Subgroups — #72323167

Order 2048.

820 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321739	C₂	1	2	yes	maximal
2	#72321735	C₂	1	4	yes	maximal, complemented
3	C₂×#94509	C₂	1	2	yes	maximal, complemented
4	#72234379	C₂	1	2	yes	maximal, complemented
5	#72231854	C₂	1	1	yes	maximal, characteristic
6	#72225405	C₂	1	4	yes	maximal, complemented
7	#512545	C₄	1	1	yes	characteristic
8	#512545	C₂²	1	1	yes	characteristic
9	#479899	C₄	1	1	yes	characteristic
10	#153583	C₄	1	8	yes	—
11	#153582	C₂²	1	8	yes	—
12	#153570	C₄	1	4	yes	—
13	#153430	C₄	1	4	yes	—
14	C₂×#7511	C₂²	1	4	yes	—
15	#149496	C₄	1	2	yes	—
16	#149496	C₄	1	1	yes	characteristic
17	#149496	C₂²	1	2	yes	—
18	#149496	C₂²	1	1	yes	characteristic
19	#149495	C₄	1	1	yes	characteristic
20	#149495	C₂²	1	2	yes	—
21	C₂×#7330	C₄	1	2	yes	—
22	C₂×#7330	C₂²	1	1	yes	characteristic
23	#94509	C₂²	1	4	yes	—
24	#94509	C₂²	1	4	yes	—
25	#94506	C₂²	1	8	yes	—
26	C₂×#2258	C₈	1	2	yes	—
27	C₂×#2258	C₂×C₄	1	2	yes	—
28	C₄×#504	C₈	1	1	yes	characteristic
29	C₄×#504	C₂×C₄	1	1	yes	characteristic
30	C₈×#131	C₈	1	1	yes	characteristic
31	C₂×M₁₂₈	C₂×C₄	1	8	yes	—
32	#13549	C₈	1	4	yes	—
33	#13549	C₈	1	2	yes	—
34	#13549	C₂×C₄	1	4	yes	—
35	#13549	C₂×C₄	1	2	yes	—
36	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
37	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
38	C₄×M₆₄	C₂×C₄	1	2	yes	—
39	C₄×M₆₄	C₂³	1	2	yes	—
40	C₂×#1397	C₂×C₄	1	2	yes	—
41	C₂×#1397	C₂×C₄	1	2	yes	—
42	C₂×#1397	C₂×C₄	1	1	yes	characteristic
43	C₂×#1397	C₂³	1	1	yes	characteristic
44	#11399	C₈	1	1	yes	characteristic
45	#11399	C₈	1	2	yes	—
46	#11399	C₂×C₄	1	2	yes	—
47	#11399	C₂×C₄	1	1	yes	characteristic
48	C₈×M₃₂	C₈	1	1	yes	characteristic
49	C₈×M₃₂	C₈	1	2	yes	—
50	C₂×#1311	C₂×C₄	1	2	yes	—
51	C₂×#1311	C₂×C₄	1	1	yes	characteristic
52	#7544	C₂×C₄	1	16	yes	—
53	#7513	C₂×C₄	1	16	yes	—
54	#7511	C₂×C₄	1	8	yes	—
55	#7511	C₂³	1	8	yes	—
56	#7330	C₂×C₄	1	1	yes	characteristic
57	#7330	C₂×C₄	1	2	yes	—
58	#7330	C₂×C₄	1	2	yes	—
59	#7330	C₂×C₄	1	2	yes	—
60	#7330	C₂×C₄	1	1	yes	characteristic
61	#7330	C₂×C₄	1	2	yes	—
62	#7330	C₂×C₄	1	2	yes	—
63	#7330	C₂³	1	2	yes	—
64	#7330	C₂³	1	1	yes	characteristic
65	#7330	C₂³	1	1	yes	characteristic
66	C₂×#504	—	2	1	no	—
67	C₂×#504	C₄²	1	1	yes	characteristic
68	C₂×#504	C₂×C₈	1	1	yes	characteristic
69	C₈×#41	—	2	1	no	—
70	C₄×#131	C₂×C₈	1	1	yes	characteristic
71	#2258	—	2	8	no	—
72	#2258	—	2	8	no	—
73	C₂×M₆₄	C₄²	1	2	yes	—
74	C₂×M₆₄	C₂×C₈	1	2	yes	—
75	C₂×M₆₄	C₂×C₈	1	4	yes	—
76	C₂×M₆₄	C₂²×C₄	1	4	yes	—
77	C₂²×C₃₂	C₄²	1	4	yes	—
78	C₂²×C₃₂	C₂×C₈	1	4	yes	—
79	C₂²×C₃₂	C₂×C₈	1	2	yes	—
80	C₂²×C₃₂	C₂²×C₄	1	2	yes	—
81	#2108	—	2	1	no	—
82	#2108	—	2	2	no	—
83	#2108	C₄²	1	2	yes	—
