Finite Group Databasebeta

← #10768849

Subgroups — #10768849

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
1#86031C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic
3C2×#4722C211yesmaximal, characteristic, complemented
4#73518C211yesmaximal, characteristic, complemented
5C3×#1501C211yesmaximal, characteristic
6#69923C211yesmaximal, characteristic, complemented
7#69445C211yesmaximal, characteristic, complemented
8#68331C214yesmaximal, complemented
9#68298C214yesmaximal, complemented
10#18269—31nomaximal, Sylow
11C3×#525C2211yescharacteristic
12#5825—21no—
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924C2211yescharacteristic
15C2×#924C2211yescharacteristic
16#5651—22no—
17#5623—21no—
18#5623C2212yes—
19#5602—21no—
20#5602C2212yes—
21#5587—22no—
22#5541—21no—
23#5541—21no—
24D4×Dic6—21no—
25D4×Dic6—21no—
26#5518—21no—
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×C4×C22⋊C4C2211yescharacteristic
29#5232—21no—
30#5207C2211yescharacteristic
31C2×#818—21no—
32C2×#816—21no—
33#5044C2212yes—
34#5043—21no—
35#5034C2212yes—
36#5033—21no—
37#4995—21no—
38#4943—22no—
39#4936—21no—
40#4929—21no—
41#4722—22no—
42#4722C2214yes—
43#4659C2214yes—
44C3×#328C2212yes—
45C3×#327C2212yes—
46#4469C2214yes—
47#4452C2214yes—
48#3538—31no—
49#3092—31no—
50C2×#484—31no—
51#2503—31no—
52#1617—34no—
53#1615—34no—
54#1508—31no—
55#1501D311yescharacteristic
56#1472—31no—
57C2×C3×#41D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6D411yescharacteristic
62C22×Dic6D411yescharacteristic
63C22×C3×C8—21no—
64C3×#123—21no—
65C3×C22⋊Q8D411yescharacteristic
66C3×C22⋊Q8C2311yescharacteristic
67C3×#121D411yescharacteristic
68C3×#121C2311yescharacteristic
69C3×#120—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C3×#117D411yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×#217—21no—
76C2×#217D412yes—
77#944—22no—
78#936—22no—
79#935—22no—
80#935—22no—
81#935—22no—
82#935—22no—
83D4×Dic3—22no—
84D4×Dic3—22no—
85#926—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94#924—21no—
95#924—21no—
96#924—21no—
97#924—21no—
98#924—21no—
99#924—21no—
100#924—21no—
101#924—21no—
102#924D412yes—
103#924C2312yes—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4C2312yes—
106C2×#208—21no—
107#889—22no—
108#889—22no—
109#889—22no—
110#889—22no—
111#888—22no—
112#888—22no—
113#879—22no—
114#879—22no—
115#879—22no—
116#879—22no—
117#878—22no—
118#878—22no—
119#878—22no—
120#878—22no—
121#877—22no—
122#877—22no—
123#870—21no—
124#869—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C3×C22⋊C8—21no—
128C3×C22⋊C8C2312yes—
129C3×#95C2312yes—
130#835—22no—
131#832—22no—
132C3⋊#95—24no—
133C3⋊#95—24no—
134#830—21no—
135#830C2312yes—
136#818—24no—
137#816—24no—
138#808—22no—
139#807—22no—
140#563—62no—
141#552—61no—
142#552—61no—
143#551—62no—
144D4×Q8—61no—
145D4×Q8—61no—
146#544—32no—
147#544—61no—
148#541—61no—
149#539—32no—
150#539—61no—
151#536—61no—
152#525C2×D311yescharacteristic
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C22×C4⋊C4—31no—
156#484—32no—
157#484—34no—
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⋊C8C2×D311yescharacteristic
168#400—34no—
169#391—61no—
170#382—61no—
171#381—62no—
172C4×C22⋊C4C2×D311yescharacteristic
173#328—34no—
174#328C2×D312yes—
175#327C2×D312yes—
176#324—34no—
177C24×C3C2×D411yescharacteristic
178C3×#41—42no—
179C2×C3×Q8C2×D411yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—42no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—44no—
198C22×Dic3—41no—
199C22×Dic3—41no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6—42no—
207C2×Dic6—42no—
208C2×Dic6—82no—
209C2×Dic6C2×D412yes—
210C2×C3×C8—22no—
211C2×C3×C8—42no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D411yescharacteristic
215C3×C4⋊C4C2×D411yescharacteristic
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—42no—
222C3×C42—21no—
223C3×C42—42no—
224#217—22no—
225#217—22no—
226#217—22no—
227#217—22no—
228#217—22no—
229#217—22no—
230#217—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—41no—
239C4⋊Dic3—41no—
240C4⋊Dic3C2×D412yes—
241Dic3⋊C4—22no—
242Dic3⋊C4—22no—
243Dic3⋊C4—22no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251C4×Dic3—22no—
252C4×Dic3—22no—
253C4×Dic3—22no—
254#208—42no—
255C2×C3⋊C8—22no—
256C2×#41D1211yescharacteristic
257C22×Q8—31no—
258C22×Q8—31no—
259C22×Q8—61no—
