Subgroups — #72322796

Order 2048.

684 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321329	C₂	1	1	yes	maximal, characteristic
2	#72321145	C₂	1	2	yes	maximal, complemented
3	#72320480	C₂	1	1	yes	maximal, characteristic
4	#72286519	C₂	1	2	yes	maximal, complemented
5	#72232024	C₂	1	1	yes	maximal, characteristic
6	#513308	—	2	1	no	—
7	#512545	C₂²	1	1	yes	characteristic
8	#494211	—	2	1	no	—
9	#483409	—	2	1	no	—
10	#153425	—	2	1	no	—
11	#153425	C₄	1	4	yes	complemented
12	C₂×#7511	—	2	2	no	—
13	C₂×#7511	C₂²	1	2	yes	—
14	#152409	C₄	1	2	yes	—
15	#152391	—	2	1	no	—
16	#152300	—	2	1	no	—
17	#152295	C₄	1	1	yes	characteristic
18	#152294	C₄	1	1	yes	characteristic
19	#151998	—	2	1	no	—
20	#151996	C₂²	1	2	yes	—
21	#151056	C₄	1	2	yes	complemented
22	#151055	C₄	1	2	yes	complemented
23	#151054	—	2	2	no	—
24	#151038	—	2	1	no	—
25	#151037	C₂²	1	1	yes	characteristic
26	C₂×#7463	C₂²	1	1	yes	characteristic
27	C₂×#2258	C₂×C₄	1	1	yes	characteristic
28	C₂×#2258	D₄	1	1	yes	characteristic
29	C₄×#504	D₄	1	1	yes	characteristic
30	#20787	—	4	1	no	—
31	#17799	—	4	1	no	—
32	#13737	—	2	8	no	—
33	C₂×M₁₂₈	—	2	4	no	—
34	C₂×M₁₂₈	—	2	2	no	—
35	#13638	—	2	2	no	—
36	C₂×#1425	C₂×C₄	1	2	yes	—
37	#13633	C₂×C₄	1	2	yes	—
38	#13549	C₂×C₄	1	1	yes	characteristic
39	#13549	C₂×C₄	1	2	yes	—
40	#13549	D₄	1	1	yes	characteristic
41	#13549	Q₈	1	2	yes	—
42	C₄×M₆₄	C₂×C₄	1	2	yes	—
43	C₄×M₆₄	D₄	1	1	yes	characteristic
44	C₂×#1397	D₄	1	2	yes	—
45	C₂×#1397	C₂³	1	1	yes	characteristic
46	#12911	—	2	2	no	—
47	#12893	—	4	1	no	—
48	#12840	—	2	1	no	—
49	#12839	—	2	1	no	—
50	#12832	—	4	1	no	—
51	#12670	—	2	2	no	—
52	#12668	—	4	1	no	—
53	#12025	—	4	2	no	—
54	#12024	—	2	2	no	—
55	#12023	—	2	2	no	—
56	#12009	—	4	1	no	—
57	#12008	—	2	1	no	—
58	C₂×#1372	—	2	1	no	—
59	#7511	—	2	4	no	—
60	#7511	—	2	2	no	—
61	#7511	C₂×C₄	1	4	yes	—
62	#7476	—	2	1	no	—
63	#7475	—	2	1	no	—
64	#7473	—	2	2	no	—
65	#7463	—	2	2	no	—
66	#7463	—	2	2	no	—
67	#7463	—	2	1	no	—
68	#7463	—	2	1	no	—
69	#7463	C₂×C₄	1	1	yes	characteristic
70	#7463	C₂×C₄	1	2	yes	—
71	#7463	C₂×C₄	1	1	yes	characteristic
72	C₂×#504	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×#504	SD₁₆	1	1	yes	characteristic
74	#2989	—	8	1	no	—
75	C₄×#131	D₈	1	1	yes	characteristic
76	#2258	—	2	4	no	—
77	#2258	—	2	2	no	—
78	#2258	—	2	2	no	—
79	C₂×M₆₄	C₄²	1	2	yes	—
80	C₂×M₆₄	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×M₆₄	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
83	C₂²×C₃₂	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₃₂	C₄⋊C₄	1	2	yes	—
85	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₃₂	C₂×D₄	1	1	yes	characteristic
87	C₂×#368	—	2	2	no	—
88	#2158	—	2	2	no	—
89	#2155	—	4	2	no	—
90	#2108	C₂²⋊C₄	1	1	yes	characteristic
