Finite Group Databasebeta

← #10779147

Subgroups — #10779147

Order 768.

670 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
1D3×#487C211yesmaximal, characteristic, complemented
2#81012C211yesmaximal, characteristic
3#80477C211yesmaximal, characteristic
4C8×#239C211yesmaximal, characteristic
5C3×#1680C211yesmaximal, characteristic, complemented
6#71232C211yesmaximal, characteristic, complemented
7#71118C211yesmaximal, characteristic, complemented
8#71054C211yesmaximal, characteristic, complemented
9#70946C211yesmaximal, characteristic, complemented
10#68539C211yesmaximal, characteristic, complemented
11#68529C211yesmaximal, characteristic, complemented
12#68445C211yesmaximal, characteristic, complemented
13#68444C211yesmaximal, characteristic, complemented
14#68422C211yesmaximal, characteristic, complemented
15#68420C211yesmaximal, characteristic, complemented
16#19275—31nomaximal, Sylow
17C2×#918—21no—
18D3×C4⋊Q8C2211yescharacteristic
19D3×C4⋊Q8C2211yescharacteristic
20C4×#239C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×C8×Q8C2211yescharacteristic
25#5182—21no—
26#5093C2211yescharacteristic
27#5087C2211yescharacteristic
28#5083C2211yescharacteristic
29#5079C2211yescharacteristic
30#5031C2211yescharacteristic
31Q8×C3⋊C8C2211yescharacteristic
32#5020C2211yescharacteristic
33#4870—21no—
34#4861—21no—
35#4846C2211yescharacteristic
36#4841C2211yescharacteristic
37#4837C2211yescharacteristic
38#4837C2211yescharacteristic
39#4834—21no—
40#4834C2211yescharacteristic
41#4833—21no—
42#4833C2211yescharacteristic
43#4818—21no—
44#4803—21no—
45#4802—21no—
46D3×#103—21no—
47D3×#103—21no—
48#4793—21no—
49#4792—21no—
50#4792—21no—
51C8×D12—21no—
52C8×D12C2211yescharacteristic
53C4×D3×C8—21no—
54C4×D3×C8C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4480C2211yescharacteristic
67#4457C2211yescharacteristic
68C2×#487—31no—
69#3079—31no—
70#3039—31no—
71C8×#41—31no—
72#1680—31no—
73#1680—31no—
74#1680—31no—
75#1680D311yescharacteristic, complemented
76#1679—31no—
77#1679—31no—
78#1679—31no—
79#1679—31no—
80#1566—31no—
81#1549—31no—
82#1546—31no—
83C2×#239D411yescharacteristic
84C2×D3×Q8—21no—
85C2×D3×Q8—21no—
86C2×C3×Q16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4×Q8C2311yescharacteristic
90Q8×Dic3D411yescharacteristic
91#944—21no—
92#944—21no—
93C2×#216—21no—
94C2×#215—21no—
95#918—44no—
96C2×#204—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8D411yescharacteristic
102C2×D3×C8D411yescharacteristic
103#893—21no—
104#893D411yescharacteristic
105#892—21no—
106#892D411yescharacteristic
107D3×C4⋊C4—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—21no—
110D3×C4⋊C4—21no—
111#888—21no—
112#888—21no—
113#888—21no—
114#888—21no—
115C4×D12—21no—
116C4×D12C2311yescharacteristic
117C42×D3—21no—
118C42×D3C2311yescharacteristic
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C3×#106—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#103—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#103—21no—
130C3×C4×C8—21no—
131C3×C4×C8C2311yescharacteristic
132#835—21no—
133#835—21no—
134#835—21no—
135#835—21no—
136#821—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#816—21no—
142#816—21no—
143#816—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146C8×Dic3—21no—
147C8×Dic3D411yescharacteristic
148C8×Dic3D411yescharacteristic
149#810—21no—
150#810—21no—
151#810—21no—
152#809—21no—
153#808—21no—
154#807—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160C4×C3⋊C8C2311yescharacteristic
