Subgroups — #72225372

Order 2048.

835 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225369	C₂	1	8	yes	maximal, complemented
2	#72225362	C₂	1	1	yes	maximal, characteristic, complemented
3	#72225353	C₂	1	2	yes	maximal, complemented
4	#72225308	C₂	1	1	yes	maximal, characteristic
5	C₂×#93752	C₂	1	2	yes	maximal, complemented
6	#72216226	C₂	1	1	yes	maximal, characteristic, complemented
7	#512773	C₄	1	1	yes	characteristic
8	#512751	—	2	2	no	—
9	C₂×#12046	C₂²	1	2	yes	—
10	#151082	C₂²	1	4	yes	—
11	#151081	C₄	1	4	yes	—
12	#151073	C₂²	1	8	yes	complemented
13	#151072	—	2	4	no	—
14	#151035	C₄	1	1	yes	characteristic
15	C₂×#7459	C₂²	1	2	yes	—
16	#150067	C₄	1	2	yes	—
17	#150051	—	2	2	no	—
18	C₈×M₆₄	C₂²	1	1	yes	characteristic
19	C₈×M₆₄	C₂²	1	1	yes	characteristic
20	C₂×#7052	C₂²	1	1	yes	characteristic
21	#93885	C₂²	1	4	yes	complemented
22	#93752	C₂²	1	8	yes	complemented
23	#93666	C₂²	1	4	yes	complemented
24	#33981	C₈	1	1	yes	characteristic
25	#33974	—	4	2	no	—
26	C₂×#2178	C₂×C₄	1	2	yes	—
27	C₂×#2170	—	2	2	no	—
28	#13656	C₈	1	4	yes	—
29	#13655	C₂×C₄	1	4	yes	—
30	#13648	—	4	4	no	—
31	#13647	—	2	8	no	—
32	#13636	C₈	1	1	yes	characteristic
33	C₂×#1422	—	2	2	no	—
34	#13620	C₈	1	2	yes	—
35	#13558	—	4	2	no	—
36	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
37	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
38	C₄×M₆₄	D₄	1	1	yes	characteristic
39	C₄×M₆₄	D₄	1	1	yes	characteristic
40	#12046	—	2	2	no	—
41	#12046	—	2	2	no	—
42	#12046	C₂×C₄	1	4	yes	—
43	#12046	C₂³	1	4	yes	—
44	C₂×#1373	C₂×C₄	1	2	yes	—
45	C₂×#1335	—	2	2	no	—
46	C₂×C₈×C₁₆	—	2	1	no	—
47	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
48	#7462	C₂×C₄	1	4	yes	—
49	#7459	—	2	2	no	—
50	#7459	C₂³	1	4	yes	—
51	#7387	C₂×C₄	1	8	yes	—
52	#7381	—	2	8	no	—
53	#7329	C₂³	1	2	yes	—
54	#7329	C₂³	1	2	yes	—
55	C₈×C₃₂	C₂×C₄	1	2	yes	—
56	C₈×C₃₂	D₄	1	2	yes	—
57	#7052	—	2	1	no	—
58	#7052	C₂³	1	2	yes	—
59	C₂×#508	C₂×C₈	1	2	yes	—
60	C₂×#504	—	4	2	no	—
61	C₂×#443	—	2	2	no	—
62	#2264	C₁₆	1	4	yes	complemented
63	#2263	C₁₆	1	4	yes	complemented
64	#2262	C₂×C₈	1	4	yes	—
65	#2258	—	4	8	no	—
66	#2258	—	8	4	no	—
67	C₂×M₆₄	—	2	1	no	—
68	C₂×M₆₄	—	2	1	no	—
69	C₂×M₆₄	—	4	2	no	—
70	C₂×M₆₄	—	8	4	no	—
71	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
72	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
73	C₂×M₆₄	D₈	1	2	yes	—
74	#2178	—	2	2	no	—
75	#2178	—	2	2	no	—
76	#2178	C₂×C₈	1	4	yes	—
77	#2178	C₂²×C₄	1	4	yes	—
78	#2170	—	2	4	no	—
79	#2170	—	2	4	no	—
80	#2170	—	4	2	no	—
81	#2170	—	4	2	no	—
82	C₂×#368	C₂×C₈	1	2	yes	—
83	C₂×#364	—	2	2	no	—
84	C₂×#359	—	2	2	no	—
85	C₂×#350	—	4	2	no	—
86	C₂×C₄×C₁₆	—	2	2	no	—
87	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈²	#41	1	1	yes	characteristic
90	#1425	C₂×C₈	1	4	yes	—
91	#1422	—	2	4	no	—
92	#1422	—	4	2	no	—
93	#1417	C₂×C₈	1	8	yes	—
94	#1401	—	4	8	no	—
95	#1397	—	2	1	no	—
96	#1397	—	2	1	no	—
97	#1397	C₂×C₈	1	2	yes	—
98	#1397	D₈	1	2	yes	—
99	C₄×C₃₂	C₂²×C₄	1	2	yes	—
100	C₄×C₃₂	C₂²×C₄	1	2	yes	—
101	C₄×C₃₂	C₂×D₄	1	2	yes	—
102	C₄×C₃₂	C₂×D₄	1	2	yes	—
103	#1373	—	2	2	no	—
104	#1373	C₂²×C₄	1	4	yes	—
105	#1335	—	2	4	no	—
106	#1335	—	2	4	no	—
107	C₈×C₁₆	—	2	2	no	—
108	C₈×C₁₆	—	2	2	no	—
109	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
110	C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
111	C₈×C₁₆	#41	1	1	yes	characteristic
