Finite Group Databasebeta

← #10943923

Subgroups — #10943923

Order 768.

3630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic
2C2×#5246C212yesmaximal, complemented
3C2×#5242C212yesmaximal, complemented
4#84093C211yesmaximal, characteristic, complemented
5#84084C212yesmaximal, complemented
6#72143C212yesmaximal, complemented
7#72140C212yesmaximal, complemented
8#72138C212yesmaximal, complemented
9#72137C211yesmaximal, characteristic, complemented
10#32893—31nomaximal, Sylow
11C3×#585—28no—
12C3×#585—28no—
13C2×C3×2−1+4—22no—
14C2×C3×2−1+4C2212yes—
15C2×C3×H(5,𝔽2)—22no—
16C2×C3×H(5,𝔽2)—22no—
17C2×C3×H(5,𝔽2)C2212yes—
18C22×C3×#41—24no—
19C22×C3×#41—22no—
20C22×C3×#41C2212yes—
21C22×C3×#41C2211yescharacteristic
22C2×#951C2214yescomplemented
23C2×#944C2211yescharacteristic, complemented
24C2×#936C2211yescharacteristic, complemented
25C2×#924C2211yescharacteristic, complemented
26#5246—28no—
27#5242—28no—
28#5239—24no—
29C2×#837C2214yescomplemented
30C2×#837C2214yescomplemented
31#5232C2214yescomplemented
32#5229C2211yescharacteristic, complemented
33#5228—22no—
34C2×#836C2212yescomplemented
35#5222—22no—
36#5221C2211yescharacteristic, complemented
37#5218—22no—
38C2×#833C2212yescomplemented
39#5210—22no—
40C4×#217C2211yescharacteristic, complemented
41#5207C2212yescomplemented
42C2×#585D311yescharacteristic
43C2×#456—32no—
44C2×#455—32no—
45#3007—32no—
46#2432—31no—
47#1793—32no—
48#1792—32no—
49#1789—32no—
50#1785—31no—
51C3×2−1+4—232no—
52C3×2−1+4—28no—
53C3×2−1+4—24no—
54C3×2−1+4D418yes—
55C3×H(5,𝔽2)—216no—
56C3×H(5,𝔽2)—232no—
57C3×H(5,𝔽2)—216no—
58C3×H(5,𝔽2)—28no—
59C3×H(5,𝔽2)—24no—
60C3×H(5,𝔽2)D418yes—
61C2×C3×#41—24no—
62C2×C3×#41—28no—
63C2×C3×#41—24no—
64C2×C3×#41—232no—
65C2×C3×#41—232no—
66C2×C3×#41—216no—
67C2×C3×#41—216no—
68C2×C3×#41—28no—
69C2×C3×#41—28no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41—24no—
74C2×C3×#41—22no—
75C2×C3×#41—22no—
76C2×C3×#41—24no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41D414yes—
80C2×C3×#41C2314yes—
81C22×C3×Q8—22no—
82C22×C3×Q8—24no—
83C22×C3×Q8D412yes—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4—28no—
86C22×C3×D4—22no—
87C22×C3×D4—24no—
88C22×C3×D4—22no—
89C22×C3×D4—24no—
90C22×C3×D4D412yes—
91C22×C3×D4D412yes—
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4—24no—
94C23×C3×C4—22no—
95C23×C3×C4D412yes—
96C23×C3×C4C2311yescharacteristic
97C22×Dic6C2311yescharacteristic
98C2×#217—21no—
99C2×#217—22no—
100C2×#217—22no—
101C2×#217—21no—
102#951—28no—
103#951—28no—
104#944—24no—
105C2×#216—24no—
106#936—24no—
107C2×#214—24no—
108C2×C4⋊Dic3C2311yescharacteristic
109#924—22no—
110#924—22no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—22no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×#208C2314yes—
116#837—28no—
117#837—28no—
118#836—24no—
119#835—24no—
120#833—24no—
121#832—24no—
122C3⋊#95—21no—
123C3⋊#95—22no—
124#830—22no—
125#830—22no—
126#585—28no—
127#585—28no—
128C2×2−1+4—22no—
129C2×2−1+4C2×D312yes—
130C2×H(5,𝔽2)—22no—
131C2×H(5,𝔽2)—22no—
132C2×H(5,𝔽2)C2×D312yes—
133C22×#41—24no—
134C22×#41—22no—
135C22×#41C2×D312yes—
136C22×#41C2×D311yescharacteristic
137C2×#137—34no—
138C2×C4⋊Q8—31no—
139C2×#124—31no—
140C2×C22⋊Q8—31no—
141#456—68no—
142#455—68no—
143#452—64no—
144#451—62no—
145#450—62no—
146C2×#104—34no—
147C2×#104—34no—
148#417—34no—
149C2×#101—32no—
150C2×#100—32no—
151#407—32no—
152#394—62no—
153#391—31no—
154#390—31no—
155#385—62no—
156C4×C22⋊C4—31no—
157C24×C3—24no—
158C24×C3—22no—
159C24×C3C2×D412yes—
160C24×C3C2×D411yescharacteristic
161C3×#41—232no—
162C3×#41—28no—
163C3×#41—28no—
164C3×#41—216no—
165C3×#41—264no—
166C3×#41—232no—
167C3×#41—216no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—28no—
173C3×#41—216no—
174C3×#41—28no—
175C3×#41—216no—
176C3×#41—216no—
177C3×#41—24no—