91	#2108	C₄⋊C₄	1	2	yes	—
92	#2108	SD₁₆	1	1	yes	characteristic
93	#2108	Q₁₆	1	2	yes	—
94	C₄×M₃₂	D₈	1	1	yes	characteristic
95	C₄×M₃₂	SD₁₆	1	2	yes	—
96	C₂×C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
98	#1841	—	8	1	no	—
99	#1833	—	4	2	no	—
100	#1827	—	8	1	no	—
101	#1813	—	4	1	no	—
102	#1812	—	4	1	no	—
103	#1772	—	4	2	no	—
104	#1757	—	8	1	no	—
105	#1747	—	8	1	no	—
106	#1569	—	8	2	no	—
107	#1568	—	4	2	no	—
108	#1567	—	4	2	no	—
109	#1565	—	8	1	no	—
110	#1564	—	4	1	no	—
111	C₂×#335	—	4	1	no	—
112	M₁₂₈	—	2	4	no	—
113	M₁₂₈	—	2	8	no	—
114	M₁₂₈	—	2	4	no	—
115	C₂×C₆₄	—	2	4	no	—
116	C₂×C₆₄	—	2	8	no	—
117	#1426	—	4	4	no	—
118	#1425	—	2	2	no	—
119	#1425	—	2	2	no	—
120	#1424	—	2	2	no	—
121	#1423	—	2	2	no	—
122	#1397	C₂²⋊C₄	1	2	yes	—
123	#1397	C₂²⋊C₄	1	2	yes	—
124	#1397	C₄⋊C₄	1	2	yes	—
125	#1397	C₂²×C₄	1	1	yes	characteristic
126	#1397	C₂²×C₄	1	1	yes	characteristic
127	#1397	C₂×D₄	1	1	yes	characteristic
128	#1397	C₂×Q₈	1	1	yes	characteristic
129	C₄×C₃₂	C₄²	1	2	yes	—
130	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
131	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
132	#1385	—	4	1	no	—
133	#1384	—	4	1	no	—
134	#1382	—	4	2	no	—
135	#1372	—	2	2	no	—
136	#1372	—	2	1	no	—
137	#1372	—	2	1	no	—
138	#1372	—	4	2	no	—
139	#1372	—	4	1	no	—
140	#1371	—	4	2	no	—
141	#1371	—	4	1	no	—
142	C₂×#131	#102	1	1	yes	characteristic
143	C₂×#131	SD₃₂	1	1	yes	characteristic
144	C₄×#41	D₁₆	1	1	yes	characteristic
145	#504	—	2	4	no	—
146	#504	—	2	2	no	—
147	#504	—	2	2	no	—
148	C₂×M₃₂	#102	1	1	yes	characteristic
149	C₂×M₃₂	#102	1	1	yes	characteristic
150	C₂×M₃₂	#103	1	2	yes	—
151	C₂×M₃₂	#106	1	2	yes	—
152	C₂²×C₁₆	#95	1	2	yes	—
153	C₂²×C₁₆	#103	1	2	yes	—
154	C₂²×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
156	#446	—	16	1	no	—
157	#436	—	16	1	no	—
158	#435	—	16	2	no	—
159	#432	—	8	2	no	—
160	#431	—	8	1	no	—
161	C₂×#108	—	4	2	no	—
162	#427	—	4	2	no	—
163	#423	—	8	2	no	—
164	C₂×C₄⋊C₈	—	8	1	no	—
165	#413	—	8	1	no	—
166	#410	—	8	1	no	—
167	#405	—	8	2	no	—
168	#405	#102	1	1	yes	characteristic
169	#405	#107	1	2	yes	—
170	#405	SD₃₂	1	1	yes	characteristic
171	#405	Q₃₂	1	2	yes	—
172	C₄×M₁₆	D₁₆	1	1	yes	characteristic
173	C₄×M₁₆	SD₃₂	1	2	yes	—
174	C₂×C₄×C₈	#102	1	2	yes	—
175	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
176	M₆₄	—	2	2	no	—
177	M₆₄	—	2	4	no	—
178	M₆₄	—	2	4	no	—
179	M₆₄	#104	1	4	yes	—
180	C₂×C₃₂	—	2	2	no	—
181	C₂×C₃₂	—	2	4	no	—
182	C₂×C₃₂	—	2	2	no	—
183	C₂×C₃₂	—	2	2	no	—
184	C₂×C₃₂	#95	1	2	yes	—
185	C₂×C₃₂	#95	1	2	yes	—
186	C₂×C₃₂	C₂×C₄²	1	1	yes	characteristic
187	C₂×C₃₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₃₂	#131	1	2	yes	—
191	C₂×C₃₂	#131	1	2	yes	—
192	#368	—	4	2	no	—
