Finite Group Databasebeta

← #10778317

Subgroups — #10778317

Order 768.

638 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#81891C211yesmaximal, characteristic, complemented
2#81070C211yesmaximal, characteristic
3#80438C211yesmaximal, characteristic
4#79538C211yesmaximal, characteristic
5C3×#1691C211yesmaximal, characteristic, complemented
6#71345C211yesmaximal, characteristic, complemented
7#71133C211yesmaximal, characteristic, complemented
8#71037C211yesmaximal, characteristic, complemented
9#70872C211yesmaximal, characteristic, complemented
10#68583C211yesmaximal, characteristic, complemented
11D3×#335C211yesmaximal, characteristic, complemented
12#68497C211yesmaximal, characteristic, complemented
13#68484C211yesmaximal, characteristic, complemented
14#68443C211yesmaximal, characteristic, complemented
15#68423C211yesmaximal, characteristic, complemented
16#19195—31nomaximal, Sylow
17C2×#907—21no—
18#5728C2211yescharacteristic
19#5728C2211yescharacteristic
20#5548C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#461C2211yescharacteristic
24C3×#436C2211yescharacteristic
25#5171—21no—
26#5163—21no—
27#5128—21no—
28#5084C2211yescharacteristic
29#5083C2211yescharacteristic
30#5078C2211yescharacteristic
31#5078C2211yescharacteristic
32#5019C2211yescharacteristic
33D4×C3⋊C8C2211yescharacteristic
34#5008C2211yescharacteristic
35#4864—21no—
36#4860—21no—
37#4860—21no—
38#4858—21no—
39#4858—21no—
40#4854—21no—
41#4842C2211yescharacteristic
42#4841C2211yescharacteristic
43#4837C2211yescharacteristic
44#4834C2211yescharacteristic
45D3×C4⋊C8C2211yescharacteristic
46D3×C4⋊C8C2211yescharacteristic
47#4807—21no—
48#4791—21no—
49#4778—21no—
50#4762—21no—
51#4727—21no—
52#67⋊C8—21no—
53#4688C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55#4680C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4546C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4479C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3246—31no—
69#3246—31no—
70#3160—31no—
71#2931—31no—
72#1691—31no—
73#1691—31no—
74#1691—31no—
75#1691D311yescharacteristic, complemented
76#1675—31no—
77#1675—31no—
78#1673—31no—
79#1673—31no—
80C2×#335—31no—
81#1550—31no—
82#1550—31no—
83C2×#235—21no—
84C2×C3×SD16—21no—
85C2×C3×M16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#946—21no—
90#946—21no—
91Q8×Dic3—21no—
92Q8×Dic3—21no—
93C2×#215—21no—
94#940D411yescharacteristic
95#938D411yescharacteristic
96C2×#214—21no—
97C4×#67—21no—
98C22×C3⋊C8—21no—
99#907—44no—
100C2×#204—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×D3×C8—21no—
104C2×D3×C8—21no—
105C2×D3×C8—21no—
106C2×D3×C8D411yescharacteristic
107C2×D3×C8D411yescharacteristic
108#897—21no—
109#897—21no—
110#895D411yescharacteristic
111#892—21no—
112#892—21no—
113#890—21no—
114#890—21no—
115#888D411yescharacteristic
116#887—21no—
117#887—21no—
118#885—21no—
119#884—21no—
120C4×D12C2311yescharacteristic
121C42×D3C2311yescharacteristic
122#869C2311yescharacteristic
123#869C2311yescharacteristic
124C3×#106—21no—
125C3×#106—21no—
126C3×#106—21no—
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×C4⋊C8C2311yescharacteristic
130C3×#103—21no—
131C3×#102—21no—
132C3×C22⋊C8—21no—
133C3×C4×C8C2311yescharacteristic
134#835—21no—
135#832—21no—
136#830—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#817—21no—
141#817—21no—
142#817—21no—
143#816—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147C8×Dic3D411yescharacteristic
148C8×Dic3D411yescharacteristic
149#810—21no—
150#809—21no—
151#808—21no—
152#808—21no—
153#807—21no—
154#807—21no—
155#807—21no—
156#807—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#131—61no—
162#533—31no—
