Finite Group Databasebeta

← #10769331

Subgroups — #10769331

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#539C211yesmaximal, characteristic
2#82742C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic
4#82181C211yesmaximal, characteristic, complemented
5#71889C214yesmaximal, complemented
6#71886C214yesmaximal, complemented
7#71688C211yesmaximal, characteristic, complemented
8#71673C211yesmaximal, characteristic
9#71552C211yesmaximal, characteristic, complemented
10#18321—31nomaximal, Sylow
11C3×#539—22no—
12C3×#539—22no—
13C3×#539—22no—
14C3×#539C2214yes—
15C2×C3×#126—21no—
16C2×C3×#125—21no—
17C2×C3×#123—21no—
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C22×C3×C4⋊C4C2211yescharacteristic
24C2×#924C2211yescharacteristic
25#5228—21no—
26#5218—21no—
27#5210—21no—
28#5210—21no—
29C4×#217C2211yescharacteristic
30#5207C2211yescharacteristic
31#5207C2211yescharacteristic
32#5052—22no—
33#5050C2212yescomplemented
34#5043C2212yescomplemented
35#5042—22no—
36#5040C2212yes—
37#5033C2212yes—
38#4981C2212yescomplemented
39#4979C2212yescomplemented
40#4976—22no—
41#4971—21no—
42#4970—21no—
43#4538C2214yescomplemented
44#4537C2214yes—
45#4528C2214yescomplemented
46C2×#539D311yescharacteristic
47#3195—31no—
48#3193—31no—
49#2583—31no—
50#1609—34no—
51#1603—34no—
52#1508—31no—
53#1507—31no—
54#1463—31no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C3×#126—24no—
64C3×#126—24no—
65C3×#125—24no—
66C3×#125—24no—
67C3×#123—24no—
68C3×#123—24no—
69C3×#123—24no—
70C3×#123—24no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8D412yes—
76C3×C22⋊Q8C2312yes—
77C3×#121—21no—
78C3×#121—21no—
79C3×#121D412yes—
80C3×#121C2312yes—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D412yes—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C42—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#216—22no—
104C2×#214—22no—
105C2×C4⋊Dic3C2311yescharacteristic
106#924—21no—
107#924—21no—
108#924—21no—
109#924—21no—
110C2×Dic3⋊C4—22no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—22no—
113C2×C4×Dic3—21no—
114C2×#208—21no—
115C2×#208—21no—
116#835—22no—
117#832—22no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120C3⋊#95C2312yes—
121#830—21no—
122#830—21no—
123#830C2312yes—
124#830C2312yes—
125#810—22no—
126#810—22no—
127#808—22no—
128#807—22no—
129#539—22no—
130#539—22no—
131#539—22no—
132#539C2×D314yes—
133C2×#126—21no—
134C2×#125—21no—
135C2×#123—21no—
136C2×#123—21no—
137C2×C22⋊Q8—21no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8C2×D311yescharacteristic
140C2×#121C2×D311yescharacteristic
141C2×C4×D4—21no—
142C22×C4⋊C4C2×D311yescharacteristic
143#484—61no—
144#483—32no—
145#483—61no—
146#480—32no—
147#470—62no—
148#468—62no—
149#451—32no—
150#451—61no—
151#450—61no—
152#448—32no—
153#407—31no—
154#407—31no—
155#398—32no—
156#395—32no—
157#389—62no—
158#385—61no—
159#385—61no—
160C4×C22⋊C4—31no—
161#328—34no—
162#327—34no—
163#324—34no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—81no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—44no—
173C2×C3×D4—81no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—21no—
183C22×C3×C4—42no—
184C22×C3×C4—81no—
185C22×C3×C4—81no—
186C22×C3×C4—82no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6—81no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C42—24no—
222#217—24no—
223#217—41no—
224#217—41no—
225#216—82no—
226#214—82no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229Dic3⋊C4—44no—
230Dic3⋊C4—41no—
231Dic3⋊C4—41no—
232Dic3⋊C4—82no—
233C4×Dic3—81no—
234#208—81no—
235#208—81no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C22×Q8—31no—
239C22×Q8#6711yescharacteristic
240C22×D4—21no—
241C22×D4#6711yescharacteristic
242C23×C4—21no—
243C23×C4#6711yescharacteristic
244C23×C4#6711yescharacteristic
245C23×C4C22×D311yescharacteristic
246C2×Q16—62no—
247C2×SD16—62no—
248C2×M16—61no—
249C2×M16—61no—
250#126—24no—
251#126—24no—
