂	C₂×C₁₆	1	4	yes	complemented
151	M₆₄	—	2	4	no	—
152	M₆₄	—	2	4	no	—
153	M₆₄	—	2	2	no	—
154	M₆₄	—	2	2	no	—
155	M₆₄	—	4	4	no	—
156	M₆₄	—	8	2	no	—
157	M₆₄	—	8	8	no	—
158	M₆₄	C₂×C₁₆	1	4	yes	—
159	M₆₄	D₁₆	1	4	yes	—
160	C₂×C₃₂	—	2	2	no	—
161	C₂×C₃₂	—	2	2	no	—
162	C₂×C₃₂	—	8	2	no	—
163	C₂×C₃₂	—	8	8	no	—
164	C₂×C₃₂	C₂²×C₈	1	2	yes	—
165	C₂×C₃₂	C₂²×C₈	1	2	yes	—
166	C₂×C₃₂	C₂×D₈	1	2	yes	—
167	C₂×C₃₂	C₂×D₈	1	2	yes	—
168	#368	—	2	2	no	—
169	#368	C₂²×C₈	1	4	yes	—
170	#364	—	2	4	no	—
171	#364	—	4	2	no	—
172	#359	—	2	4	no	—
173	#359	—	2	4	no	—
174	#350	—	4	4	no	—
175	#350	—	4	4	no	—
176	C₄×C₁₆	—	2	2	no	—
177	C₄×C₁₆	—	2	2	no	—
178	C₄×C₁₆	—	2	4	no	—
179	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
180	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
181	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
182	C₄×C₁₆	#131	1	1	yes	characteristic
183	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
184	C₄×C₁₆	#135	1	1	yes	characteristic
185	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
186	C₄×C₁₆	C₂²×D₄	1	1	yes	characteristic
187	C₈²	—	2	1	no	—
188	C₈²	C₄×D₄	1	1	yes	characteristic
189	C₈²	C₂×#41	1	1	yes	characteristic
190	C₂×#41	—	4	2	no	—
191	#135	—	2	4	no	—
192	#135	—	2	4	no	—
193	#135	—	4	2	no	—
194	#135	—	4	2	no	—
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×SD₁₆	—	4	2	no	—
197	C₂×D₈	—	2	2	no	—
198	#131	—	4	4	no	—
199	#131	—	4	4	no	—
200	#131	—	8	2	no	—
201	#131	—	8	2	no	—
202	C₂×M₁₆	—	4	2	no	—
203	C₂×M₁₆	—	8	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	8	2	no	—
206	C₂²×C₈	C₈×D₄	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₈	1	1	yes	characteristic
208	C₂×C₄²	#443	1	1	yes	characteristic
209	Q₃₂	—	2	2	no	—
210	Q₃₂	C₂²×C₁₆	1	4	yes	—
211	SD₃₂	—	2	4	no	—
212	SD₃₂	—	2	4	no	—
213	D₁₆	—	2	2	no	—
214	D₁₆	C₂²×C₁₆	1	4	yes	—
215	M₃₂	—	4	4	no	—
216	M₃₂	—	8	2	no	—
217	M₃₂	—	8	4	no	—
218	M₃₂	—	8	4	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	4	no	—
223	C₂×C₁₆	—	4	8	no	—
224	C₂×C₁₆	—	8	2	no	—
225	C₂×C₁₆	—	8	4	no	—
226	C₂×C₁₆	—	8	4	no	—
227	C₂×C₁₆	C₈×D₄	1	1	yes	characteristic
228	C₂×C₁₆	C₈×D₄	1	1	yes	characteristic
229	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
230	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×C₁₆	1	2	yes	—
232	C₂×C₁₆	#504	1	2	yes	—
233	C₂×C₁₆	C₂×D₁₆	1	2	yes	—
234	C₂×C₁₆	#508	1	2	yes	—
235	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#131	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₈	1	1	yes	characteristic
238	C₂×C₁₆	C₂×#135	1	1	yes	characteristic
239	#108	—	2	4	no	—
240	#108	—	4	2	no	—
241	#104	—	4	4	no	—
242	#104	—	4	4	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	C₈×D₄	1	1	yes	characteristic
247	C₄×C₈	C₄×D₈	1	1	yes	characteristic
248	C₄×C₈	#443	1	1	yes	characteristic
249	C₄×C₈	C₂×C₄×D₄	1	1	yes	characteristic
250	C₃₂	—	2	4	no	—
251	C₃₂	—	2	4	no	—
252	C₃₂	—	2	2	no	—
253	C₃₂	—	2	2	no	—
254	C₃₂	—	8	4	no	—
255	C₃₂	C₂²×C₁₆	1	4	yes	—
256	C₃₂	C₂×D₁₆	1	4	yes	—
