Finite Group Databasebeta

← #33056502

Subgroups — #33056502

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620487C211yesmaximal, characteristic
2#22618860C211yesmaximal, characteristic, complemented
3#22618638C211yesmaximal, characteristic
4#22608824C211yesmaximal, characteristic, complemented
5#22608822C211yesmaximal, characteristic, complemented
6#22608820C211yesmaximal, characteristic, complemented
7#22608818C211yesmaximal, characteristic, complemented
8C8×#11924017C211yesmaximal, characteristic, complemented
9C8×#11924015C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608685C211yesmaximal, characteristic, complemented
12#22608384C211yesmaximal, characteristic, complemented
13#22608380C211yesmaximal, characteristic
14C82×D73C211yesmaximal, characteristic, complemented
15C8×SD16×C73C211yesmaximal, characteristic, complemented
16C2×#14556597—21no—
17#17989141C2211yescharacteristic
18#17989134C2211yescharacteristic
19#17988984C411yescharacteristic
20#17988958C411yescharacteristic
21#17988936C411yescharacteristic
22#17988932C411yescharacteristic
23#17988819C411yescharacteristic
24#17988817C411yescharacteristic
25#17988815C411yescharacteristic
26#17988813C411yescharacteristic
27#17988718C2211yescharacteristic
28#17988716—21no—
29#17988714C2211yescharacteristic
30C8×Dic146—21no—
31C8×Dic146C2211yescharacteristic
32C4×#11924017C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35#17988665C2211yescharacteristic
36#17988664C2211yescharacteristic
37#17988662—21no—
38C8×D292C2211yescharacteristic
39C8×C22⋊D73—21no—
40#17988612—21no—
41#17988609C2211yescharacteristic
42#17988609C2211yescharacteristic
43#17988552—21no—
44#17988541—21no—
45C4×C8×D73C2211yescharacteristic
46C4×C8×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57C8×#10596831C2211yescharacteristic
58C8×#10596831C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×#11924017C2×C411yescharacteristic
69C2×#11924016C2×C411yescharacteristic
70C2×#11924015C2×C411yescharacteristic
71#14556597—44no—
72C2×#11924012—21no—
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73C2×C411yescharacteristic
79C2×C8×D73C2×C411yescharacteristic
80C2×C8×D73D411yescharacteristic
81C2×C8×D73D411yescharacteristic
82#14556590C2×C411yescharacteristic
83#14556590C2×C411yescharacteristic
84#14556589C2×C411yescharacteristic
85#14556587C2×C411yescharacteristic
86#14556581—21no—
87#14556577—21no—
88#14556571C2×C411yescharacteristic
89#14556569—21no—
90#14556567C2×C411yescharacteristic
91#14556566C2×C411yescharacteristic
92#14556564—21no—
93#14556562C2×C411yescharacteristic
94C4×Dic146—21no—
95C4×Dic146C2311yescharacteristic
96C4×D292C2311yescharacteristic
97C4×C22⋊D73—21no—
98#14556548—21no—
99C42×D73C2311yescharacteristic
100#14556523C2×C411yescharacteristic
101#14556522C2×C411yescharacteristic
102#14556522C2×C411yescharacteristic
103C4⋊#10596831—21no—
104C4⋊#10596831C2311yescharacteristic
105C4⋊#10596831C2311yescharacteristic
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#14556502—21no—
120#14556501—21no—
121C4×#10596831C2311yescharacteristic
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C8×Dic73—21no—
125C8×Dic73—21no—
126C8×Dic73C2×C411yescharacteristic
127C8×Dic73C2×C411yescharacteristic
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C22×C8×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—22no—
160#11924016—22no—
161#11924015—22no—
162#11924012—42no—
163#11924011—42no—
164#11924011—42no—
165#11924011—42no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—24no—
174C8×D73—22no—
175C8×D73—22no—
176C8×D73—22no—
177C8×D73—42no—
178C8×D73SD1614yescomplemented
179C4⋊Dic73—21no—
180C4⋊Dic73C22×C411yescharacteristic
181C4⋊Dic73C22×C411yescharacteristic
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188C22⋊Dic73—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200SD16×C73—22no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—42no—
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C4⋊C4×C73—21no—
