Finite Group Databasebeta

← #35218076

Subgroups — #35218076

Order 20864.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104707C211yesmaximal, characteristic, complemented
2C2×#19104707C211yesmaximal, characteristic, complemented
3C2×#19104704C211yesmaximal, characteristic, complemented
4C2×D4×Dic163C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic163C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781581C211yesmaximal, characteristic, complemented
8#23781579C211yesmaximal, characteristic, complemented
9#23781573C211yesmaximal, characteristic, complemented
10#23781565C211yesmaximal, characteristic, complemented
11#23781559C211yesmaximal, characteristic, complemented
12#23781545C211yesmaximal, characteristic, complemented
13#23781542C211yesmaximal, characteristic, complemented
14#23781537C211yesmaximal, characteristic, complemented
15C2×#120×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic, complemented
20C23×Dic163C2211yescharacteristic
21C23×Dic163C2211yescharacteristic
22C23×Dic163C2211yescharacteristic, complemented
23#19104707—24no—
24#19104707—24no—
25#19104704—24no—
26D4×Dic163—24no—
27C2×C4⋊Dic163C2211yescharacteristic, complemented
28C2×Dic163⋊C4C2211yescharacteristic, complemented
29C2×Dic163⋊C4C2211yescharacteristic, complemented
30C2×Dic163⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic163—24no—
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×C22⋊Dic163C2211yescharacteristic
39C2×C22⋊Dic163C2211yescharacteristic, complemented
40C2×C22⋊Dic163C2211yescharacteristic, complemented
41C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic163C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C163⋊#95C2211yescharacteristic, complemented
49C25×C163C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51C22×D4×C163C2211yescharacteristic
52C22×D4×C163C2211yescharacteristic
53#120×C163—24no—
54C2×C22⋊C4×C163C2211yescharacteristic
55C2×C22⋊C4×C163C2211yescharacteristic
56C2×C22⋊C4×C163C2211yescharacteristic
57C23×D163C2311yescharacteristic
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C2×C22⋊D163—22no—
63C2×C22⋊D163—22no—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—41no—
68C2×C22⋊D163—41no—
69C2×C22⋊D163—41no—
70C22×Dic163—22no—
71C22×Dic163—22no—
72C22×Dic163—22no—
73C22×Dic163—22no—
74C22×Dic163—22no—
75C22×Dic163—22no—
76C22×Dic163—41no—
77C22×Dic163—41no—
78C22×Dic163—41no—
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C4⋊Dic163—22no—
87Dic163⋊C4—22no—
88Dic163⋊C4—22no—
89Dic163⋊C4—41no—
90C22⋊Dic163—22no—
91C22⋊Dic163—22no—
92C22⋊Dic163—22no—
93C22⋊Dic163—22no—
94C22⋊Dic163—22no—
95C22⋊Dic163—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163D414yescomplemented
98C22⋊Dic163D414yescomplemented
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—41no—
102C4×Dic163—22no—
103C24×C163—22no—
104C24×C163—22no—
105C24×C163—22no—
106C24×C163—22no—
107C24×C163—22no—
108C24×C163—41no—
109C24×C163—41no—
110C24×C163C2311yescharacteristic
111C24×C163C2311yescharacteristic
112C24×C163C2311yescharacteristic
