Subgroups — #72323181

Order 2048.

678 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321739	C₂	1	1	yes	maximal, characteristic
2	#72321574	C₂	1	2	yes	maximal, complemented
3	#72321329	C₂	1	1	yes	maximal, characteristic
4	#72286415	C₂	1	2	yes	maximal, complemented
5	#72231861	C₂	1	1	yes	maximal, characteristic
6	#513308	C₄	1	2	yes	—
7	#512731	C₄	1	1	yes	characteristic
8	#512545	C₂²	1	1	yes	characteristic
9	#479917	C₄	1	1	yes	characteristic
10	#153583	—	2	2	no	—
11	#153582	—	2	2	no	—
12	#153570	C₄	1	2	yes	—
13	#153430	C₄	1	2	yes	—
14	#153425	—	2	2	no	—
15	#153425	C₄	1	2	yes	—
16	C₂×#7511	—	2	2	no	—
17	C₂×#7511	C₂²	1	2	yes	—
18	C₂×#7511	C₂²	1	2	yes	—
19	#149509	—	2	2	no	—
20	#149508	—	2	2	no	—
21	#149502	C₄	1	1	yes	characteristic
22	#149501	C₄	1	1	yes	characteristic
23	#149496	C₂²	1	1	yes	characteristic
24	C₂×#7330	C₂²	1	1	yes	characteristic
25	C₂×#2258	C₂×C₄	1	1	yes	characteristic
26	C₂×#2258	C₂×C₄	1	1	yes	characteristic
27	#33420	C₈	1	1	yes	characteristic
28	C₄×#504	C₂×C₄	1	1	yes	characteristic
29	#17335	C₈	1	1	yes	characteristic
30	#13737	—	2	8	no	—
31	C₂×M₁₂₈	—	2	4	no	—
32	C₂×M₁₂₈	C₂×C₄	1	4	yes	—
33	C₂×M₁₂₈	C₂×C₄	1	4	yes	—
34	#13641	—	2	2	no	—
35	#13640	—	2	2	no	—
36	#13628	C₈	1	2	yes	—
37	#13554	C₈	1	2	yes	—
38	#13549	C₂×C₄	1	1	yes	characteristic
39	#13549	C₂×C₄	1	1	yes	characteristic
40	#13549	D₄	1	2	yes	—
41	#13549	Q₈	1	2	yes	—
42	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
43	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
44	C₄×M₆₄	D₄	1	2	yes	—
45	C₂×#1397	C₂×C₄	1	1	yes	characteristic
46	C₂×#1397	D₄	1	2	yes	—
47	C₂×#1397	C₂³	1	1	yes	characteristic
48	#11409	—	2	2	no	—
49	#11408	—	2	2	no	—
50	#11402	C₈	1	1	yes	characteristic
51	#11401	C₈	1	1	yes	characteristic
52	#11399	C₂×C₄	1	1	yes	characteristic
53	C₂×#1311	C₂×C₄	1	1	yes	characteristic
54	#7544	—	2	4	no	—
55	#7513	—	2	4	no	—
56	#7511	—	2	4	no	—
57	#7511	—	2	2	no	—
58	#7511	—	2	2	no	—
59	#7511	—	2	2	no	—
60	#7511	—	2	2	no	—
61	#7332	—	2	2	no	—
62	#7332	—	2	2	no	—
63	#7330	—	2	1	no	—
64	#7330	—	2	1	no	—
65	#7330	—	2	1	no	—
66	#7330	—	2	1	no	—
67	C₂×#504	C₄²	1	1	yes	characteristic
68	C₂×#504	C₂×C₈	1	1	yes	characteristic
69	#2965	—	2	1	no	—
70	C₄×#131	C₂×C₈	1	1	yes	characteristic
71	#2258	—	2	4	no	—
72	#2258	—	2	4	no	—
73	C₂×M₆₄	C₄²	1	1	yes	characteristic
74	C₂×M₆₄	C₄²	1	1	yes	characteristic
75	C₂×M₆₄	C₂²⋊C₄	1	2	yes	—
76	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
77	C₂×M₆₄	C₂×C₈	1	2	yes	—
78	C₂²×C₃₂	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₃₂	C₄⋊C₄	1	2	yes	—
80	C₂²×C₃₂	C₂×C₈	1	2	yes	—
81	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