193	#368	—	4	2	no	—
194	#367	—	4	2	no	—
195	#366	—	4	2	no	—
196	#365	—	8	4	no	—
197	#346	#103	1	2	yes	—
198	#346	#106	1	2	yes	—
199	#346	#107	1	2	yes	—
200	C₄×C₁₆	#95	1	2	yes	—
201	C₄×C₁₆	#102	1	2	yes	—
202	C₄×C₁₆	#103	1	2	yes	—
203	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
206	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
207	#344	—	8	2	no	—
208	#339	—	8	2	no	—
209	#338	—	8	1	no	—
210	H(3,ℤ/4)	—	8	1	no	—
211	H(3,ℤ/4)	—	8	2	no	—
212	#335	—	4	2	no	—
213	#335	—	4	1	no	—
214	#335	—	4	1	no	—
215	#335	—	8	2	no	—
216	#335	—	8	1	no	—
217	#334	—	8	2	no	—
218	#334	—	8	1	no	—
219	C₆₄	—	2	8	no	—
220	C₂×#41	#357	1	1	yes	characteristic
221	#131	—	2	2	no	—
222	#131	—	2	2	no	—
223	#131	—	4	2	no	—
224	C₂×M₁₆	—	8	2	no	—
225	C₂×M₁₆	—	16	2	no	—
226	C₂×M₁₆	#357	1	1	yes	characteristic
227	C₂×M₁₆	#357	1	1	yes	characteristic
228	C₂×M₁₆	#358	1	2	yes	—
229	C₂×M₁₆	#367	1	2	yes	—
230	C₂²×C₈	—	16	1	no	—
231	C₂²×C₈	#340	1	2	yes	—
232	C₂²×C₈	#358	1	2	yes	—
233	C₂²×C₈	C₂×#102	1	1	yes	characteristic
234	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
235	C₄×Q₈	—	2	1	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	1	no	—
238	#117	—	8	2	no	—
239	#117	#357	1	1	yes	characteristic
240	#117	#366	1	2	yes	—
241	C₂×C₄²	#357	1	2	yes	—
242	C₂×C₄²	C₂×D₁₆	1	1	yes	characteristic
243	M₃₂	—	2	2	no	—
244	M₃₂	—	2	4	no	—
245	M₃₂	—	2	4	no	—
246	M₃₂	#359	1	4	yes	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	2	4	no	—
249	C₂×C₁₆	—	2	2	no	—
250	C₂×C₁₆	—	2	2	no	—
251	C₂×C₁₆	#340	1	2	yes	—
252	C₂×C₁₆	#340	1	2	yes	—
253	C₂×C₁₆	C₂×#95	1	1	yes	characteristic
254	C₂×C₁₆	#408	1	2	yes	—
255	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
256	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
257	C₂×C₁₆	C₂×#104	1	2	yes	—
258	C₂×C₁₆	C₂×#106	1	1	yes	characteristic
259	#108	—	8	2	no	—
260	#108	—	8	2	no	—
261	#107	—	8	2	no	—
262	#106	—	8	2	no	—
263	C₄⋊C₈	—	8	1	no	—
264	C₄⋊C₈	—	8	2	no	—
265	C₄⋊C₈	—	8	1	no	—
266	C₄⋊C₈	—	16	1	no	—
267	C₄⋊C₈	—	16	2	no	—
268	#101	—	16	1	no	—
269	#100	—	16	1	no	—
270	#99	—	16	2	no	—
271	C₂²⋊C₈	—	16	1	no	—
272	C₂²⋊C₈	—	16	2	no	—
273	#97	—	16	2	no	—
274	#97	—	16	2	no	—
275	#97	#358	1	2	yes	—
276	#97	#366	1	2	yes	—
277	#97	#367	1	2	yes	—
278	C₄×C₈	—	16	1	no	—
279	C₄×C₈	—	16	2	no	—
280	C₄×C₈	#340	1	2	yes	—
281	C₄×C₈	#357	1	2	yes	—
282	C₄×C₈	#358	1	2	yes	—
283	C₄×C₈	C₂×#102	1	1	yes	characteristic
284	C₄×C₈	C₂×#107	1	1	yes	characteristic
285	C₄×C₈	C₂×SD₃₂	1	1	yes	characteristic
286	C₄×C₈	C₂×Q₃₂	1	1	yes	characteristic
287	C₃₂	—	2	2	no	—
288	C₃₂	—	2	1	no	—
289	C₃₂	—	2	1	no	—
290	C₃₂	#339	1	4	yes	—
291	C₃₂	#405	1	2	yes	—
292	C₃₂	#424	1	1	yes	characteristic
293	C₃₂	#424	1	1	yes	characteristic
294	#41	—	2	2	no	—
