Finite Group Databasebeta

← #35217874

Subgroups — #35217874

Order 20864.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579698C211yesmaximal, characteristic, complemented
2C2×#19104712C211yesmaximal, characteristic, complemented
3C2×#19104707C211yesmaximal, characteristic, complemented
4C2×D4×Dic163C211yesmaximal, characteristic, complemented
5#23781581C211yesmaximal, characteristic, complemented
6#23781579C211yesmaximal, characteristic, complemented
7#23781573C211yesmaximal, characteristic, complemented
8#23781569C211yesmaximal, characteristic, complemented
9#23781565C211yesmaximal, characteristic, complemented
10#23781545C211yesmaximal, characteristic, complemented
11#23781541C211yesmaximal, characteristic, complemented
12#23781540C211yesmaximal, characteristic, complemented
13#23781539C211yesmaximal, characteristic, complemented
14#23781523C211yesmaximal, characteristic, complemented
15#392×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#19104712—24no—
28#19104707—24no—
29D4×Dic163—24no—
30C2×C4⋊Dic163C2211yescharacteristic
31C2×C4⋊Dic163C2211yescharacteristic
32C2×Dic163⋊C4C2211yescharacteristic
33C2×Dic163⋊C4C2211yescharacteristic
34C2×Dic163⋊C4C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×C22⋊Dic163C2211yescharacteristic
39C2×C22⋊Dic163C2211yescharacteristic
40C2×C22⋊Dic163C2211yescharacteristic
41C2×(C2×D163)⋊C4C2211yescharacteristic
42C2×(C2×D163)⋊C4C2211yescharacteristic
43C2×(C2×D163)⋊C4C2211yescharacteristic
44C2×C4×Dic163C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C163⋊#95C2211yescharacteristic
49C163⋊#95C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51C22×D4×C163C2211yescharacteristic
52C22×D4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C2×C22⋊C4×C163C2211yescharacteristic
56#95×C163C2211yescharacteristic
57C23×D163C2311yescharacteristic
58#14579698—232no—
59C2×Dic326—22no—
60C2×Dic326—22no—
61C2×Dic326D414yescomplemented
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—22no—
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—
90C4⋊Dic163—22no—
91C4⋊Dic163—22no—
92Dic163⋊C4—22no—
93Dic163⋊C4—22no—
94Dic163⋊C4—22no—
95C22⋊Dic163—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163—22no—
98C22⋊Dic163—22no—
99C22⋊Dic163—41no—
100C22⋊Dic163—41no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4—41no—
104C4×Dic163—22no—
105C24×C163C2311yescharacteristic
106C24×C163C2311yescharacteristic
107C24×C163C2311yescharacteristic
108C2×D4×C163—22no—
109C2×D4×C163—22no—
110C2×D4×C163—22no—
111C2×D4×C163—22no—
112C2×D4×C163—22no—
113C2×D4×C163—22no—
114C2×D4×C163—22no—
115C2×D4×C163—22no—
116C2×D4×C163—41no—
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C22⋊C4×C163—22no—
122C22⋊C4×C163—22no—
123C22⋊C4×C163—41no—
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—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—28no—
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—44no—
154C2×Dic163C2×D412yes—
155C2×Dic163C2×D412yes—
156C2×Dic163C2×D412yes—
157C2×Dic163C2×D412yes—
158C2×Dic163C2×D412yes—
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×C163—21no—
178C23×C163C2×D412yes—
179C23×C163C2×D412yes—
180C23×C163#4112yescomplemented
181C23×C163C2411yescharacteristic
182D4×C163—28no—
183D4×C163—28no—
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×C163C2×D412yes—
196C2×C4×C163#4112yescomplemented
197C22×C163—22no—
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—41no—
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×C163C22×D411yescharacteristic
219C22×C163C22×D411yescharacteristic
220C22×C163C22×D411yescharacteristic
221C22×C163C2×#4111yescharacteristic
222C22×C163C2×#4111yescharacteristic
223C22×C163H(5,𝔽2)11yescharacteristic
224C22×C1632−1+411yescharacteristic
225C2×D163—22no—
226C2×D163—22no—
227C2×D163—22no—
228C2×D163—41no—
229C2×D163—41no—
230C2×D163—41no—
231C2×D163—41no—
232C4×C163—22no—
233C4×C163—22no—
234C4×C163—41no—
235C4×C163—41no—
236Dic163—22no—
237Dic163—22no—
238Dic163—41no—
239Dic163—41no—
240Dic163—41no—
241Dic163—41no—
242Dic163#12014yes—
243C2×C163—42no—
244C2×C163—42no—
245C2×C163—42no—
246C2×C163C2×#12011yescharacteristic
247C2×C163#52511yescharacteristic
248C2×C163#53711yescharacteristic
249C2×C163#54611yescharacteristic
250C2×C163#54711yescharacteristic
251C2×C163#54811yescharacteristic
252C2×C163#55011yescharacteristic
253D163—42no—
254C163#240711yescharacteristic, complemented, Sylow
255#2407—1631nomaximal, Sylow
256C22×#41—1631no—
257C2×C22⋊Q8—1631no—
258C2×#121—1631no—