161C2×#137—61no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C4×#41—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—61no—
169#487—61no—
170#487C2×D311yescharacteristic
171#477—61no—
172#476—61no—
173#475—31no—
174#475C2×D311yescharacteristic
175#474—31no—
176#474—31no—
177#462—31no—
178#462—61no—
179#462C2×D311yescharacteristic
180#461—31no—
181#461—31no—
182#461—61no—
183C8×Q8—31no—
184C8×Q8C2×D311yescharacteristic
185C8×D4—31no—
186C8×D4—31no—
187C8×D4—61no—
188C8×D4—61no—
189#431—31no—
190#431—61no—
191#428—61no—
192#423—31no—
193#423—31no—
194#419—61no—
195#417—61no—
196C2×#103—61no—
197C2×#103—61no—
198C2×C4×C8—31no—
199C2×C4×C8—61no—
200#334—31no—
201#334—31no—
202#334—31no—
203#334C2×D311yescharacteristic
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332C2×D311yescharacteristic
211#332C2×D311yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215#239—22no—
216#239SD1614yes—
217D3×Q8—42no—
218D3×Q8—42no—
219D3×Q8—42no—
220D3×Q8—42no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—41no—
227C2×C4×D3—41no—
228C2×C4×D3C2×D411yescharacteristic
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3#4111yescharacteristic
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—41no—
234C2×Dic6—41no—
235C3×Q16—42no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C42—21no—
248C3×C42C2411yescharacteristic
249#216—42no—
250#215—42no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263Dic3⋊C4—41no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3#4111yescharacteristic
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274#204—42no—
275ΓL(1,𝔽25)—42no—
276ΓL(1,𝔽25)—42no—
277D3×C8—41no—
278D3×C8—41no—
279D3×C8—42no—
280D3×C8—42no—
281C2×#41—31no—
282C22×Q8—61no—
283C22×Q8—61no—
284#137—124no—
285C2×Q16—21no—
286C2×Q16—61no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×M16—61no—
290C2×M16—61no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8—61no—
294C22×C8—61no—
295C4⋊Q8—31no—
296C4⋊Q8—31no—
297C4⋊Q8—61no—
298C4⋊Q8—61no—
299C4⋊Q8—61no—
300C4⋊Q8—61no—
301C4⋊Q8—61no—
302C4⋊Q8—61no—
303C4⋊Q8C22×D311yescharacteristic
304C4⋊Q8C22×D311yescharacteristic
305C4×Q8—31no—
306C4×Q8C22×D311yescharacteristic
307C4×D4—31no—
308C4×D4—31no—
309C4×D4—61no—
310C4×D4—61no—
311#117—31no—
312#117—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C42—31no—
318C2×C42—61no—
319#107—61no—
320#107—61no—
321#106—21no—
322#106—61no—
323C4⋊C8—21no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—31no—
327C4⋊C8—61no—
328C4⋊C8—61no—
329C4⋊C8—61no—
330C4⋊C8C22×D311yescharacteristic
331C4⋊C8C22×D311yescharacteristic
332C4⋊C8C22×D311yescharacteristic
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#103—61no—
344#103—61no—
345#103—61no—
346#103—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352C4×C8—21no—
353C4×C8—31no—
354C4×C8—31no—
355C4×C8—31no—
356C4×C8—61no—
357C4×C8—61no—
358C4×C8C22×D311yescharacteristic
359C22×D3—21no—
360C22×D3#12111yescharacteristic
361C3×Q8—21no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8C2×SD1612yes—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3#12111yescharacteristic
378C2×Dic3#12111yescharacteristic
379C2×Dic3#12111yescharacteristic
380D12—24no—
381D12—21no—
382D12C2×SD1612yes—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—24no—
386C4×D3—22no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—42no—
390C4×D3—82no—
391C4×D3—82no—
392C4×D3C2×SD1612yes—
393C4×D3#13512yes—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—41no—
398Dic6—82no—
399Dic6—82no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410#41—34no—