113C24×C163C2311yescharacteristic
114C2×D4×C163—22no—
115C2×D4×C163—22no—
116C2×D4×C163—22no—
117C2×D4×C163—22no—
118C2×D4×C163—22no—
119C2×D4×C163—22no—
120C2×D4×C163—41no—
121C2×D4×C163—41no—
122C2×D4×C163D414yescomplemented
123C22×C4×C163C2311yescharacteristic
124C22×C4×C163C2311yescharacteristic
125C22×C4×C163C2311yescharacteristic
126C22⋊C4×C163—22no—
127C22⋊C4×C163—22no—
128C22⋊C4×C163—22no—
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—21no—
133C22×D163—21no—
134C22×D163—21no—
135C22×D163—21no—
136C22⋊D163—44no—
137C22⋊D163—44no—
138C22⋊D163—44no—
139C22⋊D163—44no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—21no—
147C2×Dic163—21no—
148C2×Dic163—21no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163—21no—
154C2×Dic163—21no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163—44no—
158C2×Dic163—44no—
159C2×Dic163—44no—
160C2×Dic163C2×D412yes—
161C2×Dic163C2×D412yes—
162C2×Dic163C2×D412yes—
163C2×Dic163C2×D412yes—
164C23×C163—21no—
165C23×C163—21no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—28no—
169C23×C163—21no—
170C23×C163—21no—
171C23×C163—21no—
172C23×C163—21no—
173C23×C163—21no—
174C23×C163—21no—
175C23×C163—21no—
176C23×C163—21no—
177C23×C163—21no—
178C23×C163—21no—
179C23×C163—21no—
180C23×C163—21no—
181C23×C163—21no—
182C23×C163—21no—
183C23×C163—21no—
184C23×C163—21no—
185C23×C163—21no—
186C23×C163—21no—
187C23×C163—44no—
188C23×C163—44no—
189C23×C163—44no—
190C23×C163—44no—
191C23×C163—44no—
192C23×C163—44no—
193C23×C163C2×D412yes—
194C23×C163C2×D412yes—
195C23×C163C2×D412yes—
196C23×C163C2×D412yes—
197C23×C163#4112yes—
198C23×C163#4112yes—
199C23×C163C2411yescharacteristic
200D4×C163—28no—
201D4×C163—44no—
202D4×C163—44no—
203C2×C4×C163—21no—
204C2×C4×C163—21no—
205C2×C4×C163—21no—
206C2×C4×C163—21no—
207C2×C4×C163—21no—
208C2×C4×C163—21no—
209C2×C4×C163—21no—
210C2×C4×C163—21no—
211C2×C4×C163C2×D412yes—
212C22×C163—22no—
213C22×C163—22no—
214C22×C163—22no—
215C22×C163—22no—
216C22×C163—22no—
217C22×C163—22no—
218C22×C163—22no—
219C22×C163—22no—
220C22×C163—22no—
221C22×C163—22no—
222C22×C163—22no—
223C22×C163—22no—
224C22×C163—22no—
225C22×C163—22no—
226C22×C163—22no—
227C22×C163—416no—
228C22×C163—41no—
229C22×C163—41no—
230C22×C163—41no—
231C22×C163—41no—
232C22×C163—41no—
233C22×C163—41no—
234C22×C163—41no—
235C22×C163—41no—
236C22×C163—41no—
237C22×C163—41no—
238C22×C163—41no—
239C22×C163—41no—
240C22×C163#12114yescomplemented
241C22×C163C22×D411yescharacteristic
242C22×C163C22×D411yescharacteristic
243C22×C163C22×D411yescharacteristic
244C22×C163C2×#4111yescharacteristic
245C22×C163C2×#4111yescharacteristic
246C22×C163H(5,𝔽2)11yescharacteristic
247C22×C163H(5,𝔽2)11yescharacteristic
248C2×D163—41no—
249C2×D163—41no—
250C2×D163—41no—
251C2×D163—41no—
252C2×D163—41no—
253C2×D163—41no—
254C2×D163—41no—
255C4×C163—22no—
256C4×C163—41no—
257C4×C163—41no—
258Dic163—22no—
259Dic163—22no—
260Dic163—22no—
261Dic163—22no—
262Dic163—41no—
263Dic163—41no—
264Dic163—41no—
265C2×C163—24no—
266C2×C163—42no—
