Finite Group Databasebeta

← #35218142

Subgroups — #35218142

Order 20864.

914 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579698C211yesmaximal, characteristic, complemented
2C2×#19104716C211yesmaximal, characteristic, complemented
3C2×#19104707C211yesmaximal, characteristic, complemented
4#23781585C211yesmaximal, characteristic, complemented
5#23781580C211yesmaximal, characteristic, complemented
6#23781577C211yesmaximal, characteristic, complemented
7#23781573C211yesmaximal, characteristic, complemented
8#23781545C211yesmaximal, characteristic, complemented
9#23781545C211yesmaximal, characteristic, complemented
10#23781544C211yesmaximal, characteristic, complemented
11#23781541C211yesmaximal, characteristic, complemented
12#23781537C211yesmaximal, characteristic, complemented
13#23781536C211yesmaximal, characteristic, complemented
14C2×C163⋊#95C211yesmaximal, characteristic, complemented
15C2×#121×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—24no—
17C2×#14579698—24no—
18C2×#14579698—24no—
19C22×Dic326C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C23×Dic163C2211yescharacteristic
24C23×Dic163C2211yescharacteristic
25C23×Dic163C2211yescharacteristic
26C22×C4×D163C2211yescharacteristic
27#19104716—24no—
28#19104707—24no—
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×Dic163⋊C4C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C163⋊#95C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51C22×D4×C163C2211yescharacteristic
52C23×C4×C163C2211yescharacteristic
53#121×C163—24no—
54C2×C4⋊C4×C163C2211yescharacteristic
55C2×C22⋊C4×C163C2211yescharacteristic
56C2×C22⋊C4×C163C2211yescharacteristic
57C23×D163C2311yescharacteristic
58#14579698—232no—
59C2×Dic326—22no—
60C2×Dic326—22no—
61C2×Dic326—22no—
62C2×C22⋊D163—22no—
63C2×C22⋊D163—22no—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—22no—
68C2×C22⋊D163—22no—
69C2×C22⋊D163—41no—
70C2×C22⋊D163D414yescomplemented
71C22×Dic163—22no—
72C22×Dic163—22no—
73C22×Dic163—22no—
74C22×Dic163—22no—
75C22×Dic163—22no—
76C22×Dic163—22no—
77C22×Dic163—22no—
78C22×Dic163—41no—
79C22×Dic163D414yescomplemented
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C22×Dic163C2311yescharacteristic
87C2×C4×D163—22no—
88C2×C4×D163—22no—
89C2×C4×D163—22no—
90Dic163⋊C4—22no—
91Dic163⋊C4—22no—
92Dic163⋊C4—22no—
93C22⋊Dic163—22no—
94C22⋊Dic163—22no—
95C22⋊Dic163—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163—22no—
98C22⋊Dic163—41no—
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—41no—
102C24×C163C2311yescharacteristic
103C24×C163C2311yescharacteristic
104C24×C163C2311yescharacteristic
105C2×D4×C163—22no—
106C2×D4×C163—22no—
107C2×D4×C163—22no—
108C2×D4×C163—22no—
109C2×D4×C163—22no—
110C2×D4×C163—22no—
111C2×D4×C163—22no—
112C2×D4×C163—41no—
113C2×D4×C163D414yescomplemented
114C22×C4×C163—22no—
115C22×C4×C163—22no—
116C22×C4×C163—41no—
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C4⋊C4×C163—22no—
122C22⋊C4×C163—22no—
123C22⋊C4×C163—22no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—21no—
128C22×D163—21no—
129C22×D163—21no—
130C22×D163C2×D412yes—
131Dic326—28no—
132C22⋊D163—28no—
133C22⋊D163—28no—
134C22⋊D163—44no—
135C2×Dic163—28no—
136C2×Dic163—28no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—21no—
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—44no—
155C2×Dic163C2×D412yes—