295	#41	—	2	2	no	—
296	C₂×Q₈	#1418	1	1	yes	characteristic
297	C₂×D₄	#1418	1	2	yes	—
298	C₂×D₄	#1418	1	1	yes	characteristic
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	#1380	1	2	yes	—
302	C₂²×C₄	C₂×#357	1	1	yes	characteristic
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	2	4	no	—
306	M₁₆	—	4	2	no	—
307	M₁₆	—	16	2	no	—
308	M₁₆	—	16	2	no	—
309	M₁₆	—	64	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	16	1	no	—
315	C₂×C₈	—	16	1	no	—
316	C₂×C₈	—	16	2	no	—
317	C₂×C₈	—	16	2	no	—
318	C₂×C₈	—	64	1	no	—
319	C₂×C₈	#1380	1	2	yes	—
320	C₂×C₈	#1380	1	2	yes	—
321	C₂×C₈	#1492	1	2	yes	—
322	C₂×C₈	C₂×#340	1	1	yes	characteristic
323	C₂×C₈	C₂×#358	1	1	yes	characteristic
324	C₂×C₈	C₂×#358	1	1	yes	characteristic
325	C₂×C₈	C₂×#359	1	2	yes	—
326	C₂×C₈	C₂×#367	1	1	yes	characteristic
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	16	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	16	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	16	2	no	—
336	C₄²	#1380	1	2	yes	—
337	C₄²	C₂×#357	1	1	yes	characteristic
338	C₄²	C₂×#366	1	1	yes	characteristic
339	C₁₆	—	2	1	no	—
340	C₁₆	—	2	1	no	—
341	C₁₆	—	2	2	no	—
342	C₁₆	#1382	1	4	yes	—
343	C₁₆	#1493	1	2	yes	—
344	C₁₆	#1729	1	1	yes	characteristic
345	C₁₆	C₂×#339	1	1	yes	characteristic
346	C₂³	#7477	1	2	yes	—
347	C₂³	C₂×#1418	1	1	yes	characteristic
348	Q₈	—	2	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	—	32	2	no	—
359	C₂×C₄	#7477	1	2	yes	—
360	C₂×C₄	#11671	1	2	yes	—
361	C₂×C₄	C₂×#1380	1	1	yes	characteristic
362	C₂×C₄	C₂×#1418	1	1	yes	characteristic
363	C₂×C₄	#13625	1	2	yes	—
364	C₈	—	2	2	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	2	2	no	—
368	C₈	—	32	1	no	—
369	C₈	—	32	2	no	—
370	C₈	—	32	1	no	—
371	C₈	—	32	2	no	—
372	C₈	#7897	1	1	yes	characteristic
373	C₈	#11672	1	2	yes	—
374	C₈	C₂×#1382	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	2	no	—
378	C₂²	#150157	1	2	yes	—
379	C₂²	C₂×#7477	1	1	yes	characteristic
380	C₄	—	2	2	no	—
381	C₄	—	4	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	1	no	—
384	C₄	—	64	2	no	—
385	C₄	#126852	1	1	yes	characteristic
386	C₄	#150157	1	2	yes	—
387	C₄	#151778	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	2	2	no	—
390	C₂	—	4	2	no	—
391	C₂	#72322794	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72322794
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#357
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ 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. #72321145	C₂	semidirect
4. #72286519	C₂	semidirect
11. #153425	C₄	semidirect
21. #151056	C₄	semidirect
22. #151055	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72321329	2
#72321145	2
#72320480	2
#72286519	2
#72232024	2