112	C₈×C₁₆	C₂⁴	1	1	yes	characteristic
113	C₂×#135	—	2	2	no	—
114	C₂×#131	—	4	2	no	—
115	#508	—	2	2	no	—
116	#508	—	2	2	no	—
117	#508	C₂×C₁₆	1	4	yes	—
118	#508	C₂²×C₈	1	4	yes	—
119	C₂×Q₃₂	C₂×C₁₆	1	2	yes	—
120	C₂×SD₃₂	—	2	2	no	—
121	C₂×D₁₆	C₂×C₁₆	1	2	yes	—
122	#504	—	4	4	no	—
123	#504	—	4	4	no	—
124	#504	—	8	2	no	—
125	#504	—	8	2	no	—
126	C₂×M₃₂	—	4	2	no	—
127	C₂×M₃₂	—	8	2	no	—
128	C₂²×C₁₆	—	4	2	no	—
129	C₂²×C₁₆	—	8	2	no	—
130	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
131	C₂²×C₁₆	#131	1	1	yes	characteristic
132	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
133	C₂²×C₁₆	#135	1	1	yes	characteristic
134	#443	—	2	4	no	—
135	#443	—	2	4	no	—
136	#443	—	4	2	no	—
137	#443	—	4	2	no	—
138	C₂×#108	—	2	2	no	—
139	C₂×#104	—	4	2	no	—
140	C₂×C₄×C₈	—	2	1	no	—
141	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
142	Q₆₄	C₂×C₁₆	1	4	yes	complemented
143	SD₆₄	C₂×C₁₆	1	8	yes	complemented
144	D₃₂	C₂×C₁₆	1	4	yes	complemented
145	M₆₄	—	2	4	no	—
146	M₆₄	—	2	2	no	—
147	M₆₄	—	2	2	no	—
148	M₆₄	—	2	2	no	—
149	M₆₄	—	4	2	no	—
150	M₆₄	—	4	4	no	—
151	M₆₄	—	8	8	no	—
152	M₆₄	—	8	2	no	—
153	M₆₄	C₂×C₁₆	1	4	yes	—
154	C₂×C₃₂	—	2	2	no	—
155	C₂×C₃₂	—	2	2	no	—
156	C₂×C₃₂	—	8	8	no	—
157	C₂×C₃₂	—	8	2	no	—
158	C₂×C₃₂	C₂²×C₈	1	2	yes	—
159	C₂×C₃₂	C₂²×C₈	1	2	yes	—
160	C₂×C₃₂	C₂×D₈	1	4	yes	—
161	#368	—	2	2	no	—
162	#368	C₂²×C₈	1	4	yes	—
163	#364	—	2	4	no	—
164	#364	—	4	2	no	—
165	#359	—	2	4	no	—
166	#359	—	2	4	no	—
167	#350	—	4	4	no	—
168	#350	—	4	4	no	—
169	C₄×C₁₆	—	2	4	no	—
170	C₄×C₁₆	—	4	2	no	—
171	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
172	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
173	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
174	C₄×C₁₆	#131	1	1	yes	characteristic
175	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
176	C₄×C₁₆	#135	1	1	yes	characteristic
177	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
178	C₄×C₁₆	C₂²×D₄	1	1	yes	characteristic
179	C₈²	—	2	1	no	—
180	C₈²	C₄×D₄	1	1	yes	characteristic
181	C₈²	C₂×#41	1	1	yes	characteristic
182	C₂×#41	—	4	2	no	—
183	#135	—	2	4	no	—
184	#135	—	2	4	no	—
185	#135	—	4	2	no	—
186	#135	—	4	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×SD₁₆	—	4	2	no	—
189	C₂×D₈	—	2	2	no	—
190	#131	—	4	4	no	—
191	#131	—	4	4	no	—
192	#131	—	8	2	no	—
193	#131	—	8	2	no	—
194	C₂×M₁₆	—	4	2	no	—
195	C₂×M₁₆	—	8	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	8	2	no	—
198	C₂²×C₈	C₈×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₄×D₈	1	1	yes	characteristic
200	C₂×C₄²	#443	1	1	yes	characteristic
201	Q₃₂	—	2	2	no	—
202	Q₃₂	C₂²×C₁₆	1	4	yes	—
203	SD₃₂	—	2	4	no	—
204	SD₃₂	—	2	4	no	—
205	D₁₆	—	2	2	no	—
206	D₁₆	C₂²×C₁₆	1	4	yes	—
207	M₃₂	—	4	4	no	—
208	M₃₂	—	8	2	no	—
209	M₃₂	—	8	4	no	—
210	M₃₂	—	8	4	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	4	4	no	—
214	C₂×C₁₆	—	8	4	no	—
215	C₂×C₁₆	—	8	2	no	—
216	C₂×C₁₆	—	8	4	no	—
217	C₂×C₁₆	—	8	4	no	—
218	C₂×C₁₆	C₈×D₄	1	1	yes	characteristic
219	C₂×C₁₆	C₈×D₄	1	1	yes	characteristic
220	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
221	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×C₁₆	1	2	yes	—
223	C₂×C₁₆	#504	1	2	yes	—
224	C₂×C₁₆	#509	1	2	yes	—
225	C₂×C₁₆	#509	1	2	yes	—
226	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
227	C₂×C₁₆	C₂×#131	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₈	1	1	yes	characteristic
