Finite Group Databasebeta

← #10769080

Subgroups — #10769080

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3459C211yesmaximal, characteristic
2#82934C212yesmaximal, complemented
3#82392C211yesmaximal, characteristic, complemented
4#71893C211yesmaximal, characteristic, complemented
5#71890C211yesmaximal, characteristic, complemented
6#71889C212yesmaximal, complemented
7#71886C212yesmaximal, complemented
8#71846C211yesmaximal, characteristic, complemented
9#71843C211yesmaximal, characteristic, complemented
10#71710C212yesmaximal, complemented
11#71600C211yesmaximal, characteristic, complemented
12#18291—31nomaximal, Sylow
13C3×#543C2211yescharacteristic
14C3×#539—22no—
15C3×#539C2212yes—
16C3×#538—22no—
17C3×#538—21no—
18C3×#534—21no—
19C3×#534C2211yescharacteristic
20C3×#529—22no—
21C3×#529—21no—
22C3×#525—22no—
23C3×#525C2212yes—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×#924C2211yescharacteristic, complemented
27#5239—22no—
28#5222—21no—
29#5211C2211yescharacteristic, complemented
30#5210—21no—
31C2×#830C2212yescomplemented
32#5051—22no—
33#5050C2212yescomplemented
34#5043C2212yescomplemented
35#5041—22no—
36#5040C2212yes—
37#5033C2212yes—
38#4996—22no—
39#4981C2211yescharacteristic, complemented
40#4979C2211yescharacteristic, complemented
41#4953C2211yescharacteristic
42#4952—21no—
43#4947C2211yescharacteristic
44#4933—21no—
45#4538C2212yescomplemented
46#4538C2212yescomplemented
47#4537C2212yes—
48#4537C2212yes—
49#4528C2212yescomplemented
50#4525C2211yescharacteristic
51#4525C2211yescharacteristic
52#3459D311yescharacteristic
53#3190—32no—
54#2638—31no—
55#1609—32no—
56#1608—31no—
57#1607—31no—
58#1603—32no—
59#1602—31no—
60#1601—31no—
61#1516—32no—
62#1459—31no—
63C2×C3×#41D412yes—
64C22×C3×D4—21no—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C3×#126—22no—
71C3×#126—22no—
72C3×#126—22no—
73C3×#125—21no—
74C3×#125—22no—
75C3×#124—22no—
76C3×#124—21no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—22no—
82C3×#123—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8D412yes—
88C3×C22⋊Q8C2312yes—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121D412yes—
94C3×#121C2312yes—
95C3×#120—22no—
96C3×#120—22no—
97C3×#120—22no—
98C3×C4×Q8—21no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—21no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×#117—22no—
108C3×#117D412yes—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42D411yescharacteristic
117C2×#217—21no—
118C2×#217—21no—
119C2×#217—21no—
120C2×#217—21no—
121C2×#216—22no—
122C2×#214—22no—
123C2×C4⋊Dic3C2311yescharacteristic
124#924—22no—
125#924—22no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C22×C3⋊C8—22no—
130#835—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95C2311yescharacteristic
138C3⋊#95C2311yescharacteristic
139#830—22no—
140#830C2312yes—
141#830C2312yes—
142#810—22no—
143#810—22no—
144#808—22no—
145#807—22no—
146#543C2×D311yescharacteristic
147#539—22no—
148#539C2×D312yes—
149#538—22no—
150#538—21no—
151#534—21no—
152#534C2×D311yescharacteristic
153#529—22no—
154#529—21no—
155#525—22no—
156#525C2×D312yes—
157C2×C22⋊Q8—31no—
158C2×C4×D4—21no—
159C2×C4×D4C2×D311yescharacteristic
160#485—62no—
161#483—32no—
162#480—32no—
163#467—62no—
164#465—62no—
165#452—62no—
166#451—32no—
167#448—32no—
168C2×C22⋊C8—32no—
169#398—31no—
170#397—31no—
171#397—61no—
172#395—31no—
173#394—61no—
174#393—31no—
175#387—61no—
176#385—61no—
177#381—31no—
178#328—32no—
179#328—32no—
180#327—32no—
181#327—32no—
182#324—32no—
183#323—31no—
184#323—31no—
185C24×C3—21no—
186C24×C3C2×D411yescharacteristic
187C24×C3C2×D411yescharacteristic
188C3×#41—82no—
189C2×C3×Q8—21no—
190C2×C3×Q8C2×D412yes—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—41no—
199C2×C3×D4—42no—
200C2×C3×D4—42no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—81no—
210C22×C3×C4—81no—
211C22×C3×C4—82no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—22no—
224C2×Dic6—81no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—21no—
251C3×C42—21no—
252#217—42no—
253#217—42no—
254#217—81no—
255#217—81no—
256#216—82no—
257#214—82no—
258C4⋊Dic3—22no—
259Dic3⋊C4—42no—
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262Dic3⋊C4—81no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—82no—
266C2×#41#6712yes—
267C22×Q8—31no—
268C22×D4—21no—
269C22×D4#6711yescharacteristic
270C23×C4—21no—
271C23×C4#6711yescharacteristic
272C23×C4C22×D311yescharacteristic
273C2×Q16—62no—
274C2×SD16—62no—
275C22×C8—62no—
276#126—22no—
277#126—22no—
278#126—22no—
279#125—21no—
280#125—22no—