252#125—24no—
253#125—24no—
254#123—24no—
255#123—24no—
256#123—24no—
257#123—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—61no—
263C22⋊Q8—61no—
264C22⋊Q8—61no—
265C22⋊Q8—61no—
266C22⋊Q8#6712yes—
267C22⋊Q8C22×D312yes—
268#121—21no—
269#121—21no—
270#121#6712yes—
271#121C22×D312yes—
272C4×D4—24no—
273C4×D4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4#6712yes—
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4#6711yescharacteristic
289C2×C4⋊C4C22×D312yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—61no—
299#107—62no—
300#106—62no—
301#103—62no—
302#103—62no—
303#103—62no—
304#102—62no—
305C22⋊C8—32no—
306C22⋊C8—32no—
307C22⋊C8—61no—
308C22⋊C8—61no—
309#95—32no—
310#95—62no—
311#95—62no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3C22×D411yescharacteristic
322C3×Q8—42no—
323C3×Q8—81no—
324C3×D4—42no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—81no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—81no—
349C2×C3×C4—84no—
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×Dic3—24no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361Dic6—42no—
362Dic6—42no—
363C3⋊C8—42no—
364C3⋊C8—42no—
365C24—21no—
366C24C2×#6711yescharacteristic
367C2×Q8—21no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—81no—
371C2×Q8—241no—
372C2×Q8C2×#6712yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—44no—
377C2×D4—81no—
378C2×D4C2×#6712yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—42no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—82no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C23×D311yescharacteristic
402Q16—242no—
403SD16—242no—
404M16—241no—
405M16—241no—
406C2×C8—62no—
407C2×C8—62no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—124no—
422C4⋊C4—121no—
423C4⋊C4—121no—
424C4⋊C4—242no—
425C4⋊C4C2×#6712yes—
426C4⋊C4C2×#6712yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—64no—
435C22⋊C4—121no—
436C22⋊C4—121no—
437C42—24no—
438C42—241no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—44no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—84no—
451C22×C3#45612yes—
452C22×C3C2×#12011yescharacteristic
453C22×C3#57512yes—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—42no—
463Dic3—44no—
464Dic3—81no—
465Dic3—81no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23#95311yescharacteristic
474C23#95311yescharacteristic
475C23C22×#6711yescharacteristic
476Q8—42no—
477Q8—81no—
478Q8—122no—
479Q8—122no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—81no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—64no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4#95311yescharacteristic
514C2×C4#95311yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C2×C4C22×#6711yescharacteristic
517C8—122no—
518C8—122no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—41no—
522C2×C3—82no—
523C2×C3#300511yescharacteristic
524C2×C3C2×#45611yescharacteristic
525C2×C3#302611yescharacteristic
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—84no—
538C22#524612yes—
539C22#582212yes—
540C22C2×#95311yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—124no—
551C4—241no—
552C4—241no—
553C3#1832111yescharacteristic, complemented, Sylow
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2#8408111yescharacteristic
559C2#8411011yescharacteristic
560C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#539C2
socleC22×C3C2×#120
prime radicalC2×C3×#121C22
hypercenterC2×#539D3
hypocenterC3#18321
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
2. #82742C2semidirect
4. #82181C2semidirect
5. #71889C2semidirect
6. #71886C2semidirect
7. #71688C2semidirect
9. #71552C2semidirect
33. #5050C22semidirect
34. #5043C22semidirect
38. #4981C22semidirect
39. #4979C22semidirect
43. #4538C22semidirect
45. #4528C22semidirect
553. C3#18321semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5392
#827422
#826312
#821812
#718892
#718862
#716882
#716732
#715522
#183213