Finite Group Databasebeta

← #33057751

Subgroups — #33057751

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620487C211yesmaximal, characteristic
2#22618873C211yesmaximal, characteristic, complemented
3#22618676C211yesmaximal, characteristic
4#22608819C211yesmaximal, characteristic, complemented
5#22608818C211yesmaximal, characteristic, complemented
6#22608773C211yesmaximal, characteristic, complemented
7#22608769C211yesmaximal, characteristic, complemented
8#22608768C211yesmaximal, characteristic, complemented
9#22608734C211yesmaximal, characteristic, complemented
10C8×#11924015C211yesmaximal, characteristic, complemented
11#334×D73C211yesmaximal, characteristic, complemented
12#22608332C211yesmaximal, characteristic, complemented
13#22608328C211yesmaximal, characteristic
14#22608210C211yesmaximal, characteristic, complemented
15C73×#1583C211yesmaximal, characteristic, complemented
16C2×#14556599—21no—
17#17989141C2211yescharacteristic
18#17989134C2211yescharacteristic
19#17988984C411yescharacteristic
20#17988958C411yescharacteristic
21#17988936C411yescharacteristic
22#17988932C411yescharacteristic
23#17988819C411yescharacteristic
24#17988817C411yescharacteristic
25#17988815C411yescharacteristic
26#17988813C411yescharacteristic
27#17988718C2211yescharacteristic
28#17988717—21no—
29#17988717C2211yescharacteristic
30#17988716—21no—
31#17988711C2211yescharacteristic
32C4×#11924017C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35#17988664C2211yescharacteristic
36#17988656—21no—
37#17988655—21no—
38C8×D292C2211yescharacteristic
39D4×#10596831C2211yescharacteristic
40#17988638C2211yescharacteristic
41#17988612—21no—
42C4⋊C8×D73C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44#17988552—21no—
45#17988541—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58C8×#10596831C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×#11924017C2×C411yescharacteristic
69C2×#11924016C2×C411yescharacteristic
70C2×#11924015C2×C411yescharacteristic
71#14556599—44no—
72C2×#11924011—21no—
73C2×#11924011D411yescharacteristic
74C2×#11924011D411yescharacteristic
75C22×#10596831—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73—21no—
79C2×C8×D73C2×C411yescharacteristic
80C2×C8×D73C2×C411yescharacteristic
81#14556590C2×C411yescharacteristic
82#14556590C2×C411yescharacteristic
83#14556589C2×C411yescharacteristic
84#14556587C2×C411yescharacteristic
85#14556581—21no—
86#14556577—21no—
87#14556571C2×C411yescharacteristic
88#14556569—21no—
89#14556567C2×C411yescharacteristic
90#14556566C2×C411yescharacteristic
91#14556564—21no—
92#14556562C2×C411yescharacteristic
93C4×Dic146—21no—
94C4×Dic146C2311yescharacteristic
95C4×D292C2311yescharacteristic
96C4×C22⋊D73—21no—
97#14556548—21no—
98C42×D73C2311yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101#14556522C2×C411yescharacteristic
102C4⋊#10596831—21no—
103C4⋊#10596831C2311yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556516C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556514C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556502—21no—
121#14556501—21no—
122#14556501D411yescharacteristic
123#14556501D411yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73C2×C411yescharacteristic
129C8×Dic73C2×C411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C2×M16×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924028—42no—
146#11924028—42no—
147C2×Dic146—21no—
148C2×Dic146—21no—
149C2×Dic146C22×C411yescharacteristic
150C2×D292—21no—
151C2×D292C22×C411yescharacteristic
152C2×C22⋊D73—21no—
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73C22×C411yescharacteristic
157C2×C4×D73C2×D411yescharacteristic
158C2×C4×D73#4111yescharacteristic
159#11924017—41no—
160#11924016—22no—
161#11924015—22no—
162#11924011—22no—
163#11924011—42no—
164#11924011SD1614yescomplemented
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—42no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—22no—
174C8×D73—41no—
