Finite Group Databasebeta

← #34031049

Subgroups — #34031049

Order 18688.

2216 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626477C211yesmaximal, characteristic, complemented
2#22626474C212yesmaximal, complemented
3#22626463C214yesmaximal, complemented
4#22626458C214yesmaximal, complemented
5#22626454C214yesmaximal, complemented
6#22626430C214yesmaximal, complemented
7#22626420C214yesmaximal, complemented
8#22626418C214yesmaximal, complemented
9#125×#64408C211yesmaximal, characteristic, complemented
10#124×#64408C212yesmaximal, complemented
11#562×D73C211yesmaximal, characteristic
12C2×Q8×#64408C2212yes—
13C2×D4×#64408C2212yes—
14C2×D4×#64408C2214yes—
15C2×#14556670C2214yes—
16(C2×#64408)⋊Q8C2212yes—
17(C2×#64408)⋊Q8C2214yes—
18#17989539C2212yes—
19#17989539C2214yes—
20#17989538C2212yes—
21#17989538C2214yes—
22#17989538C2214yes—
23#17989537C2214yes—
24#17989537C2214yes—
25#17989537C2214yes—
26#17989537C2214yes—
27#17989537C2214yes—
28#17989537C2214yes—
29#17989536C2214yes—
30#17989536C2214yes—
31#17989536C2214yes—
32#17989536C2214yes—
33#17989536C2214yes—
34#17989535C2214yes—
35#17989535C2214yes—
36#17989535C2214yes—
37#17989534C2214yes—
38#17989534C2214yes—
39#17989534C2214yes—
40#17989531C2212yes—
41#17989531C2211yescharacteristic
42#17989529C2214yes—
43C4⋊C4×#64408C2212yes—
44C4⋊C4×#64408C2214yes—
45C4⋊C4×#64408C2214yes—
46C22⋊C4×#64408C2214yes—
47C22⋊C4×#64408C2214yes—
48C22⋊C4×#64408C2214yes—
49C4×C22⋊#64408C2214yes—
50C2×(C2×#64408)⋊C4C2214yes—
51C42×#64408C2211yescharacteristic
52#17989335C411yescharacteristic, complemented
53#17989334C414yescomplemented
54#17989333C414yescomplemented
55#17989332C414yescomplemented
56#17989331C412yescomplemented
57#125×D73C2211yescharacteristic
58#124×D73C2212yes—
59#123×D73C2214yes—
60C22⋊Q8×D73C2214yes—
61#121×D73C2214yes—
62#562×C73C411yescharacteristic, complemented
63C23×#64408C2314yes—
64C23×#64408C2314yes—
65Q8×#64408—44no—
66D4×#64408—44no—
67D4×#64408—48no—
68#14556670—48no—
69C2×C4⋊#64408C2314yes—
70C2×C4⋊#64408C2314yes—
71C2×C4⋊#64408C2312yes—
72C2×C4⋊#64408C2314yes—
73C2×C4⋊#64408C2312yes—
74C2×C4⋊#64408C2314yes—
75C2×C4⋊#64408C2312yes—
76C2×C22⋊#64408C2314yes—
77C2×C22⋊#64408C2314yes—
78C2×C22⋊#64408C2314yes—
79C2×C22⋊#64408C2314yes—
80C2×C22⋊#64408C2314yes—
81C2×C22⋊#64408C2314yes—
82C2×C4×#64408C2314yes—
83C2×C4×#64408C2312yes—
84C2×C4×#64408C2311yescharacteristic
85C2×C4×#64408C2314yes—
86C2×C4×#64408C2314yes—
87C2×C4×#64408C2311yescharacteristic
88C2×C4×#64408C2312yes—
89C2×C4×#64408C2314yes—
90(C2×#64408)⋊C4C2314yes—
91(C2×#64408)⋊C4C2314yes—
92(C2×#64408)⋊C4C2314yes—
