Finite Group Databasebeta

← #35218068

Subgroups — #35218068

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×#19104706C211yesmaximal, characteristic, complemented
3C2×#19104704C211yesmaximal, characteristic, complemented
4C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic163C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781581C211yesmaximal, characteristic, complemented
8#23781579C211yesmaximal, characteristic, complemented
9#23781573C211yesmaximal, characteristic, complemented
10#23781568C211yesmaximal, characteristic, complemented
11#23781553C211yesmaximal, characteristic, complemented
12#23781545C211yesmaximal, characteristic, complemented
13#23781541C211yesmaximal, characteristic, complemented
14#23781540C211yesmaximal, characteristic, complemented
15C2×#120×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic, complemented
20C23×Dic163C2211yescharacteristic
21C23×Dic163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic, complemented
23#19104707—24no—
24#19104706—24no—
25#19104704—24no—
26(C22⋊D163)⋊C4—24no—
27C2×C4⋊Dic163C2211yescharacteristic, complemented
28C2×C4⋊Dic163C2211yescharacteristic, 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×C22⋊Dic163C2211yescharacteristic, 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—41no—
67C2×C22⋊D163—41no—
68C2×C22⋊D163—41no—
69C2×C22⋊D163D414yescomplemented
70C22×Dic163—22no—
71C22×Dic163—22no—
72C22×Dic163—22no—
73C22×Dic163—22no—
74C22×Dic163—41no—
75C22×Dic163—41no—
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C22×Dic163C2311yescharacteristic
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C2×C4×D163—22no—
84C2×C4×D163—22no—
85C2×C4×D163—41no—
86C4⋊Dic163—22no—
87C4⋊Dic163—41no—
88Dic163⋊C4—22no—
89Dic163⋊C4—22no—
90C22⋊Dic163—22no—
91C22⋊Dic163—22no—
92C22⋊Dic163—22no—
93C22⋊Dic163—22no—
94C22⋊Dic163—22no—
95C22⋊Dic163—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163—22no—
98C22⋊Dic163—41no—
99C22⋊Dic163D414yescomplemented
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—22no—
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—22no—
121C2×D4×C163—41no—
122C2×D4×C163—41no—
123C22×C4×C163C2311yescharacteristic
124C22×C4×C163C2311yescharacteristic
125C22×C4×C163C2311yescharacteristic
126C22⋊C4×C163—22no—
127C22⋊C4×C163—22no—
128C22⋊C4×C163D414yescomplemented
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—21no—
133C22×D163—21no—
134C22×D163—21no—
135C22×D163C2×D412yes—
136C22⋊D163—28no—
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—21no—
158C2×Dic163—44no—
159C2×Dic163—44no—
160C2×Dic163C2×D412yes—
161C2×Dic163C2×D412yes—
162C2×Dic163C2×D412yes—
163C4×D163—44no—
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—21no—
188C23×C163—44no—
189C23×C163—44no—
190C23×C163—44no—
191C23×C163—44no—
192C23×C163—44no—
193C23×C163—44no—
194C23×C163C2×D412yes—
195C23×C163C2×D412yes—
196C23×C163C2×D412yes—
197C23×C163#4112yes—
198C23×C163#4112yes—
199C23×C163C2411yescharacteristic
200D4×C163—44no—
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×C163C2×D412yes—
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—416no—
225C22×C163—41no—
226C22×C163—41no—
227C22×C163—41no—
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—22no—
249C2×D163—22no—
250C2×D163—22no—
251C2×D163—41no—
252C2×D163—41no—
253C2×D163—41no—
254C2×D163—41no—
255C4×C163—22no—
256C4×C163—22no—
257C4×C163—41no—
258Dic163—22no—
259Dic163—22no—
260Dic163—22no—
261Dic163—41no—
262Dic163—41no—
263Dic163—41no—
264Dic163—41no—
265C2×C163—24no—
266C2×C163—42no—
267C2×C163—42no—
268C2×C163—81no—
269C2×C163C2×#12111yescharacteristic
270C2×C163#52511yescharacteristic
271C2×C163#53811yescharacteristic
272C2×C163D4211yescharacteristic
273C2×C163D4211yescharacteristic
274C2×C163#54611yescharacteristic
275C2×C163#54611yescharacteristic
276D163—42no—
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—22no—
354C2×D4—41no—
355C2×D4—41no—
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×D4—1634no—
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—3262no—
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⋊C4C22⋊D16314yescomplemented
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—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—3261no—
438C23—3261no—
439C23—3261no—
440C23—3261no—
441C23—3261no—
442C23—3261no—
443C23—1632no—
444C23D4×D16312yes—
445C23D4×D16312yes—
446C23C2×C22⋊D16312yes—
447C23#1457969812yes—
448C23#1457969812yes—
449C23C23×D16311yescharacteristic
450D4—44no—
451D4—44no—
452D4—44no—
453D4—3268no—
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—3261no—
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×C4C2×C22⋊D16312yes—
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—416no—
503C22—41no—
504C22—41no—
505C22—41no—
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—3262no—
519C22—3262no—
520C22—3262no—
521C22—6521no—
522C22—6521no—
523C22—6521no—
524C22—6521no—
525C22#1910470614yescomplemented
526C22C2×D4×D16311yescharacteristic
527C22C2×D4×D16311yescharacteristic
528C22C22×C22⋊D16311yescharacteristic
529C22C2×#1457969811yescharacteristic
530C22C2×#1457969811yescharacteristic
531C22#1910478011yescharacteristic
532C22#1910478011yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—3262no—
537C4—3262no—
538C4—3262no—
539C4—6521no—
540C4—6521no—
541C4—6521no—
542C4—6521no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—6522no—
548C2C2×#1910470611yescharacteristic
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×#19104706C2semidirect
3. C2×#19104704C2semidirect
4. C2×(C22⋊D163)⋊C4C2semidirect
5. C22×C22⋊Dic163C2semidirect
6. #23781588C2semidirect
7. #23781581C2semidirect
8. #23781579C2semidirect
9. #23781573C2semidirect
10. #23781568C2semidirect
11. #23781553C2semidirect
12. #23781545C2semidirect
13. #23781541C2semidirect
14. #23781540C2semidirect
15. C2×#120×C163C2semidirect
19. C22×C22⋊D163C22semidirect
22. C22×C4×D163C22semidirect
27. C2×C4⋊Dic163C22semidirect
28. C2×C4⋊Dic163C22semidirect
29. C2×Dic163⋊C4C22semidirect
30. C2×Dic163⋊C4C22semidirect
39. C2×C22⋊Dic163C22semidirect
40. C2×C22⋊Dic163C22semidirect
41. C2×C22⋊Dic163C22semidirect
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
69. C2×C22⋊D163D4semidirect
99. C22⋊Dic163D4semidirect
128. C22⋊C4×C163D4semidirect
240. C22×C163#121semidirect
277. C163#2424semidirect
282. C2×#120D163semidirect
405. C22⋊C4C22⋊D163semidirect
525. C22#19104706semidirect

Maximal subgroups

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

subgroupindex
C2×#191047072
C2×#191047062
C2×#191047042
C2×(C22⋊D163)⋊C42
C22×C22⋊Dic1632
#237815882
#237815812
#237815792
#237815732
#237815682
#237815532
#237815452
#237815412
#237815402
C2×#120×C1632
#2424163