Finite Group Databasebeta

← #35217512

Subgroups — #35217512

Order 20864.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104705C211yesmaximal, characteristic
2C2×#19104697C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
4C22×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
6#23781566C211yesmaximal, characteristic, complemented
7#23781555C211yesmaximal, characteristic, complemented
8#23781554C211yesmaximal, characteristic, complemented
9#23781543C211yesmaximal, characteristic, complemented
10#23781543C211yesmaximal, characteristic, complemented
11#23781542C211yesmaximal, characteristic, complemented
12#23781536C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic163C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#385×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C23×Dic163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic
23C22×C4×D163C2211yescharacteristic, complemented
24#19104705C418yescomplemented
25#19104697—24no—
26C4×C22⋊D163—24no—
27C2×Dic163⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic163C2211yescharacteristic
29C2×(C2×D163)⋊C4—24no—
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
41C22⋊C4×D163—24no—
42C2×C4×Dic163C2211yescharacteristic, complemented
43C2×C4×Dic163C2211yescharacteristic, complemented
44C2×C4×Dic163C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C163⋊#95C2211yescharacteristic, complemented
49C23×C4×C163C2211yescharacteristic
50C2×C4⋊C4×C163C2211yescharacteristic, complemented
51C2×C4⋊C4×C163C2211yescharacteristic, complemented
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic, complemented
55#95×C163C2211yescharacteristic, complemented
56C23×D163—22no—
57C23×D163—22no—
58C23×D163—22no—
59C23×D163—22no—
60C23×D163—41no—
61C23×D163—41no—
62C23×D163C2×C414yescomplemented
63C23×D163C2311yescharacteristic
64C23×D163C2311yescharacteristic
65C23×D163C2311yescharacteristic
66C2×D652—22no—
67C2×D652—22no—
68C2×D652—41no—
69C2×D652—41no—
70C2×D652C2×C414yescomplemented
71C2×D652C2×C414yescomplemented
72C2×C22⋊D163—22no—
73C2×C22⋊D163—22no—
74C2×C22⋊D163C2×C414yescomplemented
75C22×Dic163—22no—
76C22×Dic163—22no—
77C22×Dic163—41no—
78C22×Dic163C2311yescharacteristic
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C2×C4×D163—22no—
84C2×C4×D163—22no—
85C2×C4×D163—22no—
86C2×C4×D163—22no—
87C2×C4×D163—22no—
88C2×C4×D163—22no—
89C2×C4×D163—41no—
90C2×C4×D163—41no—
91C2×C4×D163—41no—
92Dic163⋊C4—22no—
93C22⋊Dic163D414yescomplemented
94(C2×D163)⋊C4—22no—
95(C2×D163)⋊C4—22no—
96(C2×D163)⋊C4—22no—
97(C2×D163)⋊C4—22no—
98(C2×D163)⋊C4—22no—
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—41no—
102(C2×D163)⋊C4C2×C414yescomplemented
103(C2×D163)⋊C4C2×C414yescomplemented
104(C2×D163)⋊C4D414yescomplemented
105C4×Dic163—22no—
106C4×Dic163—22no—
107C4×Dic163—41no—
108C24×C163C2311yescharacteristic
109C22×C4×C163—22no—
110C22×C4×C163—22no—
111C22×C4×C163—41no—
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C22×C4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2311yescharacteristic
118C4⋊C4×C163—22no—