163#533—31no—
164#532—31no—
165#500—31no—
166#500—31no—
167#499—61no—
168#499—61no—
169#498—31no—
170#498C2×D311yescharacteristic
171#476—31no—
172#476—31no—
173#476—31no—
174#476C2×D311yescharacteristic
175#475—61no—
176#475—61no—
177#465—61no—
178#465—61no—
179#465—61no—
180#465—61no—
181#461—31no—
182#461—31no—
183#461—31no—
184#461C2×D311yescharacteristic
185#460—61no—
186#460—61no—
187#436—31no—
188#436—61no—
189#436C2×D311yescharacteristic
190C8×D4—31no—
191C8×D4—31no—
192C8×D4—61no—
193#427—61no—
194#427—61no—
195#423—31no—
196C2×C4⋊C8—31no—
197C2×C4⋊C8—31no—
198#408—61no—
199C2×C4×C8—31no—
200#335—31no—
201#335—31no—
202#335—31no—
203#335C2×D311yescharacteristic
204#333—31no—
205#333—31no—
206#333—31no—
207#333C2×D311yescharacteristic
208#332—31no—
209#332—31no—
210#332—31no—
211#332C2×D311yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×D4C2×D411yescharacteristic
215C22×C3×C4—21no—
216C2×#67—21no—
217C2×#67—21no—
218C2×#67—21no—
219#235—42no—
220#235—42no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3#4111yescharacteristic
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C3×SD16—81no—
231C3×M16—42no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C42C2411yescharacteristic
244#217—21no—
245#215—81no—
246#214—81no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—41no—
250(C2×D3)⋊C4—41no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—41no—
262C4×Dic3—41no—
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—42no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272#204—81no—
273ΓL(1,𝔽25)—42no—
274ΓL(1,𝔽25)—42no—
275D3×C8—22no—
276D3×C8—42no—
277D3×C8—81no—
278D3×C8—81no—
279D3×C8SD1614yescomplemented
280C2×#41—61no—
281C2×SD16—21no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×SD16—61no—
285#131—124no—
286C2×M16—21no—
287C2×M16—61no—
288C2×M16—61no—
289C22×C8—31no—
290C22×C8—31no—
291C22×C8—61no—
292C22×C8—61no—
293C22×C8—61no—
294C22×C8—61no—
295C4⋊Q8—31no—
296C4⋊Q8—31no—
297C4⋊Q8—31no—
298C4⋊Q8C22×D311yescharacteristic
299C4⋊Q8C22×D311yescharacteristic
300#127—31no—
301#125—61no—
302#125—61no—
303#124—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306C22⋊Q8—61no—
307C22⋊Q8—61no—
308#121—31no—
309#121—31no—
310#121—61no—
311C4×Q8—61no—
312C4×Q8—61no—
313C4×Q8—61no—
314C4×Q8—61no—
315C4×D4—31no—
316C4×D4—61no—
317C4×D4C22×D311yescharacteristic
318#117—61no—
319#117—61no—
320C2×C42—31no—
321#107—61no—
322#107—61no—
323#107—61no—
324#107—61no—
325#106—21no—
326#106—21no—
327#106—21no—
328#106—61no—
329#106—61no—
330#106—61no—
331#106—61no—
332#106—61no—
333C4⋊C8—31no—
334C4⋊C8—31no—
335C4⋊C8—31no—
336C4⋊C8—61no—
337C4⋊C8—61no—
338C4⋊C8—61no—
339C4⋊C8C22×D311yescharacteristic
340C4⋊C8C22×D311yescharacteristic
341C4⋊C8C22×D311yescharacteristic
342#103—21no—
343#103—61no—
344#103—61no—
345#103—61no—
346#102—21no—
347#102—61no—
348#102—61no—
349#102—61no—
350C22⋊C8—21no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353C22⋊C8—61no—
354C4×C8—31no—
355C4×C8—31no—
356C4×C8—31no—
357C4×C8C22×D311yescharacteristic
358C23×C3—21no—
359C22×D3—21no—
360C22×D3#12111yescharacteristic
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—81no—
364C3×D4—21no—
365C3×D4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—42no—
375#67—42no—
376#67—42no—
377C2×Dic3—21no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3#12111yescharacteristic
381C2×Dic3#12111yescharacteristic
382C2×Dic3#12111yescharacteristic
383D12—21no—
384D12—21no—
385D12—42no—
386C4×D3—21no—
387C4×D3—21no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3C2×SD1612yes—
392C4×D3#13712yes—
393Dic6—42no—
394Dic6—41no—
395Dic6—41no—
396Dic6—81no—
