Finite Group Databasebeta

← #10768848

Subgroups — #10768848

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3538C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic
3#82776C211yesmaximal, characteristic, complemented
4C2×#5044C211yesmaximal, characteristic, complemented
5#71899C214yesmaximal, complemented
6#71887C214yesmaximal, complemented
7#71707C211yesmaximal, characteristic
8#71691C211yesmaximal, characteristic, complemented
9#71688C211yesmaximal, characteristic, complemented
10#18269—31nomaximal, Sylow
11C3×#563—22no—
12C3×#552—21no—
13C3×#552—21no—
14C3×#551—22no—
15C3×D4×Q8—21no—
16C3×D4×Q8—21no—
17C3×#544—21no—
18C3×#544C2212yes—
19C3×#541—21no—
20C3×#539—21no—
21C3×#539C2212yes—
22C3×#536—21no—
23C3×#525C2211yescharacteristic
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×C22⋊Q8C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27#5232—21no—
28#5229—21no—
29C2×#835—21no—
30C4×#217C2211yescharacteristic
31#5207C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5044—22no—
34#5044C2214yes—
35#5044C2212yes—
36#5043—21no—
37#5034C2212yes—
38#5033—21no—
39#4995—21no—
40#4986C2214yes—
41#4977—21no—
42C2×#807—21no—
43#4943—22no—
44#4538C2214yes—
45#4538C2212yes—
46#4537C2212yes—
47#4528C2214yes—
48#3538D311yescharacteristic
49#3092—31no—
50C2×#484—31no—
51#2503—31no—
52#1617—34no—
53#1615—34no—
54#1508—31no—
55#1501—31no—
56#1472—31no—
57C2×C3×#41D411yescharacteristic
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×#126—22no—
68C3×#126—22no—
69C3×#126—22no—
70C3×#126—22no—
71C3×#125—22no—
72C3×#125—21no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#124—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—21no—
81C3×#123—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8D412yes—
95C3×C22⋊Q8D411yescharacteristic
96C3×C22⋊Q8C2312yes—
97C3×C22⋊Q8C2311yescharacteristic
98C3×#121D411yescharacteristic
99C3×#121C2311yescharacteristic
100C3×#120—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×D4—22no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—22no—
107C3×#117—21no—
108C3×#117—22no—
109C3×#117—22no—
110C3×#117D411yescharacteristic
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4D411yescharacteristic
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4D412yes—
117C2×#217—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2312yes—
124C2×C4×Dic3—21no—
125C2×#208—21no—
126C22×C3⋊C8—21no—
127#835—24no—
128#835—22no—
129#832—22no—
130C3⋊#95—24no—
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#563—22no—
141#552—21no—
142#552—21no—
143#551—22no—
144D4×Q8—21no—
145D4×Q8—21no—
146#544—21no—
147#544C2×D312yes—
148#541—21no—
149#539—21no—
150#539C2×D312yes—
151#536—21no—
152#525C2×D311yescharacteristic
153C2×C22⋊Q8C2×D311yescharacteristic
154C2×C22⋊Q8C2×D311yescharacteristic
155C22×C4⋊C4—31no—
156#484—34no—
157#484—32no—
158#484—62no—
159#483—61no—
160#481—32no—
161#480—61no—
162#428—61no—
163C2×#107—61no—
164#417—61no—
165C2×#103—61no—
166#407—31no—
167C2×C22⋊C8—31no—
168#400—34no—
169#391—61no—
170#382—61no—
171#381—62no—
172C4×C22⋊C4—31no—
173#328—34no—
174#328—32no—
175#327—32no—
176#324—34no—
177C24×C3C2×D411yescharacteristic
178C3×#41—42no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—42no—
184C2×C3×Q8—42no—
185C2×C3×Q8—82no—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4C2×D411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—42no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—44no—
208C22×Dic3—41no—
209C22×Dic3—41no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4C2×D412yes—
228C3×C4⋊C4C2×D411yescharacteristic
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—21no—
246#217—42no—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3—41no—
250C4⋊Dic3—41no—
251C4×Dic3—42no—
252#208—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—42no—
256C2×#41#6711yescharacteristic
257C22×Q8—21no—
258C22×Q8#6711yescharacteristic
259C22×Q8#6711yescharacteristic
260C23×C4—61no—
261C23×C4C22×D311yescharacteristic
262C2×M16—61no—
263C22×C8—61no—
264C4⋊Q8—21no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#126—22no—
269#126—22no—
270#126—22no—
271#126—22no—
272#125—21no—
273#125—22no—
274#125—22no—
275#125—22no—
276#125—22no—