267C2×C163—42no—
268C2×C163—42no—
269C2×C163C2×#12111yescharacteristic
270C2×C163#52511yescharacteristic
271C2×C163#53811yescharacteristic
272C2×C163D4211yescharacteristic
273C2×C163D4211yescharacteristic
274C2×C163#54611yescharacteristic
275C2×C163#54611yescharacteristic
276D163—81no—
277C163#242411yescharacteristic, complemented, Sylow
278#2424—1631nomaximal, Sylow
279C2×#121—1631no—
280C2×#121—1631no—
281C2×#120—1631no—
282C2×#120D16311yescharacteristic, complemented
283C2×C4×D4—1631no—
284C22×C22⋊C4—1631no—
285#396—1631no—
286#394—1631no—
287#394—1631no—
288#392—1631no—
289#390—1631no—
290#388—1631no—
291#386—1631no—
292#386—1631no—
293#385—1631no—
294C25C2×D16311yescharacteristic
295C22×D4—1631no—
296C22×D4—1631no—
297C22×D4—1631no—
298C22×D4—1631no—
299C22×D4C2×D16311yescharacteristic
300C22×D4C2×D16311yescharacteristic
301C22×D4C2×D16311yescharacteristic
302C23×C4—1631no—
303C23×C4—1631no—
304C23×C4—1631no—
305#121—3264no—
306#121—3264no—
307#120—24no—
308#120—3264no—
309C4×D4—3264no—
310C2×C4⋊C4—1631no—
311C2×C4⋊C4—1631no—
312C2×C4⋊C4—1631no—
313C2×C4⋊C4—1631no—
314C2×C22⋊C4—3264no—
315C2×C22⋊C4—1631no—
316C2×C22⋊C4—1631no—
317C2×C22⋊C4—1631no—
318C2×C22⋊C4—1631no—
319C2×C22⋊C4—1631no—
320C2×C22⋊C4—1631no—
321C2×C22⋊C4—1631no—
322C2×C22⋊C4—1631no—
323C2×C22⋊C4—1631no—
324C2×C22⋊C4—1631no—
325C2×C22⋊C4—1631no—
326C2×C22⋊C4—1631no—
327C2×C22⋊C4C2×D16311yescharacteristic
328C2×C22⋊C4C2×D16311yescharacteristic
329C2×C22⋊C4C2×D16311yescharacteristic
330C2×C42—1631no—
331#95—1631no—
332#95—1631no—
333#95—1631no—
334#95—1631no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—1631no—
343C24C22×D16311yescharacteristic
344C24C22×D16311yescharacteristic
345C24C22×D16311yescharacteristic
346C24C22×D16311yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—3262no—
356C2×D4—3262no—
357C2×D4—3262no—
358C2×D4—3262no—
359C2×D4—3262no—
360C2×D4—3262no—
361C2×D4—3262no—
362C2×D4—3262no—
363C2×D4—3262no—
364C2×D4—6521no—
365C2×D4—6521no—
366C2×D4—6521no—
367C2×D4C22⋊D16314yescomplemented
368C22×C4—3262no—
369C22×C4—3262no—
370C22×C4—3262no—
371C22×C4—3262no—
372C22×C4—3262no—
373C22×C4—3262no—
374C22×C4—6521no—
375C22×C4—6521no—
376C22×C4—6521no—
377C22×C4—1631no—
378C22×C4—1631no—
379C22×C4—1631no—
380C22×C4—1631no—
381C22×C4—1631no—
382C22×C4—1631no—
383C22×C4—1631no—
384C22×C4C22×D16311yescharacteristic
385C22×C4C22×D16311yescharacteristic
386C22×C4C22×D16311yescharacteristic
387C4⋊C4—3262no—
388C4⋊C4—3262no—
389C4⋊C4—3262no—
390C4⋊C4—6521no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—3262no—
395C22⋊C4—3262no—
396C22⋊C4—3262no—
397C22⋊C4—3262no—
398C22⋊C4—3262no—
399C22⋊C4—3262no—
400C22⋊C4—3262no—
401C22⋊C4—3262no—
402C22⋊C4—3262no—
403C22⋊C4—6521no—
404C22⋊C4—1634no—
405C22⋊C4—1634no—
406C42—3262no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—3261no—
437C23—3261no—
438C23—3261no—
439C23—3261no—
440C23—3261no—
441C23—3261no—
442C23—3261no—
443C23D4×D16312yes—
444C23D4×D16312yes—
445C23C2×C22⋊D16312yes—
446C23C2×C22⋊D16312yes—
447C23#1457969812yes—