83	#2164	C₁₆	1	4	yes	—
84	#2162	—	2	2	no	—
85	#2151	—	2	1	no	—
86	#2113	—	2	1	no	—
87	#2108	C₄²	1	1	yes	characteristic
88	#2108	C₄⋊C₄	1	2	yes	—
89	#2108	C₂×C₈	1	1	yes	characteristic
90	#2108	M₁₆	1	2	yes	—
91	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
92	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
93	C₄×M₃₂	M₁₆	1	2	yes	—
94	C₂×C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
97	#1455	C₁₆	1	2	yes	—
98	#1453	C₁₆	1	2	yes	—
99	#1451	—	2	1	no	—
100	#1450	C₂×C₈	1	1	yes	characteristic
101	C₈×M₁₆	—	2	2	no	—
102	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
103	M₁₂₈	—	2	8	no	—
104	M₁₂₈	—	2	4	no	—
105	M₁₂₈	—	2	4	no	—
106	M₁₂₈	—	2	4	no	—
107	M₁₂₈	M₁₆	1	8	yes	complemented
108	C₂×C₆₄	—	2	8	no	—
109	C₂×C₆₄	—	2	4	no	—
110	C₂×C₆₄	—	2	4	no	—
111	C₄⋊C₃₂	—	2	4	no	—
112	C₂²⋊C₃₂	—	2	4	no	—
113	#1397	—	2	1	no	—
114	#1397	—	2	1	no	—
115	#1397	—	2	1	no	—
116	#1397	C₂²⋊C₄	1	2	yes	—
117	#1397	C₂²⋊C₄	1	2	yes	—
118	#1397	C₄⋊C₄	1	2	yes	—
119	#1397	C₂×D₄	1	1	yes	characteristic
120	#1397	C₂×D₄	1	1	yes	characteristic
121	#1397	C₂×D₄	1	1	yes	characteristic
122	#1397	C₂×Q₈	1	1	yes	characteristic
123	C₄×C₃₂	—	2	1	no	—
124	C₄×C₃₂	C₂²⋊C₄	1	2	yes	—
125	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
126	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
127	#1312	—	2	2	no	—
128	#1312	—	2	2	no	—
129	#1311	—	2	1	no	—
130	#1311	—	2	1	no	—
131	#1311	—	2	1	no	—
132	C₈×C₁₆	—	2	1	no	—
133	C₂×#131	C₄×C₈	1	1	yes	characteristic
134	C₂×#131	M₃₂	1	1	yes	characteristic
135	C₄×#41	M₃₂	1	1	yes	characteristic
136	#504	—	2	2	no	—
137	#504	—	2	2	no	—
138	#504	—	2	4	no	—
139	C₂×M₃₂	—	2	1	no	—
140	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
141	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
142	C₂×M₃₂	#97	1	2	yes	—
143	C₂×M₃₂	C₄⋊C₈	1	2	yes	—
144	C₂²×C₁₆	—	2	1	no	—
145	C₂²×C₁₆	#95	1	2	yes	—
146	C₂²×C₁₆	C₂²⋊C₈	1	2	yes	—
147	C₂²×C₁₆	C₂×C₄²	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
149	#446	—	4	1	no	—
150	#436	—	4	1	no	—
151	#435	—	4	2	no	—
152	#431	—	2	2	no	—
153	#431	—	2	1	no	—
154	#405	C₄×C₈	1	1	yes	characteristic
155	#405	C₄⋊C₈	1	2	yes	—
156	#405	C₂×C₁₆	1	2	yes	—
157	#405	M₃₂	1	1	yes	characteristic
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	C₂×C₁₆	1	2	yes	—
160	C₄×M₁₆	M₃₂	1	1	yes	characteristic
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	C₂²⋊C₈	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₈	1	1	yes	characteristic
164	M₆₄	—	2	2	no	—
165	M₆₄	—	2	2	no	—
166	M₆₄	—	2	4	no	—
167	M₆₄	—	2	4	no	—
168	M₆₄	—	2	4	no	—
169	C₂×C₃₂	—	2	4	no	—
170	C₂×C₃₂	—	2	4	no	—
171	C₂×C₃₂	—	2	2	no	—
172	C₂×C₃₂	—	2	2	no	—