397C3×C8—21no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8C2×SD1612yes—
402C3⋊C8—21no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8C2×SD1612yes—
407#41—122no—
408#41—122no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×D4—31no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4D3×D411yescharacteristic
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—61no—
425SD16—81no—
426SD16—241no—
427SD16—241no—
428SD16—241no—
429M16—42no—
430M16—122no—
431M16—122no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—31no—
436C2×C8—34no—
437C2×C8—31no—
438C2×C8—62no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—122no—
443C2×C8—122no—
444C2×C8—241no—
445C2×C8—241no—
446C2×C8D3×D411yescharacteristic
447C2×C8D3×D411yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—31no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—121no—
459C4⋊C4—121no—
460C4⋊C4—121no—
461C4⋊C4—121no—
462C4⋊C4D3×D411yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—121no—
468C22⋊C4—121no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—121no—
473C42—121no—
474C42C23×D311yescharacteristic
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3C2×#12111yescharacteristic
479C2×D3—21no—
480C2×D3—21no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3#46112yes—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4C22×SD1611yescharacteristic
490C3×C4C2×#13711yescharacteristic
491C3×C4#57611yescharacteristic
492C3×C4#57811yescharacteristic
493Dic3—21no—
494Dic3—21no—
495Dic3—41no—
496Dic3—81no—
497Dic3—81no—
498Dic3#46112yes—
499C23—21no—
500C23—31no—
501C23—61no—
502Q8—41no—
503Q8—41no—
504Q8—81no—
505Q8—122no—
506Q8—121no—
507Q8—121no—
508Q8—241no—
509D4—21no—
510D4—21no—
511D4—61no—
512D4—61no—
513D4—122no—
514D4—122no—
515D4—122no—
516D4—122no—
517C2×C4—21no—
518C2×C4—31no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—31no—
522C2×C4—31no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4D3×#4111yescharacteristic
538C8—21no—
539C8—32no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—61no—
544C8—121no—
545C8—121no—
546C8—121no—
547C8D3×SD1612yes—
548C2×C3—81no—
549C2×C3C2×#46111yescharacteristic
550C2×C3#306111yescharacteristic
551C2×C3#306711yescharacteristic
552D3—22no—
553D3—81no—
554C22—32no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—61no—
559C22—61no—
560C22—121no—
561C22—121no—
562C22—121no—
563C22D3×#12111yescharacteristic
564C4—21no—
565C4—32no—
566C4—41no—
567C4—61no—
568C4—61no—
569C4—81no—
570C4—81no—
571C4—121no—
572C4—241no—
573C4—241no—
574C4C2×D3×SD1611yescharacteristic
575C4#577411yescharacteristic
576C4D3×#13711yescharacteristic
577C4#577911yescharacteristic
578C3#1919511yescharacteristic, complemented, Sylow
579C2—62no—
580C2—81no—
581C2—241no—
582C2D3×#46111yescharacteristic
583C2#8080811yescharacteristic
584C2#8092911yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1691C2
socleC22×C3C2×#121
prime radical#5548C22
hypercenter#1691D3
hypocenterC3#19195
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1691

Complements

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

NH ≅ G/Ntype
1. #81891C2semidirect
5. C3×#1691C2semidirect
6. #71345C2semidirect
7. #71133C2semidirect
8. #71037C2semidirect
9. #70872C2semidirect
10. #68583C2semidirect
11. D3×#335C2semidirect
12. #68497C2semidirect
13. #68484C2semidirect
14. #68443C2semidirect
15. #68423C2semidirect
75. #1691D3semidirect
279. D3×C8SD16semidirect
578. C3#19195semidirect

Maximal subgroups

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

subgroupindex
#818912
#810702
#804382
#795382
C3×#16912
#713452
#711332
#710372
#708722
#685832
D3×#3352
#684972
#684842
#684432
#684232
#191953