277#124—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—21no—
282#123—22no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8#6712yes—
296C22⋊Q8#6711yescharacteristic
297C22⋊Q8C22×D312yes—
298C22⋊Q8C22×D311yescharacteristic
299#121#6711yescharacteristic
300#121C22×D311yescharacteristic
301#120—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×D4—22no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—22no—
308#117—21no—
309#117—22no—
310#117—22no—
311#117#6711yescharacteristic
312C2×C4⋊C4—21no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4#6711yescharacteristic
321C2×C4⋊C4C22×D312yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4#6712yes—
325C2×C42—61no—
326#107—64no—
327#107—62no—
328#106—62no—
329#103—64no—
330#103—62no—
331#102—62no—
332C22⋊C8—32no—
333C22⋊C8—32no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336#95—32no—
337#95—64no—
338#95—64no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—41no—
343C23×C3#12011yescharacteristic
344C23×C3#12011yescharacteristic
345C23×C3C22×D411yescharacteristic
346C3×Q8—42no—
347C3×Q8—41no—
348C3×Q8—42no—
349C3×Q8—42no—
350C3×Q8—81no—
351C3×Q8—82no—
352C3×D4—41no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—81no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—81no—
374C2×C3×C4—82no—
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—44no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—82no—
385C2×Dic3—82no—
386C3⋊C8—42no—
387C3⋊C8—42no—
388C24C2×#6711yescharacteristic
389#41—42no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8C2×#6712yes—
398C2×Q8C2×#6711yescharacteristic
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C2×#6711yescharacteristic
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—124no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4C2×#6712yes—
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6712yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421M16—122no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—122no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4C2×#6711yescharacteristic
447C4⋊C4C2×#6712yes—
448C4⋊C4C2×#6711yescharacteristic
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—122no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—21no—
466C42—122no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—44no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3#45412yes—
479C22×C3C2×#12011yescharacteristic
480C22×C3#57812yes—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4#45812yes—
492Dic3—82no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—41no—
498C23#95311yescharacteristic
499C23#95311yescharacteristic
500C23C22×#6711yescharacteristic
501Q8—42no—
502Q8—41no—
503Q8—42no—
504Q8—42no—
505Q8—81no—
506Q8—82no—
507D4—41no—
508D4—42no—
509D4—42no—
510D4—81no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—124no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—242no—
536C2×C4—242no—
537C2×C4#95311yescharacteristic
538C2×C4#95311yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C2×C4C22×#6711yescharacteristic
541C8—122no—
542C8—122no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—41no—
546C2×C3—81no—
547C2×C3#301111yescharacteristic
548C2×C3C2×#45411yescharacteristic
549C2×C3C2×#45811yescharacteristic
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—44no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#524512yes—
562C22#583812yes—
563C22C2×#95311yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—242no—
575C4—242no—
576C4#524312yes—
577C3#1826911yescharacteristic, complemented, Sylow
578C2—22no—
579C2—22no—
580C2—41no—
581C2—81no—
582C2C2×#524311yescharacteristic
583C2C2×#524511yescharacteristic
584C2#8411711yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3538C2
socleC22×C3C2×#120
prime radicalC3×#525C22
hypercenter#3538D3
hypocenterC3#18269
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
3. #82776C2semidirect
4. C2×#5044C2semidirect
5. #71899C2semidirect
6. #71887C2semidirect
8. #71691C2semidirect
9. #71688C2semidirect
577. C3#18269semidirect

Maximal subgroups

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

subgroupindex
C3×#35382
#829142
#827762
C2×#50442
#718992
#718872
#717072
#716912
#716882
#182693