229	C₂×C₁₆	C₂×#135	1	1	yes	characteristic
230	#108	—	2	4	no	—
231	#108	—	4	2	no	—
232	#104	—	4	4	no	—
233	#104	—	4	4	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	C₈×D₄	1	1	yes	characteristic
238	C₄×C₈	C₄×D₈	1	1	yes	characteristic
239	C₄×C₈	#443	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄×D₄	1	1	yes	characteristic
241	C₃₂	—	2	4	no	—
242	C₃₂	—	2	2	no	—
243	C₃₂	—	2	4	no	—
244	C₃₂	—	4	2	no	—
245	C₃₂	—	8	4	no	—
246	C₃₂	C₂²×C₁₆	1	4	yes	—
247	#41	—	4	4	no	—
248	#41	—	4	4	no	—
249	#41	—	8	2	no	—
250	#41	—	8	2	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	8	2	no	—
256	C₂²×C₄	C₈×D₈	1	1	yes	characteristic
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	4	4	no	—
261	D₈	—	2	4	no	—
262	D₈	—	4	2	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	—	8	2	no	—
265	M₁₆	—	8	4	no	—
266	M₁₆	—	8	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	4	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	—	8	4	no	—
272	C₂×C₈	—	8	4	no	—
273	C₂×C₈	C₈×D₈	1	1	yes	characteristic
274	C₂×C₈	C₈×D₈	1	1	yes	characteristic
275	C₂×C₈	D₄×C₁₆	1	2	yes	—
276	C₂×C₈	#2170	1	2	yes	—
277	C₂×C₈	#2177	1	2	yes	—
278	C₂×C₈	#2179	1	2	yes	—
279	C₂×C₈	C₂×C₈×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₈	1	1	yes	characteristic
281	C₄²	—	2	1	no	—
282	C₄²	C₈×D₈	1	1	yes	characteristic
283	C₄²	C₂×#443	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	8	4	no	—
289	C₁₆	—	8	4	no	—
290	C₁₆	D₄×C₁₆	1	2	yes	—
291	C₁₆	#2177	1	1	yes	characteristic
292	C₁₆	#2177	1	1	yes	characteristic
293	C₁₆	C₂³×C₁₆	1	1	yes	characteristic
294	C₁₆	C₂×#504	1	1	yes	characteristic
295	C₁₆	C₂×#509	1	2	yes	—
296	C₂³	—	8	2	no	—
297	C₂³	#12047	1	1	yes	characteristic
298	Q₈	—	4	4	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	C₁₆×D₈	1	2	yes	—
311	C₂×C₄	#12045	1	2	yes	—
312	C₂×C₄	#12047	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₈	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	4	2	no	—
316	C₈	—	8	4	no	—
317	C₈	C₁₆×D₈	1	2	yes	—
318	C₈	#12045	1	2	yes	—
319	C₈	C₂×D₄×C₁₆	1	1	yes	characteristic
320	C₈	C₂×#2170	1	1	yes	characteristic
321	C₈	C₂×#2177	1	1	yes	characteristic
322	C₈	C₂×#2179	1	1	yes	characteristic
323	C₂²	—	4	1	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	32	2	no	—
326	C₂²	#122946	1	2	yes	—
327	C₂²	C₂×#12047	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	4	1	no	—
330	C₄	—	8	4	no	—
331	C₄	#122946	1	2	yes	—
332	C₄	C₂×C₁₆×D₈	1	1	yes	characteristic
333	C₄	C₂×#12045	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	2	no	—
336	C₂	—	16	4	no	—
337	C₂	C₂×#122946	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#2177
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂⁴
socle	C₂	C₂×#122946
prime radical	#33981	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ 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. #72225369	C₂	semidirect
2. #72225362	C₂	semidirect
3. #72225353	C₂	semidirect
5. C₂×#93752	C₂	semidirect
6. #72216226	C₂	semidirect
12. #151073	C₂²	semidirect
21. #93885	C₂²	semidirect
22. #93752	C₂²	semidirect
23. #93666	C₂²	semidirect
62. #2264	C₁₆	semidirect
63. #2263	C₁₆	semidirect
142. Q₆₄	C₂×C₁₆	semidirect
143. SD₆₄	C₂×C₁₆	semidirect
144. 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
#72225369	2
#72225362	2
#72225353	2
#72225308	2
C₂×#93752	2
#72216226	2