93(C2×#64408)⋊C4C2314yes—
94(C2×#64408)⋊C4C2314yes—
95(C2×#64408)⋊C4C2314yes—
96(C2×#64408)⋊C4C2314yes—
97(C2×#64408)⋊C4C2314yes—
98(C2×#64408)⋊C4C2314yes—
99(C2×#64408)⋊C4C2314yes—
100(C2×#64408)⋊C4C2314yes—
101C2×Q8×D73C2312yes—
102C2×D4×D73C2314yes—
103C2×D4×D73C2312yes—
104C22×C4×D73C2314yes—
105#14556583C2×C411yescharacteristic, complemented
106#14556582C2×C412yescomplemented
107#14556581C2×C414yescomplemented
108#14556580C2×C412yescomplemented
109#14556579C2×C412yescomplemented
110#14556578C2×C412yescomplemented
111#14556577C2×C414yescomplemented
112#14556577C2×C414yescomplemented
113#14556576C2×C414yescomplemented
114#14556576C2×C414yescomplemented
115#14556575C2×C414yescomplemented
116#14556574C2×C414yescomplemented
117#14556573C2×C414yescomplemented
118#14556573C2×C414yescomplemented
119#14556572C2×C414yescomplemented
120#14556571C2×C414yescomplemented
121#14556570C2×C414yescomplemented
122#14556569C2×C414yescomplemented
123#14556569C2×C414yescomplemented
124#14556568C2×C414yescomplemented
125#14556568C2×C414yescomplemented
126#14556567C2×C414yescomplemented
127#14556566C2×C414yescomplemented
128#14556565C2×C414yescomplemented
129#14556564C2×C414yescomplemented
130#14556564C2×C414yescomplemented
131#14556563C2×C414yescomplemented
132#14556563C2×C414yescomplemented
133#14556562C2×C414yescomplemented
134C4⋊C4×D73C2314yes—
135C4⋊C4×D73C2314yes—
136C4⋊C4×D73C2312yes—
137C22⋊C4×D73C2314yes—
138C22⋊C4×D73C2314yes—
139C22⋊C4×D73C2314yes—
140C42×D73C2311yescharacteristic
141#125×C73C2×C411yescharacteristic, complemented
142#124×C73C2×C412yescomplemented
143#123×C73C2×C414yescomplemented
144C22⋊Q8×C73C2×C414yescomplemented
145#121×C73C2×C414yescomplemented
146C22×#64408—28no—
147C22×#64408—44no—
148C22×#64408—44no—
149C22×#64408—44no—
150C22×#64408—44no—
151C22×#64408—44no—
152C22×#64408C2414yes—
153C22×#64408C2414yes—
154C22×#64408C2411yescharacteristic
155C22×#64408C2414yes—
156C22×#64408C2412yes—
157C22×#64408C2411yescharacteristic
158C4⋊#64408—28no—
159C4⋊#64408—44no—
160C4⋊#64408—42no—
161C4⋊#64408—44no—
162C4⋊#64408—42no—
163C4⋊#64408—44no—
164C4⋊#64408—42no—
165C22⋊#64408—28no—
166C22⋊#64408—28no—
167C22⋊#64408—44no—
168C22⋊#64408—44no—
169C22⋊#64408—44no—
170C22⋊#64408—44no—
171C4×#64408—28no—
172C4×#64408—24no—
173C4×#64408—22no—
174C4×#64408—44no—
175C4×#64408—42no—
176C4×#64408—41no—
177C4×#64408—44no—
178C4×#64408—44no—
179C23×D73C2414yes—
180C2×Dic146C22×C412yes—
181C2×Dic146C22×C414yes—
182Q8×D73—42no—
183Q8×D73—42no—
184C2×D292C22×C412yes—
185C2×D292C22×C414yes—
186C2×C22⋊D73C22×C414yes—
187C2×C22⋊D73C22×C414yes—
188C2×C22⋊D73C22×C414yes—
189D4×D73—44no—