178C3×#41—24no—
179C3×#41C2×D418yes—
180C3×#41C2×D418yes—
181C2×C3×Q8—24no—
182C2×C3×Q8—216no—
183C2×C3×Q8—216no—
184C2×C3×Q8—216no—
185C2×C3×Q8—28no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8C2×D414yes—
190C2×C3×Q8C2×D414yes—
191C2×C3×D4—28no—
192C2×C3×D4—24no—
193C2×C3×D4—264no—
194C2×C3×D4—216no—
195C2×C3×D4—216no—
196C2×C3×D4—28no—
197C2×C3×D4—28no—
198C2×C3×D4—216no—
199C2×C3×D4—24no—
200C2×C3×D4—24no—
201C2×C3×D4—216no—
202C2×C3×D4—24no—
203C2×C3×D4—24no—
204C2×C3×D4—216no—
205C2×C3×D4—28no—
206C2×C3×D4—22no—
207C2×C3×D4—24no—
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4C2×D414yes—
211C2×C3×D4C2×D414yes—
212C22×C3×C4—24no—
213C22×C3×C4—28no—
214C22×C3×C4—232no—
215C22×C3×C4—216no—
216C22×C3×C4—22no—
217C22×C3×C4—24no—
218C22×C3×C4—24no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—22no—
224C22×C3×C4—22no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237#217—42no—
238#217—44no—
239#217—42no—
240#217—44no—
241#216—48no—
242#214—48no—
243C4⋊Dic3—22no—
244Dic3⋊C4—41no—
245Dic3⋊C4—41no—
246Dic3⋊C4—44no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249#208—24no—
250#208—24no—
251C2×C3⋊C8—24no—
2522−1+4—232no—
2532−1+4—28no—
2542−1+4—24no—
2552−1+4#6718yes—
256H(5,𝔽2)—216no—
257H(5,𝔽2)—232no—
258H(5,𝔽2)—216no—
259H(5,𝔽2)—28no—
260H(5,𝔽2)—24no—
261H(5,𝔽2)#6718yes—
262C2×#41—24no—
263C2×#41—28no—
264C2×#41—24no—
265C2×#41—232no—
266C2×#41—232no—
267C2×#41—216no—
268C2×#41—216no—
269C2×#41—28no—
270C2×#41—28no—
271C2×#41—24no—
272C2×#41—24no—
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—24no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41#6714yes—
281C2×#41C22×D314yes—
282C22×Q8—22no—
283C22×Q8—24no—
284C22×Q8—31no—
285C22×Q8#6712yes—
286C22×Q8C22×D311yescharacteristic
287C22×D4—28no—
288C22×D4—22no—
289C22×D4—24no—
290C22×D4—22no—
291C22×D4—24no—
292C22×D4#6712yes—
293C22×D4#6712yes—
294C22×D4C22×D311yescharacteristic
295C23×C4—24no—
296C23×C4—22no—
297C23×C4#6712yes—
298C23×C4C22×D311yescharacteristic
299#137—68no—
300#137—68no—
301C2×Q16—64no—
302C2×SD16—64no—
303C2×M16—34no—
304C4⋊Q8—64no—
305#124—64no—
306C22⋊Q8—62no—
307C22⋊Q8—62no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—62no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C42—31no—
316C2×C42—31no—
317#104—68no—
318#104—68no—
319#103—64no—
320#102—64no—
321#101—64no—
322#100—64no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—61no—
326#95—62no—
327C23×C3—28no—
328C23×C3—216no—
329C23×C3—22no—
330C23×C3—24no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—82no—
339C23×C3—82no—
340C23×C3—84no—
341C23×C3—84no—
342C23×C3#12012yes—
343C23×C3#12014yes—
344C23×C3C22×D411yescharacteristic
345C3×Q8—216no—
346C3×Q8—28no—
347C3×Q8—24no—
348C3×Q8—28no—
349C3×Q8#12018yes—
350C3×D4—232no—
351C3×D4—216no—
352C3×D4—216no—
353C3×D4—216no—
354C3×D4—28no—
355C3×D4—28no—
356C3×D4—28no—
357C3×D4—28no—
358C3×D4—24no—
359C3×D4#12018yes—
360C2×C3×C4—216no—
361C2×C3×C4—28no—
362C2×C3×C4—216no—
363C2×C3×C4—28no—
364C2×C3×C4—232no—
365C2×C3×C4—28no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—28no—
373C2×C3×C4—28no—
374C2×C3×C4—24no—
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12014yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×Dic3—22no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392Dic6—44no—
393C3⋊C8—44no—
394C24—24no—
395C24—22no—
396C24C2×#6712yes—
397C24C2×#6711yescharacteristic
398#41—232no—
399#41—28no—
400#41—28no—
401#41—216no—
402#41—264no—
403#41—232no—
404#41—216no—
405#41—216no—
406#41—216no—
407#41—216no—
408#41—216no—
409#41—28no—
410#41—216no—
411#41—28no—
412#41—216no—
413#41—216no—
414#41—24no—
415#41—24no—
416#41C2×#6718yes—
417#41C2×#6718yes—
418C2×Q8—24no—
419C2×Q8—216no—
420C2×Q8—216no—
421C2×Q8—216no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8C2×#6714yes—