173	C₂×C₃₂	—	2	4	no	—
174	C₂×C₃₂	#95	1	2	yes	—
175	C₂×C₃₂	#95	1	2	yes	—
176	C₂×C₃₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₃₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₃₂	C₂×M₁₆	1	4	yes	—
181	C₂×C₃₂	#131	1	4	yes	—
182	C₄⋊C₁₆	—	2	4	no	—
183	C₂²⋊C₁₆	—	2	4	no	—
184	#346	#97	1	2	yes	—
185	#346	C₄⋊C₈	1	2	yes	—
186	#346	C₄⋊C₈	1	2	yes	—
187	#346	C₂×C₁₆	1	2	yes	—
188	C₄×C₁₆	—	2	1	no	—
189	C₄×C₁₆	—	2	1	no	—
190	C₄×C₁₆	#95	1	2	yes	—
191	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
192	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
193	C₄×C₁₆	C₂×C₁₆	1	2	yes	—
194	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
197	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
198	#322	—	2	2	no	—
199	#322	—	2	2	no	—
200	#322	C₂×C₁₆	1	2	yes	—
201	C₈²	—	2	1	no	—
202	C₈²	—	2	1	no	—
203	C₈²	C₂×C₁₆	1	1	yes	characteristic
204	C₈²	C₂×C₁₆	1	1	yes	characteristic
205	C₆₄	—	2	8	no	—
206	C₆₄	—	4	4	no	—
207	C₂×#41	#346	1	1	yes	characteristic
208	#131	—	2	2	no	—
209	#131	—	2	2	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	4	2	no	—
212	C₂×M₁₆	C₄×C₁₆	1	2	yes	—
213	C₂×M₁₆	#346	1	1	yes	characteristic
214	C₂×M₁₆	#346	1	1	yes	characteristic
215	C₂×M₁₆	C₄⋊C₁₆	1	2	yes	—
216	C₂²×C₈	—	4	1	no	—
217	C₂²×C₈	#336	1	2	yes	—
218	C₂²×C₈	C₂²⋊C₁₆	1	2	yes	—
219	C₂²×C₈	C₂×C₄×C₈	1	1	yes	characteristic
220	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
221	C₄×Q₈	—	2	1	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	1	no	—
224	#117	#346	1	1	yes	characteristic
225	#117	C₄⋊C₁₆	1	2	yes	—
226	C₂×C₄²	C₂²⋊C₁₆	1	2	yes	—
227	C₂×C₄²	C₂×M₃₂	1	1	yes	characteristic
228	M₃₂	—	2	2	no	—
229	M₃₂	—	2	2	no	—
230	M₃₂	—	2	4	no	—
231	M₃₂	—	2	4	no	—
232	M₃₂	—	4	2	no	—
233	C₂×C₁₆	—	2	4	no	—
234	C₂×C₁₆	—	2	2	no	—
235	C₂×C₁₆	—	2	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	—	2	2	no	—
238	C₂×C₁₆	—	4	2	no	—
239	C₂×C₁₆	#336	1	2	yes	—
240	C₂×C₁₆	#336	1	2	yes	—
241	C₂×C₁₆	C₂×#95	1	1	yes	characteristic
242	C₂×C₁₆	C₂×#97	1	1	yes	characteristic
243	C₂×C₁₆	C₄×M₁₆	1	2	yes	—
244	C₂×C₁₆	#405	1	2	yes	—
245	C₂×C₁₆	C₂×C₂²⋊C₈	1	1	yes	characteristic
246	C₂×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
247	C₂×C₁₆	#504	1	4	yes	—
248	C₄⋊C₈	—	4	2	no	—
249	C₄⋊C₈	—	4	1	no	—
250	C₂²⋊C₈	—	4	1	no	—
251	C₂²⋊C₈	—	4	2	no	—
252	#97	—	4	2	no	—
253	#97	C₄×C₁₆	1	2	yes	—
254	#97	C₄⋊C₁₆	1	2	yes	—
255	#97	C₄⋊C₁₆	1	2	yes	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	4	1	no	—
260	C₄×C₈	#336	1	2	yes	—
261	C₄×C₈	C₂²⋊C₁₆	1	2	yes	—
262	C₄×C₈	C₂²⋊C₁₆	1	2	yes	—
263	C₄×C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
264	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