119C4⋊C4×C163—41no—
120C22⋊C4×C163—22no—
121C22⋊C4×C163C2×C414yescomplemented
122C22×D163—21no—
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—28no—
127C22×D163—21no—
128C22×D163—21no—
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—21no—
133C22×D163—21no—
134C22×D163—21no—
135C22×D163—21no—
136C22×D163—21no—
137C22×D163—21no—
138C22×D163—21no—
139C22×D163—28no—
140C22×D163—44no—
141C22×D163—44no—
142C22×D163—44no—
143C22×D163—44no—
144C22×D163—44no—
145C22×D163C22×C412yes—
146C22×D163C22×C412yes—
147C22×D163C22×C412yes—
148C22×D163C2×D412yes—
149C22×D163#4112yes—
150D652—28no—
151D652—44no—
152C22⋊D163—28no—
153C2×Dic163—21no—
154C2×Dic163—21no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—21no—
160C2×Dic163—21no—
161C2×Dic163—21no—
162C2×Dic163—21no—
163C2×Dic163—21no—
164C2×Dic163—44no—
165C2×Dic163C22×C412yes—
166C2×Dic163C2×D412yes—
167C2×Dic163C2×D412yes—
168C2×Dic163C2×D412yes—
169C4×D163—44no—
170C4×D163—44no—
171C4×D163—44no—
172C23×C163—21no—
173C23×C163—21no—
174C23×C163—21no—
175C23×C163—21no—
176C23×C163C22×C412yes—
177C23×C163C2×D412yes—
178C23×C163#4112yes—
179C23×C163C2411yescharacteristic
180C2×C4×C163—21no—
181C2×C4×C163—21no—
182C2×C4×C163—21no—
183C2×C4×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163—21no—
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163—21no—
193C2×C4×C163—21no—
194C2×C4×C163—21no—
195C2×C4×C163—44no—
196C2×C4×C163C22×C412yes—
197C2×C4×C163C22×C412yes—
198C2×C4×C163C2×D412yes—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—22no—
205C22×C163C4×D414yescomplemented
206C22×C163C23×C411yescharacteristic
207C22×C163C22×D411yescharacteristic
208C22×C163C22×D411yescharacteristic
209C22×C163C2×#4111yescharacteristic
210C22×C163C2×#4111yescharacteristic
211C22×C163H(5,𝔽2)11yescharacteristic
212C22×C163H(5,𝔽2)11yescharacteristic
213C2×D163—22no—
214C2×D163—22no—
215C2×D163—22no—
216C2×D163—22no—
217C2×D163—22no—
218C2×D163—22no—
219C2×D163—22no—
220C2×D163—22no—
221C2×D163—22no—
222C2×D163—22no—
223C2×D163—22no—
224C2×D163—22no—
225C2×D163—416no—
226C2×D163—41no—
227C2×D163—41no—
228C2×D163—41no—
229C2×D163—41no—
230C2×D163—41no—
231C2×D163—41no—
232C2×D163—41no—
233C2×D163—41no—
234C2×D163C4×D414yescomplemented
235C4×C163—22no—
236C4×C163—22no—
237C4×C163—22no—
238C4×C163—41no—
239C4×C163—41no—
240C4×C163—41no—
241Dic163—22no—
242Dic163—22no—
243Dic163—22no—
244Dic163—22no—
245Dic163—41no—
246C2×C163—24no—
247C2×C163C2×C4×D411yescharacteristic
248C2×C163C2×C4×D411yescharacteristic
249C2×C163#51811yescharacteristic
250C2×C163D4211yescharacteristic
251C2×C163#54611yescharacteristic
252C2×C163#54611yescharacteristic
253C2×C163#55111yescharacteristic
254D163—24no—
255D163—42no—
256D163—42no—
257C163#235811yescharacteristic, complemented, Sylow
258#2358—1631nomaximal, Sylow
259C2×#120—1631no—
260C2×C4×D4—1631no—
261C2×C4×D4—1631no—
262C22×C22⋊C4—1631no—
263C22×C22⋊C4—1631no—
264#390—1631no—
265#388—1631no—
266#388—1631no—
267#386—1631no—
268#386—1631no—
269#386—1631no—
270#385—1631no—
271#385D16311yescharacteristic, complemented