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	#41	—	8	2	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	8	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	8	2	no	—
266	C₂²×C₄	C₈×D₈	1	1	yes	characteristic
267	Q₁₆	—	2	4	no	—
268	Q₁₆	—	4	2	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	D₈	—	2	4	no	—
272	D₈	—	4	2	no	—
273	M₁₆	—	4	4	no	—
274	M₁₆	—	8	2	no	—
275	M₁₆	—	8	4	no	—
276	M₁₆	—	8	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	4	no	—
280	C₂×C₈	—	8	2	no	—
281	C₂×C₈	—	8	4	no	—
282	C₂×C₈	—	8	4	no	—
283	C₂×C₈	C₈×D₈	1	1	yes	characteristic
284	C₂×C₈	C₈×D₈	1	1	yes	characteristic
285	C₂×C₈	D₄×C₁₆	1	2	yes	—
286	C₂×C₈	#2170	1	2	yes	—
287	C₂×C₈	C₄×D₁₆	1	2	yes	—
288	C₂×C₈	#2178	1	2	yes	—
289	C₂×C₈	C₂×C₈×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₈	1	1	yes	characteristic
291	C₄²	—	2	1	no	—
292	C₄²	C₈×D₈	1	1	yes	characteristic
293	C₄²	C₂×#443	1	1	yes	characteristic
294	C₁₆	—	2	8	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	1	no	—
299	C₁₆	—	4	4	no	—
300	C₁₆	—	8	4	no	—
301	C₁₆	D₄×C₁₆	1	2	yes	—
302	C₁₆	C₄×D₁₆	1	2	yes	—
303	C₁₆	C₂³×C₁₆	1	1	yes	characteristic
304	C₁₆	C₂×#504	1	1	yes	characteristic
305	C₁₆	C₂²×D₁₆	1	1	yes	characteristic
306	C₁₆	C₂×#508	1	1	yes	characteristic
307	C₂³	—	8	2	no	—
308	C₂³	#12046	1	1	yes	characteristic
309	Q₈	—	4	4	no	—
310	Q₈	—	8	2	no	—
311	D₄	—	4	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₁₆×D₈	1	2	yes	—
322	C₂×C₄	C₈×D₁₆	1	2	yes	—
323	C₂×C₄	#12046	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₈	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	8	4	no	—
328	C₈	C₁₆×D₈	1	2	yes	—
329	C₈	C₈×D₁₆	1	2	yes	—
330	C₈	C₂×D₄×C₁₆	1	1	yes	characteristic
331	C₈	C₂×#2170	1	1	yes	characteristic
332	C₈	C₂×C₄×D₁₆	1	1	yes	characteristic
333	C₈	C₂×#2178	1	1	yes	characteristic
334	C₂²	—	4	1	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	32	2	no	—
337	C₂²	C₁₆×D₁₆	1	2	yes	—
338	C₂²	C₂×#12046	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	8	4	no	—
342	C₄	C₁₆×D₁₆	1	2	yes	—
343	C₄	C₂×C₁₆×D₈	1	1	yes	characteristic
344	C₄	C₂×C₈×D₁₆	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	2	no	—
347	C₂	—	16	4	no	—
348	C₂	C₂×C₁₆×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²×D₁₆
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂⁴
socle	C₂	C₂×C₁₆×D₁₆
prime radical	#33981	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ 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. #72225356	C₂	semidirect
2. #72225355	C₂	semidirect
3. #72225353	C₂	semidirect
4. #72225352	C₂	semidirect
6. C₂×#93751	C₂	semidirect
7. C₁₆×M₆₄	C₂	semidirect
14. #151072	C₂²	semidirect
22. #93882	C₂²	semidirect
23. #93751	C₂²	semidirect
24. #93751	C₂²	semidirect
25. #93664	C₂²	semidirect
26. C₁₆×C₃₂	C₂²	semidirect
66. #2264	C₁₆	semidirect
67. #2263	C₁₆	semidirect
148. Q₆₄	C₂×C₁₆	semidirect
149. SD₆₄	C₂×C₁₆	semidirect
150. D₃₂	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72225356	2
#72225355	2
#72225353	2
#72225352	2
#72225308	2
C₂×#93751	2
C₁₆×M₆₄	2