84	#2108	C₄²	1	1	yes	characteristic
85	#2108	C₂×C₈	1	2	yes	—
86	#2108	C₂×C₈	1	1	yes	characteristic
87	C₄×M₃₂	C₂×C₈	1	2	yes	—
88	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
89	C₄×M₃₂	C₂×C₈	1	1	yes	characteristic
90	C₄×M₃₂	C₂×C₈	1	2	yes	—
91	C₂×C₄×C₁₆	C₂×C₈	1	2	yes	—
92	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₆	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
95	#1450	C₂×C₈	1	2	yes	—
96	#1450	C₂×C₈	1	1	yes	characteristic
97	C₈×M₁₆	—	2	2	no	—
98	C₈×M₁₆	—	2	1	no	—
99	C₂×C₈²	C₂×C₈	1	2	yes	—
100	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
101	M₁₂₈	—	2	8	no	—
102	M₁₂₈	C₂²×C₄	1	16	yes	—
103	C₂×C₆₄	—	2	8	no	—
104	#1397	C₄²	1	4	yes	—
105	#1397	C₂×C₈	1	4	yes	—
106	#1397	C₂×C₈	1	2	yes	—
107	#1397	C₂²×C₄	1	2	yes	—
108	#1397	C₂²×C₄	1	1	yes	characteristic
109	#1397	C₂²×C₄	1	2	yes	—
110	#1397	C₂²×C₄	1	1	yes	characteristic
111	#1397	C₂²×C₄	1	2	yes	—
112	#1397	C₂²×C₄	1	1	yes	characteristic
113	#1397	C₂⁴	1	1	yes	characteristic
114	C₄×C₃₂	C₄²	1	2	yes	—
115	C₄×C₃₂	C₂×C₈	1	2	yes	—
116	C₄×C₃₂	C₂×C₈	1	4	yes	—
117	C₄×C₃₂	C₂²×C₄	1	4	yes	—
118	#1311	C₂×C₈	1	2	yes	—
119	#1311	C₂×C₈	1	1	yes	characteristic
120	#1311	C₂×C₈	1	2	yes	—
121	#1311	C₂×C₈	1	1	yes	characteristic
122	#1311	C₂²×C₄	1	1	yes	characteristic
123	#1311	C₂²×C₄	1	2	yes	—
124	#1311	C₂²×C₄	1	1	yes	characteristic
125	C₈×C₁₆	C₂×C₈	1	2	yes	—
126	C₈×C₁₆	C₂×C₈	1	2	yes	—
127	C₈×C₁₆	C₂×C₈	1	2	yes	—
128	C₂×#131	C₄×C₈	1	1	yes	characteristic
129	C₂×#131	M₃₂	1	1	yes	characteristic
130	C₄×#41	M₃₂	1	1	yes	characteristic
131	#504	—	2	4	no	—
132	#504	—	2	4	no	—
133	#504	—	4	4	no	—
134	C₂×M₃₂	—	2	1	no	—
135	C₂×M₃₂	—	2	2	no	—
136	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
137	C₂×M₃₂	C₄×C₈	1	2	yes	—
138	C₂×M₃₂	C₄×C₈	1	1	yes	characteristic
139	C₂×M₃₂	C₄×C₈	1	2	yes	—
140	C₂²×C₁₆	—	2	1	no	—
141	C₂²×C₁₆	—	2	2	no	—
142	C₂²×C₁₆	C₂×C₄²	1	2	yes	—
143	C₂²×C₁₆	C₂×C₄²	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₈	1	2	yes	—
145	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
146	C₈×Q₈	—	4	1	no	—
147	C₈×D₄	—	4	2	no	—
148	C₈×D₄	—	4	1	no	—
149	#431	—	2	2	no	—
150	#431	—	2	1	no	—
151	#405	C₄×C₈	1	2	yes	—
152	#405	C₄×C₈	1	1	yes	characteristic
153	#405	M₃₂	1	1	yes	characteristic
154	#405	M₃₂	1	2	yes	—
155	C₄×M₁₆	M₃₂	1	2	yes	—
156	C₄×M₁₆	M₃₂	1	1	yes	characteristic
157	C₂×C₄×C₈	—	2	2	no	—
158	C₂×C₄×C₈	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₈	1	1	yes	characteristic
161	M₆₄	—	2	4	no	—
162	M₆₄	—	2	4	no	—
163	M₆₄	—	2	8	no	—
164	M₆₄	—	2	8	no	—
165	C₂×C₃₂	—	2	8	no	—
166	C₂×C₃₂	—	2	8	no	—
167	C₂×C₃₂	M₃₂	1	8	yes	—
168	C₂×C₃₂	C₂×C₄²	1	4	yes	—
169	C₂×C₃₂	C₂×C₄²	1	2	yes	—
170	C₂×C₃₂	C₂²×C₈	1	4	yes	—