265	C₄×C₈	C₂²×C₁₆	1	1	yes	characteristic
266	C₄×C₈	C₂²×C₁₆	1	1	yes	characteristic
267	C₃₂	—	2	2	no	—
268	C₃₂	—	2	2	no	—
269	C₃₂	—	4	2	no	—
270	C₃₂	—	4	2	no	—
271	C₃₂	#407	1	2	yes	—
272	C₃₂	#408	1	2	yes	—
273	C₃₂	#423	1	2	yes	—
274	C₃₂	#424	1	2	yes	—
275	#41	—	2	2	no	—
276	#41	—	2	2	no	—
277	C₂×Q₈	#1398	1	1	yes	characteristic
278	C₂×D₄	#1398	1	2	yes	—
279	C₂×D₄	#1398	1	1	yes	characteristic
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	#1374	1	2	yes	—
283	C₂²×C₄	C₂×#346	1	1	yes	characteristic
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	4	no	—
287	M₁₆	—	4	2	no	—
288	M₁₆	—	16	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	—	16	1	no	—
297	C₂×C₈	#1374	1	2	yes	—
298	C₂×C₈	#1374	1	2	yes	—
299	C₂×C₈	C₈×M₁₆	1	2	yes	—
300	C₂×C₈	C₂×#336	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
302	C₂×C₈	#2108	1	2	yes	—
303	C₂×C₈	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄⋊C₁₆	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	#1374	1	2	yes	—
312	C₄²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
313	C₄²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
314	C₁₆	—	2	2	no	—
315	C₁₆	—	2	2	no	—
316	C₁₆	—	4	2	no	—
317	C₁₆	—	4	2	no	—
318	C₁₆	#1451	1	2	yes	—
319	C₁₆	#1728	1	1	yes	characteristic
320	C₁₆	#1729	1	1	yes	characteristic
321	C₁₆	#2113	1	2	yes	—
322	C₁₆	#2151	1	2	yes	—
323	C₂³	#7466	1	2	yes	—
324	C₂³	C₂×#1398	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	D₄	—	2	4	no	—
327	D₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	#7466	1	2	yes	—
336	C₂×C₄	#11408	1	2	yes	—
337	C₂×C₄	C₂×#1374	1	1	yes	characteristic
338	C₂×C₄	C₂×#1398	1	1	yes	characteristic
339	C₂×C₄	#13551	1	2	yes	—
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	2	no	—
344	C₈	—	8	1	no	—
345	C₈	—	8	1	no	—
346	C₈	—	8	2	no	—
347	C₈	#7848	1	1	yes	characteristic
348	C₈	C₁₆×M₁₆	1	2	yes	—
349	C₈	#12545	1	1	yes	characteristic
350	C₂²	—	2	2	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	2	no	—
353	C₂²	#149508	1	2	yes	—
354	C₂²	C₂×#7466	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	1	no	—
359	C₄	#126560	1	1	yes	characteristic
360	C₄	#149508	1	2	yes	—
361	C₄	#151756	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	2	no	—
364	C₂	—	4	2	no	—
365	C₂	#72322703	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#7848
commutator	C₂×C₈	C₂×C₄×C₁₆
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72322703
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72321574	C₂	semidirect
4. #72286415	C₂	semidirect
107. M₁₂₈	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72321739	2
#72321574	2
#72321329	2
#72286415	2
#72231861	2