190D4×D73—42no—
191D4×D73—44no—
192D4×D73—42no—
193C22×Dic73C22×C414yes—
194C2×C4×D73—44no—
195C2×C4×D73—44no—
196C2×C4×D73C22×C414yes—
197C2×C4×D73C22×C414yes—
198C2×C4×D73C2411yescharacteristic
199C2×C4×D73C2412yes—
200C2×C4×D73C2414yes—
201C2×C4×D73C2414yes—
202C4⋊Dic73C22×C414yes—
203C4⋊Dic73C22×C414yes—
204C4⋊Dic73C22×C412yes—
205#11924007C22×C414yes—
206#11924007C22×C414yes—
207#11924007C22×C414yes—
208#11924007C22×C414yes—
209#11924007C22×C414yes—
210C22⋊Dic73C22×C414yes—
211C22⋊Dic73C22×C414yes—
212C22⋊Dic73C22×C414yes—
213#11924005C22×C414yes—
214#11924005C22×C414yes—
215#11924005C22×C414yes—
216#11924005C22×C414yes—
217#11924005C22×C414yes—
218#11924005C22×C414yes—
219C4×Dic73C22×C411yescharacteristic
220C4×Dic73C22×C412yes—
221C2×Q8×C73C22×C412yes—
222C2×D4×C73C22×C414yes—
223C2×D4×C73C22×C412yes—
224C22×C4×C73C22×C414yes—
225C4⋊C4×C73C22×C414yes—
226C4⋊C4×C73C22×C414yes—
227C4⋊C4×C73C22×C412yes—
228C22⋊C4×C73C22×C414yes—
229C22⋊C4×C73C22×C414yes—
230C22⋊C4×C73C22×C414yes—
231C42×C73C22×C411yescharacteristic
232C23×C73C23×C414yes—
233C22×D73—24no—
234C22×D73—24no—
235C22×D73—24no—
236C22×D73—24no—
237C22×D73—24no—
238C22×D73—24no—
239C22×D73C23×C414yes—
240C22×D73C2511yescharacteristic
241C2×#64408—22no—
242C2×#64408—22no—
243C2×#64408—22no—
244C2×#64408—22no—
245C2×#64408—24no—
246C2×#64408—24no—
247C2×#64408—24no—
248C2×#64408—24no—
249C2×#64408—21no—
250C2×#64408—24no—
251C2×#64408—24no—
252C2×#64408—24no—
253C2×#64408—24no—
254C2×#64408—24no—
255C2×#64408—88no—
256C2×#64408—88no—
257C2×#64408H(5,𝔽2)14yes—
258C2×#644082−1+412yes—
259Q8×C73—42no—
260D4×C73—42no—
261D4×C73—44no—
262C2×C4×C73—44no—
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C23×C412yes—
265C2×C4×C73C23×C414yes—
266C2×C4×C73C23×C414yes—
267C22⋊D73—44no—
268C22⋊D73—44no—
269C22⋊D73—44no—
270C2×Dic73—44no—
271C2×Dic73C23×C411yescharacteristic
272C2×Dic73C23×C412yes—
273C2×Dic73C23×C414yes—
274C2×Dic73C23×C414yes—
275D292—42no—
276D292—44no—
277C4×D73—24no—
278C4×D73—24no—
279C4×D73—22no—
280C4×D73—21no—
281C4×D73—24no—
282C4×D73—24no—
283C4×D73—22no—
284C4×D73—21no—
285C4×D73—44no—
286C4×D73—44no—
287Dic146—42no—
288Dic146—44no—
289C22×C73—24no—
290C22×C73—24no—
291C22×C73—24no—
292C22×C73C24×C411yescharacteristic
293C2×D73—24no—
294C2×D73—24no—
295C2×D73—24no—
296C2×D73—44no—
297C2×D73—44no—
298C2×D73—44no—
299C2×D73—44no—
300C2×D73C2×H(5,𝔽2)12yes—
301C2×D73C2×2−1+411yescharacteristic
302C2×D73#58511yescharacteristic
303C2×D73#58512yes—
304#64408—24no—
305#64408—22no—
306#64408—44no—
307#64408—44no—
308#64408—41no—
309#64408—44no—
310C4×C73—24no—
311C4×C73—22no—