259C2×C4×D4—1631no—
260#394—1631no—
261#394—1631no—
262#392—1631no—
263#392D16311yescharacteristic, complemented
264#391—1631no—
265#390—1631no—
266#386—1631no—
267#386—1631no—
268#385—1631no—
269#385—1631no—
270#380—1631no—
271C2×#41—3264no—
272C2×#41—3264no—
273C2×#41—3264no—
274C22×Q8—1631no—
275C22×D4—1631no—
276C22×D4—1631no—
277C22×D4—1631no—
278C22×D4C2×D16311yescharacteristic
279C22×D4C2×D16311yescharacteristic
280C22×D4C2×D16311yescharacteristic
281C23×C4—1631no—
282C23×C4—1631no—
283C23×C4—1631no—
284C23×C4—1631no—
285C22⋊Q8—3264no—
286#121—3264no—
287C4×D4—3264no—
288C2×C4⋊C4—1631no—
289C2×C4⋊C4—1631no—
290C2×C4⋊C4—1631no—
291C2×C4⋊C4—1631no—
292C2×C4⋊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⋊C4C2×D16311yescharacteristic
303C2×C22⋊C4C2×D16311yescharacteristic
304C2×C22⋊C4C2×D16311yescharacteristic
305C2×C42—1631no—
306#95—1631no—
307#95—1631no—
308#95—1631no—
309#95—1631no—
310#95—1631no—
311#95C2×D16311yescharacteristic
312C24—1631no—
313C24C22×D16311yescharacteristic
314C24C22×D16311yescharacteristic
315C24C22×D16311yescharacteristic
316#41—32632no—
317C2×Q8—3262no—
318C2×Q8—3262no—
319C2×Q8—1634no—
320C2×D4—22no—
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—
338C22×C4—3262no—
339C22×C4—3262no—
340C22×C4—3262no—
341C22×C4—3262no—
342C22×C4—3262no—
343C22×C4—3262no—
344C22×C4—3262no—
345C22×C4—3262no—
346C22×C4—3262no—
347C22×C4—3262no—
348C22×C4—3262no—
349C22×C4—1631no—
350C22×C4—1631no—
351C22×C4—1631no—
352C22×C4—1631no—
353C22×C4—1634no—
354C22×C4—1631no—
355C22×C4—1631no—
356C22×C4—1631no—
357C22×C4C22×D16311yescharacteristic
358C22×C4C22×D16311yescharacteristic
359C22×C4C22×D16311yescharacteristic
360C22×C4C22×D16311yescharacteristic
361C4⋊C4—3262no—
362C4⋊C4—3262no—
363C4⋊C4—3262no—
364C4⋊C4—3262no—
365C4⋊C4—3262no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—3262no—
370C22⋊C4—3262no—
371C22⋊C4—3262no—
372C22⋊C4—3262no—
373C22⋊C4—3262no—
374C22⋊C4—3262no—
375C22⋊C4—6521no—
376C22⋊C4—6521no—
377C22⋊C4—6521no—
378C42—3262no—
379C23—21no—
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—
406C23#1457969812yescomplemented
407C23C23×D16311yescharacteristic
408Q8—3268no—
409D4—28no—
410D4—28no—
411D4—44no—
412D4—3268no—
413D4—3268no—
414D4—6524no—
415C2×C4—21no—
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—3268no—
426C2×C4—3268no—
427C2×C4—3261no—
428C2×C4—3261no—
429C2×C4—3261no—
430C2×C4—3261no—
431C2×C4—3268no—
432C2×C4—3261no—
433C2×C4—3261no—
434C2×C4—3261no—
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—6524no—
445C2×C4—1632no—
446C2×C4—1632no—
447C2×C4—1632no—
448C2×C4—1632no—
449C2×C4—1632no—
450C2×C4D4×D16312yes—
451C2×C4#1457969812yescomplemented
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—3262no—
474C22—3262no—
475C22—3262no—
476C22—6521no—
477C22—6521no—
478C22—6521no—
479C22—6521no—
480C22C2×D4×D16311yescharacteristic
481C22C2×D4×D16311yescharacteristic
482C22C2×D4×D16311yescharacteristic
483C22C2×#1457969811yescharacteristic
484C22C2×#1457969811yescharacteristic
485C22#1910478011yescharacteristic
486C22#1910478211yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—3262no—
492C4—3262no—
493C4—6521no—
494C4—6521no—
495C4—6521no—
496C4—6521no—
497C4—1634no—
498C2—42no—
499C2—42no—
500C2—42no—
501C2—6522no—
502C2#120×D16311yescharacteristic
503C2#2378217111yescharacteristic
504C2#2378224211yescharacteristic
505C2#2378230211yescharacteristic
506C2#2378231511yescharacteristic
507C2#2378232111yescharacteristic
508C2#2378233511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#392×C163C2
hypercenter#392D163
hypocenterC163#2407
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#14579698C2semidirect
2. C2×#19104712C2semidirect
3. C2×#19104707C2semidirect
4. C2×D4×Dic163C2semidirect
5. #23781581C2semidirect
6. #23781579C2semidirect
7. #23781573C2semidirect
8. #23781569C2semidirect
9. #23781565C2semidirect
10. #23781545C2semidirect
11. #23781541C2semidirect
12. #23781540C2semidirect
13. #23781539C2semidirect
14. #23781523C2semidirect
15. #392×C163C2semidirect
61. C2×Dic326D4semidirect
70. C2×C22⋊D163D4semidirect
79. C22×Dic163D4semidirect
180. C23×C163#41semidirect
196. C2×C4×C163#41semidirect
254. C163#2407semidirect
263. #392D163semidirect
406. C23#14579698semidirect
451. 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×#191047122
C2×#191047072
C2×D4×Dic1632
#237815812
#237815792
#237815732
#237815692
#237815652
#237815452
#237815412
#237815402
#237815392
#237815232
#392×C1632
#2407163