156C2×Dic163C2×D412yes—
157C2×Dic163C2×D412yes—
158C2×Dic163#4112yescomplemented
159C4×D163—28no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165C23×C163—21no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—21no—
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×C163C2×D412yes—
178C23×C163C2×D412yes—
179C23×C163C2×D412yes—
180C23×C163C2×D412yes—
181C23×C163C2411yescharacteristic
182D4×C163—28no—
183D4×C163—44no—
184D4×C163—44no—
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×C163C2×D412yes—
197C2×C4×C163#4112yescomplemented
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—22no—
205C22×C163—22no—
206C22×C163—22no—
207C22×C163—41no—
208C22×C163—41no—
209C22×C163—41no—
210C22×C163—41no—
211C22×C163—41no—
212C22×C163—41no—
213C22×C163—41no—
214C22×C163—41no—
215C22×C163—41no—
216C22×C163—41no—
217C22×C163—41no—
218C22×C163#12014yes—
219C22×C163C22×D411yescharacteristic
220C22×C163C22×D411yescharacteristic
221C22×C163C22×D411yescharacteristic
222C22×C163C2×#4111yescharacteristic
223C22×C163C2×#4111yescharacteristic
224C22×C163H(5,𝔽2)11yescharacteristic
225C22×C1632−1+411yescharacteristic
226C2×D163—22no—
227C2×D163—22no—
228C2×D163—22no—
229C2×D163—41no—
230C2×D163—41no—
231C2×D163—41no—
232C2×D163—41no—
233C4×C163—22no—
234C4×C163—22no—
235C4×C163—41no—
236C4×C163—41no—
237Dic163—22no—
238Dic163—22no—
239Dic163—22no—
240Dic163—22no—
241Dic163—41no—
242Dic163—41no—
243Dic163—41no—
244C2×C163—24no—
245C2×C163—42no—
246C2×C163—81no—
247C2×C163C2×#12011yescharacteristic
248C2×C163#53211yescharacteristic
249C2×C163#53711yescharacteristic
250C2×C163#54611yescharacteristic
251C2×C163#54611yescharacteristic
252C2×C163#54711yescharacteristic
253C2×C163#54711yescharacteristic
254D163—42no—
255C163#243011yescharacteristic, complemented, Sylow
256#2430—1631nomaximal, Sylow
257C22×#41—1631no—
258C2×#123—1631no—
259C2×#121—1631no—
260C2×#121D16311yescharacteristic, complemented
261#395—1631no—
262#394—1631no—
263#393—1631no—
264#392—1631no—
265#386—1631no—
266#386—1631no—
267#386—1631no—
268#386—1631no—
269#385—1631no—
270#385—1631no—
271C2×#95—1631no—
272C2×#41—3264no—
273C2×#41—3264no—
274C2×#41—3264no—
275C22×Q8—1631no—
276C22×D4—1631no—
277C22×D4—1631no—
278C22×D4—1631no—
279C22×D4C2×D16311yescharacteristic
280C22×D4C2×D16311yescharacteristic
281C22×D4C2×D16311yescharacteristic
282C23×C4—1631no—
283C23×C4—1631no—
284C23×C4—1631no—
285C23×C4—1631no—
286C23×C4C2×D16311yescharacteristic
287#123—3264no—
288#121—24no—
289#121—3264no—
290C2×C4⋊C4—1631no—
291C2×C4⋊C4—1631no—
292C2×C4⋊C4—1631no—
293C2×C4⋊C4C2×D16311yescharacteristic
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
305#95—1631no—
306#95—1631no—
307#95—1631no—
308#95—1631no—
309#95—1631no—
310#95—1631no—
311#95—1631no—
312#95—1631no—
313C24—1631no—
314C24C22×D16311yescharacteristic
315C24C22×D16311yescharacteristic
316C24C22×D16311yescharacteristic
317#41—32632no—
318C2×Q8—3262no—
319C2×Q8—3262no—
320C2×Q8—3262no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—3262no—
330C2×D4—3262no—
331C2×D4—3262no—
332C2×D4—3262no—
333C2×D4—3262no—
334C2×D4—3262no—
335C2×D4—3262no—
336C2×D4—6521no—
337C2×D4—1634no—
338C2×D4C22⋊D16314yescomplemented
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—3262no—
343C22×C4—3262no—
344C22×C4—3262no—
345C22×C4—3262no—
346C22×C4—3262no—
347C22×C4—3262no—
348C22×C4—3262no—
349C22×C4—3262no—