171	C₂×C₃₂	C₂²×C₈	1	2	yes	—
172	C₂×C₃₂	C₂²×C₈	1	2	yes	—
173	C₂×C₃₂	#131	1	8	yes	—
174	C₂×C₃₂	C₂³×C₄	1	2	yes	—
175	#346	—	2	2	no	—
176	#346	—	2	1	no	—
177	#346	C₄×C₈	1	2	yes	—
178	#346	C₄×C₈	1	2	yes	—
179	#346	C₄×C₈	1	2	yes	—
180	C₄×C₁₆	—	2	2	no	—
181	C₄×C₁₆	—	2	1	no	—
182	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
183	C₄×C₁₆	C₂×C₄²	1	2	yes	—
184	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
185	C₄×C₁₆	C₂²×C₈	1	2	yes	—
186	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
187	C₄×C₁₆	C₂²×C₈	1	2	yes	—
188	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
189	C₄×C₁₆	C₂²×C₈	1	2	yes	—
190	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
191	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
192	#322	—	2	1	no	—
193	#322	—	2	2	no	—
194	C₈²	—	2	1	no	—
195	C₈²	—	2	2	no	—
196	C₈²	C₂²×C₈	1	1	yes	characteristic
197	C₈²	C₂²×C₈	1	2	yes	—
198	C₈²	C₂²×C₈	1	1	yes	characteristic
199	C₆₄	—	2	16	no	—
200	C₂×#41	#346	1	1	yes	characteristic
201	#131	—	2	4	no	—
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	C₂×M₁₆	#346	1	1	yes	characteristic
205	C₂×M₁₆	#346	1	2	yes	—
206	C₂×M₁₆	#346	1	2	yes	—
207	C₂×M₁₆	#346	1	1	yes	characteristic
208	C₂²×C₈	—	4	2	no	—
209	C₂²×C₈	—	4	1	no	—
210	C₂²×C₈	C₂×C₄×C₈	1	2	yes	—
211	C₂²×C₈	C₂×C₄×C₈	1	1	yes	characteristic
212	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
213	C₂²×C₈	C₂×M₃₂	1	2	yes	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	1	no	—
217	#117	#346	1	2	yes	—
218	#117	#346	1	1	yes	characteristic
219	C₂×C₄²	C₂×M₃₂	1	2	yes	—
220	C₂×C₄²	C₂×M₃₂	1	1	yes	characteristic
221	M₃₂	—	2	2	no	—
222	M₃₂	—	2	2	no	—
223	M₃₂	—	2	4	no	—
224	M₃₂	—	2	4	no	—
225	M₃₂	—	4	2	no	—
226	M₃₂	—	4	4	no	—
227	C₂×C₁₆	—	2	4	no	—
228	C₂×C₁₆	—	2	4	no	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	—	2	1	no	—
231	C₂×C₁₆	—	2	2	no	—
232	C₂×C₁₆	—	4	2	no	—
233	C₂×C₁₆	—	4	4	no	—
234	C₂×C₁₆	C₂×C₄×C₈	1	2	yes	—
235	C₂×C₁₆	C₂×C₄×C₈	1	2	yes	—
236	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
237	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
238	C₂×C₁₆	#405	1	4	yes	—
239	C₂×C₁₆	C₂×M₃₂	1	4	yes	—
240	C₂×C₁₆	C₂²×C₄²	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
242	C₄⋊C₈	—	4	2	no	—
243	C₄⋊C₈	—	4	1	no	—
244	C₂²⋊C₈	—	4	2	no	—
245	C₂²⋊C₈	—	4	1	no	—
246	#97	#346	1	2	yes	—
247	#97	#346	1	2	yes	—
248	#97	#346	1	2	yes	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	—	4	2	no	—
253	C₄×C₈	—	4	1	no	—
254	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
255	C₄×C₈	C₂×C₄×C₈	1	2	yes	—
256	C₄×C₈	C₂×M₃₂	1	2	yes	—
257	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
258	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