210C4⋊C4×C73C22×C411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C22⋊C4×C73—21no—
213C42×C73—21no—
214C42×C73C2411yescharacteristic
215C23×C73—21no—
216C22×D73—21no—
217C22×D73C4×D411yescharacteristic
218Q8×C73—21no—
219Q8×C73—21no—
220D4×C73—21no—
221D4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C23×C411yescharacteristic
228C2×C4×C73C22×D411yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C22⋊D73—42no—
231C22⋊D73—42no—
232C22⋊D73—42no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C4×D411yescharacteristic
239D292—21no—
240D292—21no—
241D292—42no—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—22no—
245C4×D73—41no—
246C4×D73—41no—
247C4×D73C2×SD1612yes—
248C4×D73#13512yes—
249Dic146—21no—
250Dic146—21no—
251Dic146—42no—
252Dic146—42no—
253C8×C73—22no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73#13112yescomplemented
259C8×C73C2×SD1612yes—
260#10596831—22no—
261#10596831—21no—
262#10596831—21no—
263#10596831—41no—
264#10596831—41no—
265#10596831#13112yescomplemented
266#10596831C2×SD1612yes—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73C2×C4×D411yescharacteristic
271C2×D73—21no—
272C2×D73—21no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73C4×SD1612yes—
277C4×C73—21no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73C2×#13111yescharacteristic
282C4×C73C2×#13111yescharacteristic
283C4×C73C22×SD1611yescharacteristic
284C4×C73C2×#13511yescharacteristic
285Dic73—21no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289Dic73C4×SD1612yes—
290Dic73#44312yes—
291#17957—731nomaximal, Sylow
292C2×C73—81no—
293C2×C73#293111yescharacteristic
294C2×C73C2×C4×SD1611yescharacteristic
295C2×C73C2×#44311yescharacteristic
296D73—22no—
297D73—81no—
298#3167—731no—
299#2960—731no—
300#2931—731no—
301#1590—731no—
302#1590—731no—
303#1590—731no—
304#1590—731no—
305C8×SD16—731no—
306C8×SD16—731no—
307C8×SD16—731no—
308C8×SD16D7311yescharacteristic, complemented
309#1566—731no—
310#1493—731no—
311#1492—731no—
312C2×C82—731no—
313C73#1795711yescharacteristic, complemented, Sylow
314C2×#131—1461no—
315#533—731no—
316#532—731no—
317#498—731no—
318#492—731no—
319#487—731no—
320#486—731no—
321#465—731no—
322#465—731no—
323#465—731no—
324#465—731no—
325#446—731no—
326#446—731no—
327#446—1461no—
328C8×Q8—731no—
329C8×Q8—1461no—
330C8×Q8C2×D7311yescharacteristic
331C4×SD16—731no—
332C4×SD16—731no—
333C4×SD16—731no—
334C4×SD16C2×D7311yescharacteristic
335#436—731no—
336#436—731no—
337#436—1461no—
338C8×D4—731no—
339C8×D4—1461no—
340C8×D4C2×D7311yescharacteristic
341#424—1461no—
342#423—731no—
343#423—731no—
344#408—1461no—
345#405—1461no—
346C2×C4×C8—731no—
347C2×C4×C8—731no—
348C2×C4×C8—731no—
349#334—731no—
350#334—731no—
351#334—731no—
352#334C2×D7311yescharacteristic
353Q8⋊C8—731no—
354Q8⋊C8—731no—
355Q8⋊C8—731no—
356Q8⋊C8C2×D7311yescharacteristic
357D4⋊C8—731no—
358D4⋊C8—731no—
359D4⋊C8—731no—
360D4⋊C8C2×D7311yescharacteristic
361C82—731no—
362C82—731no—
363C82—731no—
364C82C2×D7311yescharacteristic
365C2×#41—1461no—
366C2×SD16—731no—
367C2×SD16—731no—
368C2×SD16—731no—
369C2×SD16C4×D7311yescharacteristic
370#131—2924no—
371C2×M16—1461no—
372C2×M16—1461no—
373C2×M16—1461no—
374C2×M16—1461no—
375C22×C8—21no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—1461no—
381C22×C8—1461no—
382C4⋊Q8—731no—
383C4⋊Q8—731no—
384C4⋊Q8—731no—
385#127—731no—
386#125—1461no—
387#124—1461no—
388C22⋊Q8—731no—
389C22⋊Q8—731no—
390C22⋊Q8—1461no—
391#121—731no—
392#121—731no—
393#121—1461no—
394C4×Q8—731no—
395C4×Q8—1461no—
396C4×Q8C22×D7311yescharacteristic
397C4×D4—731no—
398C4×D4—1461no—
399C4×D4C22×D7311yescharacteristic
400#117—1461no—
401C2×C42—731no—
402#106—731no—
403#106—731no—
404#106—731no—
405#106C4×D7311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—731no—
408C4⋊C8—731no—
409C4⋊C8—1461no—
410C4⋊C8—1461no—
411C4⋊C8—1461no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8C22×D7311yescharacteristic
415C4⋊C8C22×D7311yescharacteristic
