Finite Group Databasebeta

← #10770367

Subgroups — #10770367

Order 768.

972 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3558C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic, complemented
3#83123C211yesmaximal, characteristic, complemented
4#82978C211yesmaximal, characteristic, complemented
5#71882C214yesmaximal, complemented
6#71843C214yesmaximal, complemented
7#71812C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic, complemented
9#71720C211yesmaximal, characteristic, complemented
10#18436—31nomaximal, Sylow
11C3×#562—21no—
12C3×H(3,𝔽4)—21no—
13C3×#559—21no—
14C3×#548—21no—
15C3×#546—21no—
16C3×#546—22no—
17C3×D42—21no—
18C3×#542—21no—
19C3×#538—22no—
20C3×#538C2212yes—
21C3×#535—21no—
22C3×#534C2212yes—
23C3×#532C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×#869C2211yescharacteristic, complemented
27#5229—21no—
28#5221—21no—
29#5078—21no—
30#5070—21no—
31#5064—21no—
32#5055—21no—
33#5043C2212yescomplemented
34#5033C2212yes—
35#5033C2214yescomplemented
36#5021—21no—
37#5008—21no—
38#5007—22no—
39#4953C2214yescomplemented
40#4947—22no—
41#4932C2211yescharacteristic, complemented
42#4926C2211yescharacteristic, complemented
43#4925C2211yescharacteristic, complemented
44#4538C2212yescomplemented
45#4537C2212yes—
46#4537C2214yescomplemented
47#4525C2214yescomplemented
48#3558D311yescharacteristic
49#3238—31no—
50#3226—31no—
51#2640—31no—
52#1612—34no—
53#1607—34no—
54#1505—31no—
55#1500—31no—
56#1469—31no—
57C2×C3×#41—21no—
58C2×C3×#41—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4D411yescharacteristic
63C22×Dic6—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×#127—21no—
66C3×#127D411yescharacteristic
67C3×#126—22no—
68C3×#126—21no—
69C3×#125D411yescharacteristic
70C3×#124—21no—
71C3×#124—22no—
72C3×#124—22no—
73C3×#124—22no—
74C3×#124—21no—
75C3×#124—21no—
76C3×#124—21no—
77C3×#124—21no—
78C3×#124—21no—
79C3×#124D411yescharacteristic
80C3×#123—22no—
81C3×#123—21no—
82C3×#123—22no—
83C3×#123—22no—
84C3×#123—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8C2311yescharacteristic
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121—22no—
94C3×#121—21no—
95C3×#121—22no—
96C3×#121C2311yescharacteristic
97C3×#121C2312yes—
98C3×#120—22no—
99C3×#120—22no—
100C3×#120—22no—
101C3×#120—22no—
102C3×#120—22no—
103C3×C4×Q8—21no—
104C3×C4×Q8D411yescharacteristic
105C3×C4×D4—22no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
108C3×C4×D4—22no—
109C3×C4×D4—21no—
110C3×C4×D4D412yes—
111C3×C4×D4D411yescharacteristic
112C3×#117—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4D412yes—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×#217—24no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2312yes—
120C2×C4×Dic3—21no—
121#869—21no—
122#869—21no—
123#869—21no—
124#869—21no—
125#835—22no—
126#832—24no—
127#832—22no—
128C3⋊#95—21no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#808—22no—
134#807—24no—
135#807—22no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139C4×C3⋊C8—21no—
140#562—21no—
141H(3,𝔽4)—21no—
142#559—21no—
143#548—21no—
144#546—21no—
145#546—22no—
146D42—21no—
147#542—21no—
148#538—22no—
149#538C2×D312yes—
150#535—21no—
151#534C2×D312yes—
152#532C2×D311yescharacteristic
153C2×C4⋊Q8—31no—
154#529C2×D311yescharacteristic
155C2×#124C2×D311yescharacteristic
156#501—61no—
157#500—61no—
158#489—61no—
159#486—61no—
160#483—32no—
161#480—32no—
162#480—34no—
163#477—61no—
164#476—61no—
165#474—62no—
166#432—31no—
167#431—31no—
168#397—34no—
169#393—62no—
170#391—61no—
171#390—61no—
172#376—31no—
173#328—32no—
174#327—32no—
175#327—34no—
176#323—34no—
177C24×C3—22no—
178C24×C3C2×D412yes—
179C3×#41—82no—
180C3×#41—82no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—42no—
185C2×C3×Q8C2×D411yescharacteristic
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—42no—
198C2×C3×D4—82no—
199C2×C3×D4—84no—
200C2×C3×D4C2×D411yescharacteristic
201C2×C3×D4C2×D412yes—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—22no—
211C22×Dic3—22no—
212C2×Dic6—41no—
213C2×Dic6—41no—
214C2×Dic6—82no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D411yescharacteristic
224C3×C4⋊C4C2×D411yescharacteristic
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247#217—84no—
248C4⋊Dic3—22no—
249C4⋊Dic3—22no—
250C4⋊Dic3—41no—
251C4⋊Dic3—41no—
252C4×Dic3—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×#41—21no—
256C2×#41—21no—
257C22×Q8—61no—
258C22×Q8#6711yescharacteristic
259C22×D4—21no—
260C22×D4—22no—
261C22×D4#6711yescharacteristic
262C4⋊Q8—61no—
263C4⋊Q8—61no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8#6711yescharacteristic
267#127—21no—
268#127#6711yescharacteristic
269#126—22no—
270#126—21no—
271#125#6711yescharacteristic
272#124—21no—
273#124—22no—
274#124—22no—
275#124—22no—
276#124—21no—
277#124—21no—
278#124—21no—
279#124—21no—
280#124—21no—