448C23#1457969812yes—
449C23C23×D16311yescharacteristic
450D4—28no—
451D4—44no—
452D4—44no—
453D4—6524no—
454D4—6524no—
455D4—6524no—
456D4—6524no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—3261no—
466C2×C4—3261no—
467C2×C4—3261no—
468C2×C4—3261no—
469C2×C4—3261no—
470C2×C4—3261no—
471C2×C4—3261no—
472C2×C4—3261no—
473C2×C4—3261no—
474C2×C4—3261no—
475C2×C4—3261no—
476C2×C4—3261no—
477C2×C4—3261no—
478C2×C4—3261no—
479C2×C4—3261no—
480C2×C4—3261no—
481C2×C4—3261no—
482C2×C4—6524no—
483C2×C4—6524no—
484C2×C4—6524no—
485C2×C4—1632no—
486C2×C4—1632no—
487C2×C4—1632no—
488C2×C4—1632no—
489C2×C4C2×C22⋊D16312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—416no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—6521no—
519C22—6521no—
520C22—6521no—
521C22—6521no—
522C22—6521no—
523C22—6521no—
524C22—6521no—
525C22#1910470714yescomplemented
526C22C2×D4×D16311yescharacteristic
527C22C2×D4×D16311yescharacteristic
528C22C22×C22⋊D16311yescharacteristic
529C22C2×#1457969811yescharacteristic
530C22C2×#1457969811yescharacteristic
531C22#1910478011yescharacteristic
532C22#1910478011yescharacteristic
533C4—22no—
534C4—41no—
535C4—41no—
536C4—3262no—
537C4—3262no—
538C4—3262no—
539C4—3262no—
540C4—6521no—
541C4—6521no—
542C4—6521no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—42no—
547C2—13041no—
548C2C2×#1910470711yescharacteristic
549C2#2378217111yescharacteristic
550C2#2378225011yescharacteristic
551C2#2378229711yescharacteristic
552C2D4×C22⋊D16311yescharacteristic
553C2#2378230211yescharacteristic
554C2#2378230911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×#120×C163C2
hypercenterC2×#120D163
hypocenterC163#2424
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#19104707C2semidirect
2. C2×#19104707C2semidirect
3. C2×#19104704C2semidirect
4. C2×D4×Dic163C2semidirect
5. C22×C22⋊Dic163C2semidirect
6. #23781588C2semidirect
7. #23781581C2semidirect
8. #23781579C2semidirect
9. #23781573C2semidirect
10. #23781565C2semidirect
11. #23781559C2semidirect
12. #23781545C2semidirect
13. #23781542C2semidirect
14. #23781537C2semidirect
15. C2×#120×C163C2semidirect
19. C22×C22⋊D163C22semidirect
22. C23×Dic163C22semidirect
27. C2×C4⋊Dic163C22semidirect
28. C2×Dic163⋊C4C22semidirect
29. C2×Dic163⋊C4C22semidirect
30. C2×Dic163⋊C4C22semidirect
39. C2×C22⋊Dic163C22semidirect
40. C2×C22⋊Dic163C22semidirect
41. C2×(C2×D163)⋊C4C22semidirect
42. C2×(C2×D163)⋊C4C22semidirect
43. C2×(C2×D163)⋊C4C22semidirect
44. C2×C4×Dic163C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C163⋊#95C22semidirect
97. C22⋊Dic163D4semidirect
98. C22⋊Dic163D4semidirect
122. C2×D4×C163D4semidirect
240. C22×C163#121semidirect
277. C163#2424semidirect
282. C2×#120D163semidirect
367. C2×D4C22⋊D163semidirect
525. C22#19104707semidirect

Maximal subgroups

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

subgroupindex
C2×#191047072
C2×#191047042
C2×D4×Dic1632
C22×C22⋊Dic1632
#237815882
#237815812
#237815792
#237815732
#237815652
#237815592
#237815452
#237815422
#237815372
C2×#120×C1632
#2424163