175C8×D73—42no—
176C8×D73—42no—
177C8×D73—42no—
178C4⋊Dic73—21no—
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199SD16×C73—41no—
200M16×C73—42no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C22⋊C4×C73—21no—
212C42×C73—21no—
213C42×C73C2411yescharacteristic
214C23×C73—21no—
215C22×D73—21no—
216C22×D73C4×D411yescharacteristic
217Q8×C73—21no—
218Q8×C73—21no—
219D4×C73—21no—
220D4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73C23×C411yescharacteristic
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C22⋊D73—42no—
230C22⋊D73—42no—
231C22⋊D73—81no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238D292—21no—
239D292—21no—
240D292—42no—
241C4×D73—21no—
242C4×D73—21no—
243C4×D73—41no—
244C4×D73—41no—
245C4×D73—41no—
246C4×D73C2×SD1612yes—
247C4×D73#13512yes—
248Dic146—21no—
249Dic146—21no—
250Dic146—42no—
251Dic146—81no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—21no—
255C8×C73—41no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73C2×SD1612yes—
259#10596831—21no—
260#10596831—21no—
261#10596831—41no—
262#10596831—41no—
263#10596831—41no—
264#10596831#13112yescomplemented
265#10596831C2×SD1612yes—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73C2×C4×D411yescharacteristic
270C2×D73—21no—
271C2×D73—21no—
272C2×D73—41no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73C4×SD1612yes—
276C4×C73—21no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73C2×#13111yescharacteristic
281C4×C73#56811yescharacteristic
282C4×C73C22×SD1611yescharacteristic
283C4×C73C2×#13511yescharacteristic
284Dic73—21no—
285Dic73—21no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289Dic73C4×SD1612yes—
290#18067—731nomaximal, Sylow
291C2×C73—81no—
292C2×C73#293511yescharacteristic
293C2×C73C2×C4×SD1611yescharacteristic
294C2×C73#295711yescharacteristic
295D73—22no—
296D73—81no—
297#3167—731no—
298#2962—731no—
299#2935—731no—
300#1590—731no—
301#1590—731no—
302#1583—731no—
303#1583D7311yescharacteristic, complemented
304#1582—731no—
305#1582—731no—
306C8×SD16—731no—
307C8×SD16—731no—
308C2×#334—731no—
309#1481—731no—
310#1480—731no—
311C8×M16—731no—
312C73#1806711yescharacteristic, complemented, Sylow
313C2×#131—1461no—
314#533—731no—
315#532—731no—
316#498—731no—
317#492—731no—
318#487—731no—
319#486—731no—
320#465—731no—
321#465—731no—
322#465—731no—
323#465—731no—
324#446—731no—
325#446—731no—
326#446—1461no—
327#446—1461no—
328C8×Q8—731no—
329C8×Q8C2×D7311yescharacteristic
330C4×SD16—731no—
331C4×SD16—731no—
332C4×SD16—731no—
333C4×SD16C2×D7311yescharacteristic
334#436—731no—
335#436C2×D7311yescharacteristic
336#435—1461no—
337#435—1461no—
338C8×D4—731no—
339C8×D4—731no—
340#431—731no—
341#424—1461no—
342C2×C4⋊C8—731no—
343C2×C4⋊C8—731no—
344#408—1461no—
345#405—1461no—
346C4×M16—731no—
347C2×C4×C8—731no—
348#334—731no—
349#334—731no—
350#334—731no—
351#334C2×D7311yescharacteristic
352Q8⋊C8—731no—
353Q8⋊C8—731no—
354Q8⋊C8—731no—
355Q8⋊C8C2×D7311yescharacteristic
356D4⋊C8—731no—
357D4⋊C8—731no—
358D4⋊C8—731no—
359D4⋊C8C2×D7311yescharacteristic
360#322—731no—
361#322C2×D7311yescharacteristic
362C82—731no—
363C82—731no—
364C2×#41—1461no—
365C2×SD16—731no—
366C2×SD16—731no—
367C2×SD16—731no—
368C2×SD16C4×D7311yescharacteristic
369#131—2924no—
370C2×M16—21no—
371C2×M16—731no—
372C2×M16—731no—
373C2×M16—1461no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—1461no—
377C22×C8—1461no—
378C22×C8—1461no—
379C22×C8—1461no—
380C4⋊Q8—731no—
381C4⋊Q8—731no—
382C4⋊Q8—731no—
383#127—731no—
384#125—1461no—
385#124—1461no—
386C22⋊Q8—731no—
387C22⋊Q8—731no—
388C22⋊Q8—1461no—
389#121—731no—
390#121—731no—
391#121—1461no—
392C4×Q8—731no—
393C4×Q8—1461no—
394C4×Q8C22×D7311yescharacteristic
395C4×D4—731no—
396C4×D4—1461no—
397C4×D4C22×D7311yescharacteristic
398#117—1461no—
399C2×C42—731no—
400#106—731no—
401#106—731no—
402#106—731no—
403#106C4×D7311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—731no—
406C4⋊C8—731no—
407C4⋊C8—1461no—
408C4⋊C8—1461no—