281#124#6711yescharacteristic
282#123—22no—
283#123—21no—
284#123—22no—
285#123—22no—
286#123—22no—
287C22⋊Q8—22no—
288C22⋊Q8—21no—
289C22⋊Q8—22no—
290C22⋊Q8C22×D311yescharacteristic
291#121—22no—
292#121—22no—
293#121—22no—
294#121—21no—
295#121—22no—
296#121—21no—
297#121—22no—
298#121C22×D311yescharacteristic
299#121C22×D312yes—
300#120—22no—
301#120—22no—
302#120—22no—
303#120—22no—
304#120—22no—
305C4×Q8—21no—
306C4×Q8#6711yescharacteristic
307C4×D4—22no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—22no—
311C4×D4—21no—
312C4×D4#6711yescharacteristic
313C4×D4#6712yes—
314#117—21no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—61no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—64no—
319C2×C22⋊C4#6712yes—
320C2×C22⋊C4C22×D312yes—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—64no—
324#107—62no—
325#106—62no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328#103—62no—
329#102—64no—
330#102—62no—
331C22⋊C8—32no—
332C22⋊C8—32no—
333#97—61no—
334C4×C8—61no—
335#95—32no—
336#95—61no—
337#95—64no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—24no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—82no—
345C23×C3—82no—
346C23×C3—82no—
347C23×C3C22×D411yescharacteristic
348C3×Q8—41no—
349C3×Q8—42no—
350C3×Q8—81no—
351C3×Q8—81no—
352C3×D4—42no—
353C3×D4—41no—
354C3×D4—42no—
355C3×D4—44no—
356C3×D4—42no—
357C3×D4—42no—
358C3×D4—81no—
359C3×D4—81no—
360C3×D4—82no—
361C3×D4—82no—
362C3×D4—82no—
363C3×D4—82no—
364C3×D4—82no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—82no—
376C2×C3×C4—82no—
377C2×C3×C4—82no—
378C2×C3×C4#12011yescharacteristic
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—44no—
389Dic6—82no—
390Dic6—82no—
391C3⋊C8—42no—
392C3⋊C8—42no—
393C24—22no—
394C24C2×#6712yes—
395#41—82no—
396#41—82no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—42no—
401C2×Q8—121no—
402C2×Q8—121no—
403C2×Q8—242no—
404C2×Q8C2×#6711yescharacteristic
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—82no—
418C2×D4—84no—
419C2×D4C2×#6711yescharacteristic
420C2×D4C2×#6712yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C2×#6712yes—
430C22×C4C23×D311yescharacteristic
431C2×C8—62no—
432C2×C8—62no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4C2×#6711yescharacteristic
445C4⋊C4C2×#6712yes—
446C4⋊C4C2×#6711yescharacteristic
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—244no—
464C42—22no—
465C42—21no—
466C42—122no—
467C42C2×#6711yescharacteristic
468C42C2×#6711yescharacteristic
469C42C2×#6711yescharacteristic
470C42C2×#6711yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—84no—
483C22×C3—164no—
484C22×C3C2×#12011yescharacteristic
485C22×C3#57612yes—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—41no—
492C3×C4—42no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4#45512yes—
497Dic3—82no—
498Dic3—82no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23C22×#6711yescharacteristic
509Q8—41no—
510Q8—42no—
511Q8—81no—
512Q8—81no—
513Q8—242no—
514Q8—242no—
515D4—42no—
516D4—41no—
517D4—42no—
518D4—44no—
519D4—42no—
520D4—42no—
521D4—81no—
522D4—81no—
523D4—82no—
524D4—82no—
525D4—82no—
526D4—82no—
527D4—82no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—124no—
545C2×C4—122no—
546C2×C4#95311yescharacteristic
547C2×C4#95311yescharacteristic
548C2×C4#95311yescharacteristic
549C2×C4#95311yescharacteristic
550C2×C4C22×#6711yescharacteristic
551C2×C4C22×#6711yescharacteristic
552C8—122no—
553C8—122no—
554C2×C3—22no—
555C2×C3—82no—
556C2×C3—82no—
557C2×C3—81no—
558C2×C3#300811yescharacteristic
559C2×C3C2×#45511yescharacteristic
560C2×C3#302811yescharacteristic
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—84no—
573C22—164no—
574C22#583512yes—
575C22C2×#95311yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—242no—
587C4—242no—
588C4#524212yes—
589C3#1843611yescharacteristic, complemented, Sylow
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2#8410111yescharacteristic
595C2C2×#524211yescharacteristic
596C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3558C2
socleC22×C3C2×#120
hypercenter#3558D3
hypocenterC3#18436
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3558

Complements

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

NH ≅ G/Ntype
2. #83270C2semidirect
3. #83123C2semidirect
4. #82978C2semidirect
5. #71882C2semidirect
6. #71843C2semidirect
7. #71812C2semidirect
8. #71754C2semidirect
9. #71720C2semidirect
26. C2×#869C22semidirect
33. #5043C22semidirect
35. #5033C22semidirect
39. #4953C22semidirect
41. #4932C22semidirect
42. #4926C22semidirect
43. #4925C22semidirect
44. #4538C22semidirect
46. #4537C22semidirect
47. #4525C22semidirect
589. C3#18436semidirect

Maximal subgroups

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

subgroupindex
C3×#35582
#832702
#831232
#829782
#718822
#718432
#718122
#717542
#717202
#184363