Finite Group Databasebeta

← #10768803

Subgroups — #10768803

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3537C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic, complemented
3#82665C211yesmaximal, characteristic, complemented
4C2×#5033C211yesmaximal, characteristic, complemented
5#71882C214yesmaximal, complemented
6#71843C214yesmaximal, complemented
7#71707C211yesmaximal, characteristic, complemented
8#71676C211yesmaximal, characteristic, complemented
9#71673C211yesmaximal, characteristic, complemented
10#18264—31nomaximal, Sylow
11C3×#560—22no—
12C3×#553—21no—
13C3×#553—21no—
14C3×D42—22no—
15C3×D42—21no—
16C3×D42—21no—
17C3×#540—21no—
18C3×#538—21no—
19C3×#538C2212yes—
20C3×#535—21no—
21C3×#534—21no—
22C3×#534C2212yes—
23C3×#525C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic, complemented
27#5232—21no—
28#5221—21no—
29#5220—22no—
30C2×#832—21no—
31C4×#217C2211yescharacteristic, complemented
32#5207C2211yescharacteristic, complemented
33C2×#830C2211yescharacteristic, complemented
34#5044—21no—
35#5043C2212yescomplemented
36#5034—21no—
37#5033—22no—
38#5033C2212yescomplemented
39#5033C2214yescomplemented
40#4995—21no—
41#4977—21no—
42C2×#807—21no—
43#4953C2214yescomplemented
44#4538C2212yescomplemented
45#4537C2212yescomplemented
46#4537C2214yescomplemented
47#4525C2214yescomplemented
48#3537D311yescharacteristic
49#3092—31no—
50C2×#480—31no—
51#2504—31no—
52#1612—34no—
53#1607—34no—
54#1507—31no—
55#1501—31no—
56#1471—31no—
57C2×C3×#41D411yescharacteristic
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4D412yes—
63C22×C3×D4D411yescharacteristic
64C22×C3×D4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C3×#127—22no—
68C3×#127—21no—
69C3×#126—22no—
70C3×#126—22no—
71C3×#125—21no—
72C3×#124—22no—
73C3×#123—22no—
74C3×#123—21no—
75C3×#123—22no—
76C3×#123—22no—
77C3×#123—22no—
78C3×C22⋊Q8D411yescharacteristic
79C3×C22⋊Q8C2311yescharacteristic
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#121—21no—
85C3×#121—21no—
86C3×#121—22no—
87C3×#121—21no—
88C3×#121—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#121—22no—
94C3×#121D412yes—
95C3×#121D411yescharacteristic
96C3×#121C2312yes—
97C3×#121C2311yescharacteristic
98C3×#120—22no—
99C3×#120—22no—
100C3×#120—22no—
101C3×#120—21no—
102C3×#120—22no—
103C3×#120—22no—
104C3×#120—22no—
105C3×C4×D4—22no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×C4×D4—21no—
110C3×C4×D4—22no—
111C3×#117—21no—
112C3×#117D411yescharacteristic
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4D411yescharacteristic
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4C2312yes—
117C2×#217—21no—
118C2×#217—24no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3C2312yes—
125C2×C4×Dic3—21no—
126C2×#208—21no—
127C22×C3⋊C8—21no—
128#835—22no—
129#832—24no—
130#832—22no—
131C3⋊#95—24no—
132C3⋊#95C2312yes—
133#830—21no—
134#830—21no—
135#830C2312yes—
136#830C2312yes—
137#808—22no—
138#807—24no—
139#807—22no—
140#560—22no—
141#553—21no—
142#553—21no—
143D42—22no—
144D42—21no—
145D42—21no—
146#540—21no—
147#538—21no—
148#538C2×D312yes—
149#535—21no—
150#534—21no—
151#534C2×D312yes—
152#525C2×D311yescharacteristic
153C2×#121C2×D311yescharacteristic
154C2×#121C2×D311yescharacteristic
155C22×C4⋊C4—31no—
156#484—61no—
157#483—32no—
158#481—61no—
159#480—34no—
160#480—32no—
161#480—62no—
162#428—61no—
163C2×#107—61no—
164#417—61no—
165C2×#102—61no—
166#407—31no—
167C2×C22⋊C8—31no—
168#397—34no—
169#390—61no—
170#386—62no—
171#382—61no—
172C4×C22⋊C4—31no—
173#328—32no—
174#327—32no—
175#327—34no—
176#323—34no—
177C24×C3—22no—
178C24×C3—21no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D412yes—
181C3×#41—42no—
182C2×C3×Q8C2×D411yescharacteristic
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—44no—
201C2×C3×D4—82no—
202C2×C3×D4—82no—
203C2×C3×D4—84no—
204C2×C3×D4—84no—
205C2×C3×D4C2×D411yescharacteristic
206C2×C3×D4C2×D412yes—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—41no—
222C22×Dic3—41no—
223C22×Dic3—82no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D411yescharacteristic
233C3×C4⋊C4C2×D411yescharacteristic
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—21no—
248C3×C42—22no—
249C3×C42—21no—
250#217—42no—
251#217—84no—
252C4⋊Dic3—22no—
253C4⋊Dic3—22no—
254C4⋊Dic3—41no—
255C4⋊Dic3—41no—
256C4×Dic3—42no—
257#208—42no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—42no—
261C2×#41#6711yescharacteristic
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—22no—
266C22×D4#6712yes—
267C22×D4#6711yescharacteristic
268C22×D4#6711yescharacteristic
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×M16—61no—
272C22×C8—61no—
273#127—22no—
274#127—21no—
275#126—22no—
276#126—22no—
277#125—21no—
278#124—22no—
279#123—22no—
280#123—21no—
281#123—22no—
282#123—22no—
283#123—22no—
284C22⋊Q8#6711yescharacteristic
285C22⋊Q8C22×D311yescharacteristic
286#121—22no—
287#121—21no—
288#121—21no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—22no—
293#121—21no—
294#121—21no—