272C4×C22⋊C4—1631no—
273C4×C22⋊C4—1631no—
274C25—1631no—
275C22×D4—1631no—
276C22×D4—1631no—
277C22×D4—1631no—
278C23×C4—1631no—
279C23×C4—1631no—
280C23×C4—1631no—
281C23×C4—1631no—
282C23×C4C2×D16311yescharacteristic
283#120—1638no—
284C4×D4—3264no—
285C4×D4—3264no—
286C2×C4⋊C4—1631no—
287C2×C4⋊C4C2×D16311yescharacteristic, complemented
288C2×C4⋊C4C2×D16311yescharacteristic, complemented
289C2×C22⋊C4—3264no—
290C2×C22⋊C4—3264no—
291C2×C22⋊C4—1631no—
292C2×C22⋊C4—1631no—
293C2×C22⋊C4—1631no—
294C2×C22⋊C4—1631no—
295C2×C22⋊C4—1631no—
296C2×C22⋊C4—1631no—
297C2×C22⋊C4—1631no—
298C2×C22⋊C4—1631no—
299C2×C22⋊C4—1631no—
300C2×C22⋊C4—1631no—
301C2×C22⋊C4—1631no—
302C2×C22⋊C4—1631no—
303C2×C22⋊C4C2×D16311yescharacteristic
304C2×C22⋊C4C2×D16311yescharacteristic
305C2×C42—1631no—
306C2×C42—1631no—
307C2×C42—1631no—
308#95—1631no—
309#95—1631no—
310#95—1631no—
311#95—1631no—
312#95C2×D16311yescharacteristic, complemented
313#95C2×D16311yescharacteristic, complemented
314C24—3262no—
315C24—3262no—
316C24—3262no—
317C24—3262no—
318C24—6521no—
319C24—6521no—
320C24—1631no—
321C24—1631no—
322C24—1634no—
323C24—1631no—
324C24C22×D16311yescharacteristic
325C2×D4—3262no—
326C2×D4—3262no—
327C2×D4—3262no—
328C2×D4—3262no—
329C2×D4—6521no—
330C2×D4—6521no—
331C2×D4—1634no—
332C2×D4—1634no—
333C2×D4—1634no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—3262no—
338C22×C4—3262no—
339C22×C4—3262no—
340C22×C4—3262no—
341C22×C4—3262no—
342C22×C4—3262no—
343C22×C4—3262no—
344C22×C4—3262no—
345C22×C4—6521no—
346C22×C4—6521no—
347C22×C4—6521no—
348C22×C4—6521no—
349C22×C4—1631no—
350C22×C4—1631no—
351C22×C4—1631no—
352C22×C4—1631no—
353C22×C4—1631no—
354C22×C4C22×D16311yescharacteristic
355C22×C4C22×D16311yescharacteristic
356C22×C4C22×D16311yescharacteristic
357C22×C4C22×D16311yescharacteristic
358C22×C4C22×D16311yescharacteristic
359C22×C4C22×D16311yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—3262no—
363C22⋊C4—22no—
364C22⋊C4—3262no—
365C22⋊C4—3262no—
366C22⋊C4—3262no—
367C22⋊C4—3262no—
368C22⋊C4—3262no—
369C22⋊C4—3262no—
370C22⋊C4—3262no—
371C22⋊C4—6521no—
372C22⋊C4—1634no—
373C22⋊C4—1634no—
374C22⋊C4—1634no—
375C22⋊C4—1634no—
376C22⋊C4C4×D16314yescomplemented
377C42—3262no—
378C42—3262no—
379C42—6521no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—3268no—
385C23—3261no—
386C23—3261no—
387C23—3261no—
388C23—3261no—
389C23—3261no—
390C23—3261no—
391C23—3261no—
392C23—3261no—
393C23—3261no—
394C23—3261no—
395C23—3261no—
396C23—3261no—
397C23—3261no—
398C23—3261no—
399C23—3261no—
400C23—3261no—
401C23—3268no—
402C23—6524no—
403C23—6524no—
404C23—6524no—
405C23—6524no—
406C23—6524no—
407C23—1632no—
408C23—1632no—
409C23—1632no—
410C23—1632no—
411C23—1632no—
412C23C2×C4×D16312yes—
413C23D4×D16312yes—
414C23#1457970012yes—
415C23C23×D16311yescharacteristic
416D4—3268no—
417D4—3268no—
418D4—6524no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—3261no—
438C2×C4—3261no—
439C2×C4—3261no—
440C2×C4—3261no—
441C2×C4—3261no—
442C2×C4—3261no—
443C2×C4—3261no—
444C2×C4—3261no—
445C2×C4—3261no—