281#124—22no—
282#124—21no—
283#123—22no—
284#123—22no—
285#123—22no—
286#123—22no—
287#123—22no—
288#123—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—21no—
292C22⋊Q8—22no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8#6712yes—
296C22⋊Q8C22×D312yes—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—21no—
301#121#6712yes—
302#121C22×D312yes—
303#120—22no—
304#120—22no—
305#120—22no—
306C4×Q8—21no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—21no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315#117—22no—
316#117#6712yes—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4#6711yescharacteristic
330C2×C22⋊C4C22×D311yescharacteristic
331C2×C42—21no—
332C2×C42#6711yescharacteristic
333#107—62no—
334#106—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338#102—62no—
339C22⋊C8—32no—
340C22⋊C8—32no—
341C22⋊C8—62no—
342#95—31no—
343#95—31no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—42no—
359C23×C3—81no—
360C23×C3—81no—
361C23×C3#12012yes—
362C23×C3#13512yes—
363C23×C3C22×D411yescharacteristic
364C3×Q8—42no—
365C3×Q8—81no—
366C3×D4—42no—
367C3×D4—44no—
368C3×D4—44no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—82no—
372C3×D4—82no—
373C3×D4—81no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—44no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—81no—
392C2×C3×C4—81no—
393C2×C3×C4—82no—
394C2×C3×C4—81no—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4C22×D412yes—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410Dic6—42no—
411Dic6—42no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24—21no—
415C24C2×#6711yescharacteristic
416C24C2×#6711yescharacteristic
417#41—82no—
418C2×Q8—21no—
419C2×Q8—62no—
420C2×Q8—241no—
421C2×Q8C2×#6712yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×#6712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—82no—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6712yes—
452C22×C4C2×#6712yes—
453C22×C4C23×D311yescharacteristic
454Q16—242no—
455SD16—242no—
456C2×C8—62no—
457C2×C8—62no—
458C2×C8—242no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—62no—
469C4⋊C4—122no—
470C4⋊C4—121no—
471C4⋊C4—121no—
472C4⋊C4—241no—
473C4⋊C4C2×#6712yes—
474C4⋊C4C2×#6712yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—241no—
490C22⋊C4—241no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—84no—
511C22×C3C2×#12011yescharacteristic
512C22×C3C2×#13511yescharacteristic
513C22×C3#57511yescharacteristic
514C3×C4—22no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—42no—
523Dic3—81no—
524Dic3—81no—
525Dic3—81no—
526Dic3—81no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—81no—
538C23—81no—
539C23#95012yes—
540C23#95312yes—
541C23C22×#6711yescharacteristic
542Q8—42no—
543Q8—81no—
544Q8—122no—
545Q8—122no—
546D4—42no—
547D4—44no—
548D4—44no—
549D4—42no—
550D4—42no—
551D4—82no—
552D4—82no—
553D4—81no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4#95311yescharacteristic
588C2×C4#95311yescharacteristic
589C2×C4C22×#6712yes—
590C8—122no—
591C8—122no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—42no—
595C2×C3—42no—
596C2×C3—82no—
597C2×C3#300211yescharacteristic
598C2×C3#302211yescharacteristic
599C2×C3#302611yescharacteristic
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—84no—
616C22C2×#95011yescharacteristic
617C22#582211yescharacteristic
618C22C2×#95311yescharacteristic
619C4—22no—
620C4—21no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4—241no—
632C3#1829111yescharacteristic, complemented, Sylow
633C2—21no—
634C2—21no—
635C2—42no—
636C2—42no—
637C2—82no—
638C2#8408011yescharacteristic
639C2#8411011yescharacteristic
640C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3459C2
socleC22×C3C2×#120
hypercenter#3459D3
hypocenterC3#18291
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3459

Complements

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

NH ≅ G/Ntype
2. #82934C2semidirect
3. #82392C2semidirect
4. #71893C2semidirect
5. #71890C2semidirect
6. #71889C2semidirect
7. #71886C2semidirect
8. #71846C2semidirect
9. #71843C2semidirect
10. #71710C2semidirect
11. #71600C2semidirect
26. C2×#924C22semidirect
29. #5211C22semidirect
31. C2×#830C22semidirect
33. #5050C22semidirect
34. #5043C22semidirect
39. #4981C22semidirect
40. #4979C22semidirect
45. #4538C22semidirect
46. #4538C22semidirect
49. #4528C22semidirect
632. C3#18291semidirect

Maximal subgroups

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

subgroupindex
C3×#34592
#829342
#823922
#718932
#718902
#718892
#718862
#718462
#718432
#717102
#716002
#182913