295#121—21no—
296#121—21no—
297#121—21no—
298#121—21no—
299#121—22no—
300#121#6712yes—
301#121#6711yescharacteristic
302#121C22×D312yes—
303#121C22×D311yescharacteristic
304#120—22no—
305#120—22no—
306#120—22no—
307#120—21no—
308#120—22no—
309#120—22no—
310#120—22no—
311C4×D4—22no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—22no—
317#117—21no—
318#117#6711yescharacteristic
319C2×C4⋊C4—32no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4#6711yescharacteristic
326C2×C4⋊C4#6711yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—64no—
330C2×C22⋊C4C22×D312yes—
331C2×C42—61no—
332#107—64no—
333#107—62no—
334#106—62no—
335#103—62no—
336#102—64no—
337#102—62no—
338C22⋊C8—32no—
339C22⋊C8—32no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#95—32no—
343#95—64no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—21no—
349C23×C3—24no—
350C23×C3—41no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—81no—
355C23×C3—81no—
356C23×C3—82no—
357C23×C3—82no—
358C23×C3—82no—
359C23×C3#12011yescharacteristic
360C23×C3#12011yescharacteristic
361C23×C3C22×D411yescharacteristic
362C3×Q8—41no—
363C3×D4—42no—
364C3×D4—41no—
365C3×D4—44no—
366C3×D4—42no—
367C3×D4—44no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—81no—
372C3×D4—82no—
373C3×D4—82no—
374C3×D4—81no—
375C3×D4—82no—
376C3×D4—82no—
377C3×D4—82no—
378C3×D4—82no—
379C3×D4—82no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—81no—
394C2×C3×C4—82no—
395C2×C3×C4—82no—
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—44no—
405C2×Dic3—82no—
406C2×Dic3—82no—
407C3⋊C8—42no—
408C3⋊C8—42no—
409C24—22no—
410C24—21no—
411C24C2×#6711yescharacteristic
412C24C2×#6712yes—
413#41—42no—
414C2×Q8C2×#6711yescharacteristic
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4C2×#6711yescharacteristic
438C2×D4C2×#6712yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—121no—
447C22×C4—121no—
448C22×C4—242no—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6712yes—
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456M16—122no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—122no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—121no—
470C4⋊C4—121no—
471C4⋊C4C2×#6711yescharacteristic
472C4⋊C4C2×#6712yes—
473C4⋊C4C2×#6711yescharacteristic
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—122no—
489C22⋊C4—244no—
490C42—22no—
491C42—21no—
492C42—122no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—41no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—84no—
511C22×C3—162no—
512C22×C3—164no—
513C22×C3#45512yes—
514C22×C3C2×#12011yescharacteristic
515C22×C3#57612yes—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4#45812yes—
525Dic3—82no—
526Dic3—82no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—24no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—81no—
538C23—81no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23#95311yescharacteristic
543C23#95311yescharacteristic
544C23C22×#6711yescharacteristic
545Q8—41no—
546D4—42no—
547D4—41no—
548D4—44no—
549D4—42no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—81no—
555D4—82no—
556D4—82no—
557D4—81no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—82no—
562D4—82no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—124no—
583C2×C4—122no—
584C2×C4—242no—
585C2×C4—242no—
586C2×C4#95311yescharacteristic
587C2×C4#95311yescharacteristic
588C2×C4C22×#6711yescharacteristic
589C2×C4C22×#6711yescharacteristic
590C8—122no—
591C8—122no—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—41no—
595C2×C3—82no—
596C2×C3—82no—
597C2×C3—81no—
598C2×C3#300811yescharacteristic
599C2×C3C2×#45511yescharacteristic
600C2×C3C2×#45811yescharacteristic
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—84no—
619C22—162no—
620C22—164no—
621C22#524212yes—
622C22#583512yes—
623C22C2×#95311yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—242no—
633C4—242no—
634C4#524312yes—
635C3#1826411yescharacteristic, complemented, Sylow
636C2—22no—
637C2—22no—
638C2—41no—
639C2—82no—
640C2—82no—
641C2—81no—
642C2#8410111yescharacteristic
643C2C2×#524211yescharacteristic
644C2C2×#524311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3537C2
socleC22×C3C2×#120
hypercenter#3537D3
hypocenterC3#18264
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3537

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. #82665C2semidirect
4. C2×#5033C2semidirect
5. #71882C2semidirect
6. #71843C2semidirect
7. #71707C2semidirect
8. #71676C2semidirect
9. #71673C2semidirect
26. C22×C4⋊Dic3C22semidirect
31. C4×#217C22semidirect
32. #5207C22semidirect
33. C2×#830C22semidirect
35. #5043C22semidirect
38. #5033C22semidirect
39. #5033C22semidirect
43. #4953C22semidirect
44. #4538C22semidirect
45. #4537C22semidirect
46. #4537C22semidirect
47. #4525C22semidirect
635. C3#18264semidirect

Maximal subgroups

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

subgroupindex
C3×#35372
#829142
#826652
C2×#50332
#718822
#718432
#717072
#716762
#716732
#182643