312C4×C73—24no—
313C4×C73—21no—
314Dic73—24no—
315Dic73—22no—
316Dic73—24no—
317Dic73—21no—
318#34522—731nomaximal, Sylow
319C2×C73—44no—
320C2×C73C4×H(5,𝔽2)12yes—
321C2×C73C4×2−1+411yescharacteristic
322D73—44no—
323D73#354512yes—
324D73#354811yescharacteristic
325D73#355311yescharacteristic
326C2×#562—731no—
327#2382—731no—
328#2379—732no—
329#2368—734no—
330#2363—734no—
331#2359—734no—
332#2335—734no—
333#2325—734no—
334#2323—734no—
335C4×#125—731no—
336C4×#124—732no—
337C73#3452211yescharacteristic, complemented, Sylow
338#562—731no—
339#562—732no—
340#562—734no—
341#562—734no—
342#562—734no—
343#562#6440811yescharacteristic, complemented
344C2×#125—731no—
345C2×#124—732no—
346C2×#123—734no—
347C2×C22⋊Q8—734no—
348C2×#121—734no—
349C2×C4×Q8—732no—
350C2×C4×D4—732no—
351C2×C4×D4—734no—
352C2×#117—734no—
353#391—734no—
354#391—732no—
355#390—734no—
356#390—732no—
357#389—732no—
358#389—734no—
359#389—734no—
360#388—734no—
361#388—734no—
362#388—734no—
363#388—734no—
364#388—734no—
365#388—734no—
366#387—734no—
367#387—734no—
368#387—734no—
369#387—734no—
370#387—734no—
371#386—734no—
372#386—734no—
373#386—734no—
374#385—734no—
375#385—734no—
376#385—734no—
377#382—731no—
378#382—732no—
379#380—734no—
380C4×C4⋊C4—732no—
381C4×C4⋊C4—734no—
382C4×C4⋊C4—734no—
383C4×C22⋊C4—734no—
384C4×C22⋊C4—734no—
385C4×C22⋊C4—734no—
386C4×C22⋊C4—734no—
387C2×#95—734no—
388C43—731no—
389C22×Q8—732no—
390C22×D4—734no—
391C22×D4—732no—
392C23×C4—734no—
393C23×C4—734no—
394C23×C4—734no—
395#125—734no—
396#125—732no—
397#125—731no—
398#125C2×#6440811yescharacteristic, complemented
399#124—732no—
400#124—732no—
401#124—734no—
402#124—734no—
403#124—732no—
404#124C2×#6440812yescomplemented
405#123—734no—
406#123—734no—
407#123—734no—
408#123—734no—
409#123—734no—
410#123—734no—
411#123—734no—
412#123C2×#6440814yescomplemented
413C22⋊Q8—734no—
414C22⋊Q8—734no—
415C22⋊Q8—734no—
416C22⋊Q8—734no—
417C22⋊Q8—734no—
418C22⋊Q8—734no—
419C22⋊Q8—734no—
420C22⋊Q8C2×#6440814yescomplemented
421#121—734no—
422#121—734no—
423#121—734no—
424#121—734no—
425#121—734no—
426#121—734no—
427#121—734no—
428#121C2×#6440814yescomplemented
429C4×Q8—2924no—
430C4×D4—2928no—
431C4×D4—2924no—
432#117—2928no—
433C2×C4⋊C4—734no—
434C2×C4⋊C4—732no—
435C2×C4⋊C4—734no—
436C2×C4⋊C4—732no—
437C2×C4⋊C4—734no—
438C2×C4⋊C4—732no—
439C2×C4⋊C4—734no—
440C2×C4⋊C4—734no—
441C2×C4⋊C4—732no—
442C2×C4⋊C4—734no—
443C2×C22⋊C4—734no—
444C2×C22⋊C4—734no—
445C2×C22⋊C4—734no—
446C2×C22⋊C4—734no—
447C2×C22⋊C4—734no—
448C2×C22⋊C4—734no—
449C2×C22⋊C4—734no—
450C2×C22⋊C4—734no—
451C2×C22⋊C4—734no—
452C2×C42—734no—