416#103—731no—
417#103—731no—
418#103—731no—
419#103C4×D7311yescharacteristic
420#102—731no—
421#102—731no—
422#102—731no—
423#102C4×D7311yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—1461no—
426C22⋊C8—1461no—
427C22⋊C8—1461no—
428#97—1461no—
429#97—1461no—
430C4×C8—21no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—731no—
438C4×C8—1461no—
439C4×C8—1461no—
440C4×C8C22×D7311yescharacteristic
441C4×C8C22×D7311yescharacteristic
442C4×C8C22×D7311yescharacteristic
443#41—2922no—
444#41—2922no—
445C2×Q8—731no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8C2×C4×D7311yescharacteristic
449C2×D4—731no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4C2×C4×D7311yescharacteristic
455C22×C4—21no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—1461no—
460SD16—22no—
461SD16—1462no—
462SD16—1462no—
463SD16—1462no—
464M16—2922no—
465M16—2922no—
466M16—2922no—
467M16—2922no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—42no—
472C2×C8—2922no—
473C2×C8—731no—
474C2×C8—731no—
475C2×C8—734no—
476C2×C8—731no—
477C2×C8—731no—
478C2×C8—1462no—
479C2×C8—1462no—
480C2×C8—1464no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1462no—
485C2×C8C2×C4×D7311yescharacteristic
486C2×C8C2×C4×D7311yescharacteristic
487C2×C8D4×D7311yescharacteristic
488C2×C8D4×D7311yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—731no—
491C4⋊C4—731no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4C2×C4×D7311yescharacteristic
500C4⋊C4C2×C4×D7311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1461no—
503C22⋊C4—1461no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C42—21no—
508C42—731no—
509C42—731no—
510C42—731no—
511C42—1461no—
512C42C23×D7311yescharacteristic
513C23—21no—
514C23—731no—
515C23—1461no—
516Q8—21no—
517Q8—21no—
518Q8—2922no—
519Q8—2922no—
520Q8—1461no—
521Q8—1461no—
522D4—21no—
523D4—21no—
524D4—2922no—
525D4—2922no—
526D4—2922no—
527D4—2922no—
528D4—1461no—
529D4—1461no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—731no—
538C2×C4—731no—
539C2×C4—732no—
540C2×C4—732no—
541C2×C4—731no—
542C2×C4—1462no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4C22×C4×D7311yescharacteristic
549C2×C4C2×D4×D7311yescharacteristic
550C2×C4#41×D7311yescharacteristic
551C8—22no—
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—2921no—
557C8—2921no—
558C8—732no—
559C8—732no—
560C8—1462no—
561C8—1461no—
562C8—1461no—
563C8#1455659712yescomplemented
564C8SD16×D7312yes—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—2921no—
569C22—2921no—
570C22—2921no—
571C22—732no—
572C22—1461no—
573C22—1461no—
574C22C4×D4×D7311yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—2921no—
580C4—2921no—
581C4—2921no—
582C4—732no—
583C4—732no—
584C4—1461no—
585C4#131×D7311yescharacteristic
586C4C2×#1455659711yescharacteristic
587C4C2×SD16×D7311yescharacteristic
588C4#135×D7311yescharacteristic
589C2—81no—
590C2—5841no—
591C2—1462no—
592C2#2261863811yescharacteristic
593C2C4×SD16×D7311yescharacteristic
594C2#443×D7311yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×SD16×C73C2
socleC22×C73C2×C4×D4
prime radical#17989134C22
cosocleC42×C73C24
hypercenterC8×SD16D73
hypocenterC73#17957
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #22618860C2semidirect
4. #22608824C2semidirect
5. #22608822C2semidirect
6. #22608820C2semidirect
7. #22608818C2semidirect
8. C8×#11924017C2semidirect
9. C8×#11924015C2semidirect
10. C8×#11924016C2semidirect
11. #22608685C2semidirect
12. #22608384C2semidirect
14. C82×D73C2semidirect
15. C8×SD16×C73C2semidirect
178. C8×D73SD16semidirect
258. C8×C73#131semidirect
265. #10596831#131semidirect
308. C8×SD16D73semidirect
313. C73#17957semidirect
563. C8#14556597semidirect

Maximal subgroups

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

subgroupindex
#226204872
#226188602
#226186382
#226088242
#226088222
#226088202
#226088182
C8×#119240172
C8×#119240152
C8×#119240162
#226086852
#226083842
#226083802
C82×D732
C8×SD16×C732
#1795773