260C23×C4—61no—
261C23×C4C22×D311yescharacteristic
262C2×M16—61no—
263C22×C8—21no—
264C4⋊Q8—61no—
265C4⋊Q8—62no—
266C4⋊Q8—62no—
267C4⋊Q8—62no—
268#126—62no—
269#126—62no—
270#126—62no—
271#126—62no—
272#125—61no—
273#125—62no—
274#125—62no—
275#125—62no—
276#125—62no—
277#124—62no—
278#123—21no—
279#123—62no—
280#123—62no—
281#123—62no—
282#123—62no—
283C22⋊Q8—32no—
284C22⋊Q8—32no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C22⋊Q8—62no—
288C22⋊Q8—62no—
289C22⋊Q8—61no—
290C22⋊Q8—61no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297C22⋊Q8D1211yescharacteristic
298C22⋊Q8C22×D311yescharacteristic
299#121D1211yescharacteristic
300#121C22×D311yescharacteristic
301#120—21no—
302C4×Q8—61no—
303C4×Q8—61no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—62no—
307C4×D4—62no—
308#117—62no—
309#117—62no—
310#117—61no—
311#117D1211yescharacteristic
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—61no—
325C2×C42—21no—
326#107—64no—
327#107—62no—
328#106—62no—
329#103—64no—
330#103—62no—
331#102—62no—
332C22⋊C8—21no—
333C22⋊C8—32no—
334C22⋊C8—61no—
335C22⋊C8C22×D312yes—
336#95—64no—
337#95—64no—
338#95C22×D312yes—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—41no—
343C23×C3#12011yescharacteristic
344C23×C3#12011yescharacteristic
345C23×C3C22×D411yescharacteristic
346C3×Q8—41no—
347C3×D4—41no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—81no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—44no—
376C2×Dic3—42no—
377C2×Dic3—81no—
378C2×Dic3—82no—
379C2×Dic3—82no—
380C2×Dic3—82no—
381Dic6—42no—
382Dic6—42no—
383Dic6—42no—
384Dic6—81no—
385Dic6—82no—
386C3×C8—42no—
387C3⋊C8—42no—
388C24D3×D411yescharacteristic
389#41—42no—
390C2×Q8—32no—
391C2×Q8—62no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—122no—
396C2×Q8—122no—
397C2×Q8—242no—
398C2×Q8C2×D1211yescharacteristic
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C2×D1211yescharacteristic
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—42no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—124no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421M16—122no—
422C2×C8—22no—
423C2×C8—42no—
424C2×C8—62no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—32no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4C2×D1211yescharacteristic
448C4⋊C4C2×D1211yescharacteristic
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—42no—
455C22⋊C4—62no—
456C22⋊C4—62no—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—62no—
462C42—21no—
463C42—42no—
464C42—62no—
465C42—62no—
466C42—62no—
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—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—82no—
488C3×C4#45812yes—
489Dic3—42no—
490Dic3—42no—
491Dic3—42no—
492Dic3—42no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—41no—
498C23#88211yescharacteristic
499C23#88211yescharacteristic
500C23C2×D3×D411yescharacteristic
501Q8—41no—
502Q8—122no—
503Q8—122no—
504Q8—122no—
505Q8—241no—
506Q8—242no—
507D4—41no—
508D4—42no—
509D4—42no—
510D4—81no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—64no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—124no—
532C2×C4—122no—
533C2×C4—241no—
534C2×C4—242no—
535C2×C4—242no—
536C2×C4—242no—
537C2×C4#88211yescharacteristic
538C2×C4#88211yescharacteristic
539C2×C4C22×D1211yescharacteristic
540C2×C4C2×D3×D411yescharacteristic
541C8—42no—
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#482712yes—
562C22C2×#88211yescharacteristic
563C22#577112yes—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—82no—
571C4—122no—
572C4—122no—
573C4—122no—
574C4—122no—
575C4—242no—
576C4#474212yes—
577C3#1826911yescharacteristic, complemented, Sylow
578C2—22no—
579C2—22no—
580C2—41no—
581C2—81no—
582C2C2×#474211yescharacteristic
583C2#7883711yescharacteristic
584C2C2×#482711yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1501C2
socleC22×C3C2×#120
prime radicalC3×#525C22
hypercenter#1501D3
hypocenterC3#18269
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
3. C2×#4722C2semidirect
4. #73518C2semidirect
6. #69923C2semidirect
7. #69445C2semidirect
8. #68331C2semidirect
9. #68298C2semidirect
577. C3#18269semidirect

Maximal subgroups

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

subgroupindex
#860312
#829142
C2×#47222
#735182
C3×#15012
#699232
#694452
#683312
#682982
#182693