453C2×C42—734no—
454C2×C42—731no—
455C2×C42—732no—
456C2×C42—734no—
457C2×C42—731no—
458C2×C42—734no—
459C2×C42—732no—
460C2×C42—731no—
461#95—734no—
462#95—734no—
463#95—734no—
464#95—734no—
465#95—734no—
466#95—734no—
467#95—734no—
468#95—734no—
469#95—734no—
470#95—734no—
471#95—734no—
472C24—734no—
473C2×Q8—2922no—
474C2×Q8—2922no—
475C2×Q8—734no—
476C2×Q8—732no—
477C2×Q8C22×#6440812yes—
478C2×D4—2924no—
479C2×D4—2922no—
480C2×D4—2924no—
481C2×D4—2922no—
482C2×D4—734no—
483C2×D4—734no—
484C2×D4—734no—
485C2×D4—734no—
486C2×D4—732no—
487C2×D4C22×#6440814yes—
488C2×D4C22×#6440812yes—
489C22×C4—2924no—
490C22×C4—2924no—
491C22×C4—2924no—
492C22×C4—2924no—
493C22×C4—2924no—
494C22×C4—2924no—
495C22×C4—2924no—
496C22×C4—734no—
497C22×C4—734no—
498C22×C4—734no—
499C22×C4—731no—
500C22×C4—732no—
501C22×C4—731no—
502C22×C4—732no—
503C22×C4—734no—
504C22×C4—734no—
505C22×C4—734no—
506C22×C4—734no—
507C22×C4—734no—
508C22×C4—731no—
509C22×C4—1468no—
510C22×C4C22×#6440814yes—
511C4⋊C4—2922no—
512C4⋊C4—2924no—
513C4⋊C4—2924no—
514C4⋊C4—2922no—
515C4⋊C4—2924no—
516C4⋊C4—2922no—
517C4⋊C4—734no—
518C4⋊C4—734no—
519C4⋊C4—734no—
520C4⋊C4—734no—
521C4⋊C4—734no—
522C4⋊C4—734no—
523C4⋊C4—734no—
524C4⋊C4—732no—
525C4⋊C4—1468no—
526C4⋊C4C22×#6440814yes—
527C4⋊C4C22×#6440814yes—
528C4⋊C4C22×#6440812yes—
529C22⋊C4—2924no—
530C22⋊C4—2924no—
531C22⋊C4—2924no—
532C22⋊C4—2924no—
533C22⋊C4—734no—
534C22⋊C4—734no—
535C22⋊C4—734no—
536C22⋊C4—734no—
537C22⋊C4—734no—
538C22⋊C4—734no—
539C22⋊C4—734no—
540C22⋊C4—734no—
541C22⋊C4—734no—
542C22⋊C4—1468no—
543C22⋊C4—1468no—
544C22⋊C4C22×#6440814yes—
545C22⋊C4C22×#6440814yes—
546C22⋊C4C22×#6440814yes—
547C42—2921no—
548C42—2924no—
549C42—2922no—
550C42—2924no—
551C42—2924no—
552C42—732no—
553C42—731no—
554C42—1462no—
555C42—1468no—
556C42—1464no—
557C42C22×#6440811yescharacteristic
558C23—731no—
559C23—734no—
560C23—1464no—
561C23—1464no—
562C23—1464no—
563C23—1464no—
564C23—1464no—
565C23—1464no—
566C23C23×#6440814yes—
567Q8—42no—
568Q8—2924no—
569Q8—2922no—
570D4—44no—
571D4—42no—
572D4—2924no—
573D4—2922no—
574D4—2924no—
575D4—2924no—
576D4—2924no—
577C2×C4—44no—
578C2×C4—2924no—
579C2×C4—2924no—
580C2×C4—2924no—
581C2×C4—5848no—
582C2×C4—5848no—
583C2×C4—734no—
584C2×C4—732no—
585C2×C4—731no—
586C2×C4—732no—
587C2×C4—734no—
588C2×C4—734no—
589C2×C4—1464no—
590C2×C4—1461no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4—1462no—
594C2×C4—1462no—
595C2×C4—1461no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1462no—
602C2×C4—1464no—
603C2×C4—1461no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4C23×#6440811yescharacteristic