259	C₄×C₈	C₂×M₃₂	1	2	yes	—
260	C₄×C₈	C₂³×C₈	1	1	yes	characteristic
261	C₃₂	—	2	4	no	—
262	C₃₂	—	2	4	no	—
263	C₃₂	#405	1	4	yes	—
264	C₃₂	C₂×M₃₂	1	4	yes	—
265	C₃₂	C₂×M₃₂	1	4	yes	—
266	C₃₂	C₂×#131	1	4	yes	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	C₂×Q₈	#1398	1	1	yes	characteristic
270	C₂×D₄	#1398	1	2	yes	—
271	C₂×D₄	#1398	1	1	yes	characteristic
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	C₂×#346	1	2	yes	—
275	C₂²×C₄	C₂×#346	1	1	yes	characteristic
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	#1398	1	4	yes	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	16	1	no	—
288	C₂×C₈	—	16	2	no	—
289	C₂×C₈	#1450	1	2	yes	—
290	C₂×C₈	C₂×#346	1	1	yes	characteristic
291	C₂×C₈	C₂×#346	1	2	yes	—
292	C₂×C₈	C₂×#346	1	1	yes	characteristic
293	C₂×C₈	C₂×#346	1	2	yes	—
294	C₂×C₈	C₄×M₃₂	1	2	yes	—
295	C₂×C₈	C₂²×C₄×C₈	1	1	yes	characteristic
296	C₂×C₈	C₂²×M₃₂	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	C₂×#346	1	1	yes	characteristic
304	C₄²	C₂×#346	1	2	yes	—
305	C₄²	C₂²×M₃₂	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	4	1	no	—
309	C₁₆	—	4	1	no	—
310	C₁₆	—	4	2	no	—
311	C₁₆	#1450	1	2	yes	—
312	C₁₆	C₄×M₃₂	1	2	yes	—
313	C₁₆	C₂×#405	1	2	yes	—
314	C₁₆	C₂²×M₃₂	1	2	yes	—
315	C₂³	C₂×#1398	1	2	yes	—
316	C₂³	C₂×#1398	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	D₄	—	2	4	no	—
319	D₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	#11411	1	2	yes	—
328	C₂×C₄	C₂×#1398	1	2	yes	—
329	C₂×C₄	C₂×#1398	1	1	yes	characteristic
330	C₂×C₄	C₂×#1398	1	2	yes	—
331	C₂×C₄	C₂²×#346	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	—	8	1	no	—
336	C₈	—	8	2	no	—
337	C₈	—	8	1	no	—
338	C₈	#11411	1	2	yes	—
339	C₈	C₂×#1398	1	4	yes	—
340	C₈	C₂×#1450	1	1	yes	characteristic
341	C₈	C₂×C₄×M₃₂	1	1	yes	characteristic
342	C₂²	—	2	2	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	#149513	1	2	yes	—
346	C₂²	C₂²×#1398	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	2	no	—
351	C₄	#149513	1	2	yes	—
352	C₄	C₂×#11411	1	1	yes	characteristic
353	C₄	C₂²×#1398	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	2	no	—
356	C₂	—	4	2	no	—
357	C₂	C₂×#149513	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#1450
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	#1397	C₂⁴
socle	C₂	C₂×#149513
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #149496 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72321735	C₂	semidirect
3. C₂×#94509	C₂	semidirect
4. #72234379	C₂	semidirect
6. #72225405	C₂	semidirect

Maximal subgroups

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

subgroup	index
#72321739	2
#72321735	2
C₂×#94509	2
#72234379	2
#72231854	2
#72225405	2