409C4⋊C8—1461no—
410C4⋊C8—1461no—
411C4⋊C8—1461no—
412C4⋊C8—1461no—
413C4⋊C8C22×D7311yescharacteristic
414C4⋊C8C22×D7311yescharacteristic
415#103—731no—
416#103—731no—
417#103—731no—
418#103C4×D7311yescharacteristic
419#102—731no—
420#102—731no—
421#102—731no—
422#102C4×D7311yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426C22⋊C8—1461no—
427C22⋊C8—1461no—
428#97—731no—
429#97—731no—
430#97—1461no—
431#97—1461no—
432C4×C8—21no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—731no—
438C4×C8—1461no—
439C4×C8C22×D7311yescharacteristic
440C4×C8C22×D7311yescharacteristic
441C4×C8C22×D7311yescharacteristic
442#41—2922no—
443#41—2922no—
444C2×Q8—731no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8C2×C4×D7311yescharacteristic
448C2×D4—731no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C2×D4C2×C4×D7311yescharacteristic
454C22×C4—21no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459SD16—41no—
460SD16—2921no—
461SD16—1462no—
462SD16—1462no—
463M16—42no—
464M16—2922no—
465M16—734no—
466M16—1462no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—2921no—
471C2×C8—2922no—
472C2×C8—2922no—
473C2×C8—2922no—
474C2×C8—2922no—
475C2×C8—731no—
476C2×C8—731no—
477C2×C8—731no—
478C2×C8—731no—
479C2×C8—1462no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8C2×C4×D7311yescharacteristic
484C2×C8C2×C4×D7311yescharacteristic
485C2×C8D4×D7311yescharacteristic
486C2×C8D4×D7311yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—731no—
489C4⋊C4—731no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4C2×C4×D7311yescharacteristic
498C4⋊C4C2×C4×D7311yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1461no—
501C22⋊C4—1461no—
502C22⋊C4—1461no—
503C22⋊C4—1461no—
504C22⋊C4—1461no—
505C42—21no—
506C42—731no—
507C42—731no—
508C42—731no—
509C42—1461no—
510C42C23×D7311yescharacteristic
511C23—21no—
512C23—731no—
513C23—1461no—
514Q8—21no—
515Q8—21no—
516Q8—2922no—
517Q8—5841no—
518Q8—1461no—
519Q8—1461no—
520D4—21no—
521D4—21no—
522D4—2922no—
523D4—2922no—
524D4—2922no—
525D4—5841no—
526D4—1461no—
527D4—1461no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—731no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—731no—
540C2×C4—731no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4C22×C4×D7311yescharacteristic
547C2×C4C2×D4×D7311yescharacteristic
548C2×C4#41×D7311yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—2921no—
556C8—2921no—
557C8—2921no—
558C8—732no—
559C8—732no—
560C8—1461no—
561C8—1461no—
562C8SD16×D7312yes—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—2921no—
567C22—2921no—
568C22—2921no—
569C22—732no—
570C22—1461no—
571C22—1461no—
572C22C4×D4×D7311yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—2921no—
578C4—2921no—
579C4—2921no—
580C4—732no—
581C4—1461no—
582C4—1461no—
583C4#131×D7311yescharacteristic
584C4#1798935311yescharacteristic
585C4C2×SD16×D7311yescharacteristic
586C4#135×D7311yescharacteristic
587C2—81no—
588C2—5841no—
589C2—1462no—
590C2#2261867211yescharacteristic
591C2C4×SD16×D7311yescharacteristic
592C2#2261884511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1583C2
socleC22×C73C2×C4×D4
prime radical#17989134C22
cosocleC42×C73C24
hypercenter#1583D73
hypocenterC73#18067
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #22618873C2semidirect
4. #22608819C2semidirect
5. #22608818C2semidirect
6. #22608773C2semidirect
7. #22608769C2semidirect
8. #22608768C2semidirect
9. #22608734C2semidirect
10. C8×#11924015C2semidirect
11. #334×D73C2semidirect
12. #22608332C2semidirect
14. #22608210C2semidirect
15. C73×#1583C2semidirect
164. #11924011SD16semidirect
264. #10596831#131semidirect
303. #1583D73semidirect
312. C73#18067semidirect

Maximal subgroups

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

subgroupindex
#226204872
#226188732
#226186762
#226088192
#226088182
#226087732
#226087692
#226087682
#226087342
C8×#119240152
#334×D732
#226083322
#226083282
#226082102
C73×#15832
#1806773