612C2×C4C23×#6440812yes—
613C2×C4C23×#6440814yes—
614C2×C4C23×#6440814yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—2924no—
619C22—2924no—
620C22—2924no—
621C22—2924no—
622C22—732no—
623C22—732no—
624C22—731no—
625C22—731no—
626C22—1464no—
627C22—1464no—
628C22—1464no—
629C22C24×#6440811yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—21no—
634C4—2924no—
635C4—2924no—
636C4—2924no—
637C4—2921no—
638C4—1464no—
639C4—1462no—
640C4—1464no—
641C4—1464no—
642C4—1462no—
643C4—1461no—
644C2—44no—
645C2—2924no—
646C2—731no—
647C2—732no—
648C2—731no—
649C2H(5,𝔽2)×#6440812yes—
650C22−1+4×#6440811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#562×C73C4
prime radical#562×D73C2
cosocleC22×D73C25
hypercenter#562#64408
hypocenterC73#34522
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #22626477C2semidirect
2. #22626474C2semidirect
3. #22626463C2semidirect
4. #22626458C2semidirect
5. #22626454C2semidirect
6. #22626430C2semidirect
7. #22626420C2semidirect
8. #22626418C2semidirect
9. #125×#64408C2semidirect
10. #124×#64408C2semidirect
52. #17989335C4semidirect
53. #17989334C4semidirect
54. #17989333C4semidirect
55. #17989332C4semidirect
56. #17989331C4semidirect
62. #562×C73C4semidirect
105. #14556583C2×C4semidirect
106. #14556582C2×C4semidirect
107. #14556581C2×C4semidirect
108. #14556580C2×C4semidirect
109. #14556579C2×C4semidirect
110. #14556578C2×C4semidirect
111. #14556577C2×C4semidirect
112. #14556577C2×C4semidirect
113. #14556576C2×C4semidirect
114. #14556576C2×C4semidirect
115. #14556575C2×C4semidirect
116. #14556574C2×C4semidirect
117. #14556573C2×C4semidirect
118. #14556573C2×C4semidirect
119. #14556572C2×C4semidirect
120. #14556571C2×C4semidirect
121. #14556570C2×C4semidirect
122. #14556569C2×C4semidirect
123. #14556569C2×C4semidirect
124. #14556568C2×C4semidirect
125. #14556568C2×C4semidirect
126. #14556567C2×C4semidirect
127. #14556566C2×C4semidirect
128. #14556565C2×C4semidirect
129. #14556564C2×C4semidirect
130. #14556564C2×C4semidirect
131. #14556563C2×C4semidirect
132. #14556563C2×C4semidirect
133. #14556562C2×C4semidirect
141. #125×C73C2×C4semidirect
142. #124×C73C2×C4semidirect
143. #123×C73C2×C4semidirect
144. C22⋊Q8×C73C2×C4semidirect
145. #121×C73C2×C4semidirect
337. C73#34522semidirect
343. #562#64408semidirect
398. #125C2×#64408semidirect
404. #124C2×#64408semidirect
412. #123C2×#64408semidirect
420. C22⋊Q8C2×#64408semidirect
428. #121C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264772
#226264742
#226264632
#226264582
#226264542
#226264302
#226264202
#226264182
#125×#644082
#124×#644082
#562×D732
#3452273