411#41—62no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—121no—
417C2×Q8—121no—
418C2×Q8—122no—
419C2×Q8—122no—
420C2×Q8—122no—
421C2×Q8—122no—
422C2×Q8D3×D411yescharacteristic
423C2×D4—31no—
424C2×D4—61no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—121no—
432C22×C4—121no—
433Q16—42no—
434Q16—122no—
435SD16—122no—
436SD16—122no—
437M16—122no—
438M16—122no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—31no—
443C2×C8—31no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—121no—
448C2×C8—121no—
449C2×C8—122no—
450C2×C8—122no—
451C2×C8D3×D411yescharacteristic
452C2×C8D3×D411yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—31no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C4⋊C4—121no—
467C4⋊C4—121no—
468C4⋊C4D3×D411yescharacteristic
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C42—21no—
473C42—31no—
474C42—31no—
475C42—31no—
476C42—61no—
477C42—61no—
478C42C23×D311yescharacteristic
479C22×C3C2×#12111yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—22no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3#46512yes—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4C22×SD1611yescharacteristic
492C3×C4C2×#13511yescharacteristic
493C3×C4#57511yescharacteristic
494C3×C4#57511yescharacteristic
495Dic3—21no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499Dic3#46512yes—
500Dic3#47312yes—
501C23—31no—
502C23—61no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—121no—
509Q8—121no—
510Q8—121no—
511Q8—121no—
512Q8—242no—
513Q8—242no—
514Q8D3×SD1612yes—
515D4—32no—
516D4—64no—
517D4—61no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—31no—
521C2×C4—31no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—31no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—62no—
528C2×C4—64no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—242no—
538C2×C4—242no—
539C2×C4C2×D3×D411yescharacteristic
540C2×C4C2×D3×D411yescharacteristic
541C2×C4D3×#4111yescharacteristic
542C8—21no—
543C8—21no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—61no—
548C8—61no—
549C8—121no—
550C8—121no—
551C8—121no—
552C2×C3#304011yescharacteristic
553C2×C3C2×#46511yescharacteristic
554C2×C3C2×#47311yescharacteristic
555D3—22no—
556D3—42no—
557C22—32no—
558C22—61no—
559C22—61no—
560C22—62no—
561C22—121no—
562C22—121no—
563C22D3×#12111yescharacteristic
564C4—22no—
565C4—21no—
566C4—32no—
567C4—32no—
568C4—41no—
569C4—61no—
570C4—81no—
571C4—81no—
572C4—121no—
573C4—241no—
574C4—241no—
575C4C2×D3×SD1611yescharacteristic
576C4D3×#13511yescharacteristic
577C4#576811yescharacteristic
578C4#577711yescharacteristic
579C3#1927511yescharacteristic, complemented, Sylow
580C2—62no—
581C2—122no—
582C2#8059911yescharacteristic
583C2D3×#46511yescharacteristic
584C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1680C2
socleC22×C3C2×#121
prime radicalC2×#239D4
hypercenter#1680D3
hypocenterC3#19275
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1680

Complements

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

NH ≅ G/Ntype
1. D3×#487C2semidirect
5. C3×#1680C2semidirect
6. #71232C2semidirect
7. #71118C2semidirect
8. #71054C2semidirect
9. #70946C2semidirect
10. #68539C2semidirect
11. #68529C2semidirect
12. #68445C2semidirect
13. #68444C2semidirect
14. #68422C2semidirect
15. #68420C2semidirect
75. #1680D3semidirect
579. C3#19275semidirect

Maximal subgroups

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

subgroupindex
D3×#4872
#810122
#804772
C8×#2392
C3×#16802
#712322
#711182
#710542
#709462
#685392
#685292
#684452
#684442
#684222
#684202
#192753