446C2×C4—6524no—
447C2×C4—6524no—
448C2×C4—6524no—
449C2×C4—6524no—
450C2×C4—1632no—
451C2×C4—1632no—
452C2×C4—1632no—
453C2×C4—1632no—
454C2×C4C2×C4×D16312yes—
455C2×C4C2×C4×D16312yes—
456C2×C4D4×D16312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—3262no—
464C22—3262no—
465C22—3262no—
466C22—3262no—
467C22—3262no—
468C22—3262no—
469C22—3262no—
470C22—3262no—
471C22—3262no—
472C22—3262no—
473C22—3262no—
474C22—3262no—
475C22—65216no—
476C22—6521no—
477C22—6521no—
478C22—6521no—
479C22—6521no—
480C22—6521no—
481C22—6521no—
482C22—6521no—
483C22—6521no—
484C22—1634no—
485C22#1910469714yescomplemented
486C22C22×C4×D16311yescharacteristic
487C22C2×D4×D16311yescharacteristic
488C22C2×D4×D16311yescharacteristic
489C22#41×D16311yescharacteristic
490C22C2×#1457970011yescharacteristic
491C22#1910478011yescharacteristic
492C22#1910478111yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—3262no—
500C4—3262no—
501C4—3262no—
502C4—3262no—
503C4—6521no—
504C2—24no—
505C2—3264no—
506C2—6522no—
507C2—6522no—
508C2C4×D4×D16311yescharacteristic
509C2C2×#1910469711yescharacteristic
510C2#2378213711yescharacteristic
511C2#2378229711yescharacteristic
512C2#2378230611yescharacteristic
513C2#2378230711yescharacteristic
514C2#2378234211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC2×#19104705C2
hypercenter#385D163
hypocenterC163#2358
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#19104697C2semidirect
3. C2×C4×C22⋊D163C2semidirect
4. C22×(C2×D163)⋊C4C2semidirect
5. C2×C22⋊C4×D163C2semidirect
6. #23781566C2semidirect
7. #23781555C2semidirect
8. #23781554C2semidirect
9. #23781543C2semidirect
10. #23781543C2semidirect
11. #23781542C2semidirect
12. #23781536C2semidirect
13. C22⋊C4×Dic163C2semidirect
14. #23781512C2semidirect
15. #385×C163C2semidirect
23. C22×C4×D163C22semidirect
24. #19104705C4semidirect
27. C2×Dic163⋊C4C22semidirect
38. C2×(C2×D163)⋊C4C22semidirect
39. C2×(C2×D163)⋊C4C22semidirect
40. C2×(C2×D163)⋊C4C22semidirect
42. C2×C4×Dic163C22semidirect
43. C2×C4×Dic163C22semidirect
44. C2×C4×Dic163C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C163⋊#95C22semidirect
50. C2×C4⋊C4×C163C22semidirect
51. C2×C4⋊C4×C163C22semidirect
54. #95×C163C22semidirect
55. #95×C163C22semidirect
62. C23×D163C2×C4semidirect
70. C2×D652C2×C4semidirect
71. C2×D652C2×C4semidirect
74. C2×C22⋊D163C2×C4semidirect
93. C22⋊Dic163D4semidirect
102. (C2×D163)⋊C4C2×C4semidirect
103. (C2×D163)⋊C4C2×C4semidirect
104. (C2×D163)⋊C4D4semidirect
121. C22⋊C4×C163C2×C4semidirect
205. C22×C163C4×D4semidirect
234. C2×D163C4×D4semidirect
257. C163#2358semidirect
271. #385D163semidirect
287. C2×C4⋊C4C2×D163semidirect
288. C2×C4⋊C4C2×D163semidirect
312. #95C2×D163semidirect
313. #95C2×D163semidirect
376. C22⋊C4C4×D163semidirect
485. C22#19104697semidirect

Maximal subgroups

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

subgroupindex
C2×#191047052
C2×#191046972
C2×C4×C22⋊D1632
C22×(C2×D163)⋊C42
C2×C22⋊C4×D1632
#237815662
#237815552
#237815542
#237815432
#237815422
#237815362
C22⋊C4×Dic1632
#237815122
#385×C1632
#2358163