430C2×Q8C2×#6714yes—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—264no—
434C2×D4—216no—
435C2×D4—216no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—216no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—216no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—216no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4C2×#6714yes—
451C2×D4C2×#6714yes—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—232no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6712yes—
471C22×C4C2×#6712yes—
472C22×C4C2×#6712yes—
473C22×C4C23×D311yescharacteristic
474Q16—128no—
475SD16—128no—
476M16—64no—
477M16—64no—
478C2×C8—64no—
479C4⋊C4—62no—
480C4⋊C4—121no—
481C4⋊C4—121no—
482C4⋊C4—124no—
483C22⋊C4—122no—
484C22⋊C4—124no—
485C22⋊C4—122no—
486C22⋊C4—124no—
487C42—62no—
488C42—62no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—22no—
492C22×C3—24no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—44no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—42no—
500C22×C3—44no—
501C22×C3—81no—
502C22×C3—82no—
503C22×C3—81no—
504C22×C3—88no—
505C22×C3—816no—
506C22×C3C2×#12011yescharacteristic
507C22×C3C2×#12011yescharacteristic
508C22×C3C2×#12011yescharacteristic
509C3×C4—21no—
510C3×C4—28no—
511C3×C4—24no—
512C3×C4—21no—
513C3×C4C2×#12014yes—
514Dic3—42no—
515Dic3—81no—
516Dic3—81no—
517Dic3—81no—
518C23—28no—
519C23—216no—
520C23—22no—
521C23—24no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—82no—
530C23—82no—
531C23—84no—
532C23—84no—
533C23#95312yes—
534C23#95314yes—
535C23C22×#6711yescharacteristic
536Q8—216no—
537Q8—28no—
538Q8—24no—
539Q8—28no—
540Q8—124no—
541Q8#95318yes—
542D4—232no—
543D4—216no—
544D4—216no—
545D4—216no—
546D4—28no—
547D4—28no—
548D4—28no—
549D4—28no—
550D4—24no—
551D4#95318yes—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—216no—
555C2×C4—28no—
556C2×C4—232no—
557C2×C4—28no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—62no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—122no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4#95314yes—
579C2×C4#95311yescharacteristic
580C2×C4#95311yescharacteristic
581C2×C4C22×#6712yes—
582C2×C4C22×#6712yes—
583C2×C4C22×#6712yes—
584C8—124no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—42no—
588C2×C3—44no—
589C2×C3—48no—
590C2×C3#284611yescharacteristic
591C2×C3#301811yescharacteristic
592C2×C3#302211yescharacteristic
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—24no—
599C22—28no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—88no—
609C22—816no—
610C22C2×#95311yescharacteristic
611C22C2×#95311yescharacteristic
612C22C2×#95311yescharacteristic
613C4—21no—
614C4—28no—
615C4—24no—
616C4—21no—
617C4—122no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4C2×#95314yes—
622C3#3289311yescharacteristic, complemented, Sylow
623C2—21no—
624C2—21no—
625C2—42no—
626C2—44no—
627C2—48no—
628C2#8412111yescharacteristic
629C2#8412311yescharacteristic
630C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#585C2
socleC22×C3C2×#120
hypercenterC2×#585D3
hypocenterC3#32893
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
2. C2×#5246C2semidirect
3. C2×#5242C2semidirect
4. #84093C2semidirect
5. #84084C2semidirect
6. #72143C2semidirect
7. #72140C2semidirect
8. #72138C2semidirect
9. #72137C2semidirect
22. C2×#951C22semidirect
23. C2×#944C22semidirect
24. C2×#936C22semidirect
25. C2×#924C22semidirect
29. C2×#837C22semidirect
30. C2×#837C22semidirect
31. #5232C22semidirect
32. #5229C22semidirect
34. C2×#836C22semidirect
36. #5221C22semidirect
38. C2×#833C22semidirect
40. C4×#217C22semidirect
41. #5207C22semidirect
622. C3#32893semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C2×#52462
C2×#52422
#840932
#840842
#721432
#721402
#721382
#721372
#328933