350C22×C4—3262no—
351C22×C4—3262no—
352C22×C4—6521no—
353C22×C4—1631no—
354C22×C4—1631no—
355C22×C4—1631no—
356C22×C4—1634no—
357C22×C4—1631no—
358C22×C4—1631no—
359C22×C4—1631no—
360C22×C4—1631no—
361C22×C4C22×D16311yescharacteristic
362C22×C4C22×D16311yescharacteristic
363C22×C4C22×D16311yescharacteristic
364C22×C4C22×D16311yescharacteristic
365C4⋊C4—22no—
366C4⋊C4—3262no—
367C4⋊C4—3262no—
368C4⋊C4—3262no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—3262no—
372C22⋊C4—3262no—
373C22⋊C4—3262no—
374C22⋊C4—3262no—
375C22⋊C4—3262no—
376C22⋊C4—3262no—
377C22⋊C4—3262no—
378C22⋊C4—6521no—
379C22⋊C4—6521no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—3261no—
398C23—3261no—
399C23—3261no—
400C23—3261no—
401C23—3261no—
402C23—3261no—
403C23—1632no—
404C23D4×D16312yes—
405C23D4×D16312yes—
406C23C2×C22⋊D16312yes—
407C23C2×C22⋊D16312yes—
408C23C23×D16311yescharacteristic
409Q8—3268no—
410D4—28no—
411D4—44no—
412D4—44no—
413D4—3268no—
414D4—3268no—
415D4—6524no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—44no—
427C2×C4—3268no—
428C2×C4—3268no—
429C2×C4—3261no—
430C2×C4—3261no—
431C2×C4—3261no—
432C2×C4—3261no—
433C2×C4—3261no—
434C2×C4—3261no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—3268no—
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—3261no—
447C2×C4—6524no—
448C2×C4—1632no—
449C2×C4—1632no—
450C2×C4—1632no—
451C2×C4—1632no—
452C2×C4C2×C22⋊D16312yes—
453C2×C4#1457969812yescomplemented
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—3262no—
475C22—3262no—
476C22—3262no—
477C22—6521no—
478C22—6521no—
479C22—6521no—
480C22—6521no—
481C22#1910470314yes—
482C22C2×D4×D16311yescharacteristic
483C22C2×D4×D16311yescharacteristic
484C22C22×C22⋊D16311yescharacteristic
485C22#41×D16311yescharacteristic
486C22C2×#1457969811yescharacteristic
487C22#1910478011yescharacteristic
488C22#1910478211yescharacteristic
489C4—22no—
490C4—22no—
491C4—41no—
492C4—41no—
493C4—3262no—
494C4—3262no—
495C4—3262no—
496C4—3262no—
497C4—6521no—
498C4—6521no—
499C4—6521no—
500C2—24no—
501C2—42no—
502C2—81no—
503C2—6522no—
504C2C2×#1910470311yescharacteristic
505C2#2378221011yescharacteristic
506C2#2378224211yescharacteristic
507C2#2378230411yescharacteristic
508C2#2378230911yescharacteristic
509C2#2378231411yescharacteristic
510C2#2378231511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×#121×C163C2
hypercenterC2×#121D163
hypocenterC163#2430
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14579698C2semidirect
2. C2×#19104716C2semidirect
3. C2×#19104707C2semidirect
4. #23781585C2semidirect
5. #23781580C2semidirect
6. #23781577C2semidirect
7. #23781573C2semidirect
8. #23781545C2semidirect
9. #23781545C2semidirect
10. #23781544C2semidirect
11. #23781541C2semidirect
12. #23781537C2semidirect
13. #23781536C2semidirect
14. C2×C163⋊#95C2semidirect
15. C2×#121×C163C2semidirect
70. C2×C22⋊D163D4semidirect
79. C22×Dic163D4semidirect
113. C2×D4×C163D4semidirect
158. C2×Dic163#41semidirect
197. C2×C4×C163#41semidirect
255. C163#2430semidirect
260. C2×#121D163semidirect
338. C2×D4C22⋊D163semidirect
453. C2×C4#14579698semidirect

Maximal subgroups

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

subgroupindex
C22×#145796982
C2×#191047162
C2×#191047072
#237815852
#237815802
#237815772
#237815732
#237815452
#237815442
#237815412
#237815372
#237815362
C2×C163⋊#952
C2×#121×C1632
#2430163