Finite Group Databasebeta

← #10780337

Subgroups — #10780337

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81565C211yesmaximal, characteristic, complemented
2#80483C211yesmaximal, characteristic, complemented
3#80402C211yesmaximal, characteristic
4#79942C211yesmaximal, characteristic, complemented
5C3×#1702C211yesmaximal, characteristic, complemented
6#71201C211yesmaximal, characteristic, complemented
7#71054C211yesmaximal, characteristic, complemented
8#71023C211yesmaximal, characteristic, complemented
9#70966C211yesmaximal, characteristic, complemented
10#68539C211yesmaximal, characteristic, complemented
11#68531C211yesmaximal, characteristic, complemented
12#68453C211yesmaximal, characteristic, complemented
13#68420C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68386C211yesmaximal, characteristic, complemented
16#19383—31nomaximal, Sylow
17D3×C4⋊Q8C2211yescharacteristic
18#5716C2211yescharacteristic
19#5580C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#462C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×#446C2211yescharacteristic
24#5190—21no—
25#5186—21no—
26#5174—21no—
27#5165—21no—
28#5093C2211yescharacteristic
29#5083C2211yescharacteristic
30#5072C2211yescharacteristic
31#5070C2211yescharacteristic
32#5031C2211yescharacteristic
33Q8×C3⋊C8C2211yescharacteristic
34#5020C2211yescharacteristic
35#4846C2211yescharacteristic
36#4841C2211yescharacteristic
37#4840—21no—
38#4838—21no—
39#4838C2211yescharacteristic
40#4837C2211yescharacteristic
41#4835C2211yescharacteristic
42#4833C2211yescharacteristic
43#4806—21no—
44#4797—21no—
45#4782—21no—
46#4771—21no—
47#4771—21no—
48#4766—21no—
49#4765—21no—
50#4765—21no—
51#4708—21no—
52#4706—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54C8×D12C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4547C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3251—31no—
69#3218—31no—
70#3141—31no—
71#2969—31no—
72#1702D311yescharacteristic, complemented
73#1701—31no—
74#1700—31no—
75#1699—31no—
76#1682—31no—
77#1681—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1553—31no—
82#1546—31no—
83C2×D3×Q8—21no—
84C2×#237—21no—
85C2×C3×Q16—21no—
86C2×C3×SD16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2311yescharacteristic
89C3×C4×Q8C2311yescharacteristic
90#947D411yescharacteristic
91#946D411yescharacteristic
92#944—21no—
93C2×#216—21no—
94C2×#215—21no—
95#938—21no—
96#938—21no—
97#936—21no—
98D4×Dic3—21no—
99D4×Dic3—21no—
100C2×#213—21no—
101C2×#204—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×ΓL(1,𝔽25)D411yescharacteristic
105C2×ΓL(1,𝔽25)D411yescharacteristic
106C2×D3×C8—21no—
107#898—21no—
108#895D411yescharacteristic
109#894—21no—
110D3×C4⋊C4—21no—
111D3×C4⋊C4—21no—
112#890D411yescharacteristic
113#888—21no—
114#888—21no—
115C4×D12—21no—
116C4×D12C2311yescharacteristic
117#872—21no—
118C42×D3C2311yescharacteristic
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×#102—21no—
129C3×#97—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#821—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#818—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#815—21no—
146#815D411yescharacteristic
147#815D411yescharacteristic
148Dic3⋊C8—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#808—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#497—61no—
164#491—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486C2×D311yescharacteristic
171#478—61no—
172#475—31no—
173#474—31no—
174#469—61no—
175#468—61no—
176#463—61no—
177#462—31no—
178#462C2×D311yescharacteristic
179#461—31no—
180#461—31no—
181#460C2×D311yescharacteristic
182#459—31no—
183#451—61no—
184#448—61no—
185#446C2×D311yescharacteristic
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423—31no—
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332C2×D311yescharacteristic
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×#67—41no—
217C22×Dic3—41no—
218D3×Q8—81no—
219D3×Q8—81no—
220#237—42no—
221#237—42no—
222C2×D12—21no—
223C2×D12C2×D411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—41no—
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—41no—
232C3×Q16—81no—
233C3×SD16—81no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C42—21no—
244C3×C42C2411yescharacteristic
245#217—41no—
246#217—41no—
247#216—81no—
248#215—81no—
249#213—81no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3#4111yescharacteristic
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273#204—81no—
274ΓL(1,𝔽25)—41no—
275ΓL(1,𝔽25)—41no—
276ΓL(1,𝔽25)—42no—
277ΓL(1,𝔽25)—81no—
278D3×C8—42no—
279C2×#41—61no—
280C22×Q8—61no—
281C2×Q16—21no—
282C2×Q16—61no—
283C2×SD16—21no—
284C2×SD16—61no—
285C2×SD16—61no—
286C2×D8—61no—
287C2×M16—31no—
288C2×M16—31no—
289C2×M16—61no—
290C2×M16—61no—
291C22×C8—61no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—61no—
295C4⋊Q8—61no—
296C4⋊Q8—61no—
297C4⋊Q8C22×D311yescharacteristic
298#127C22×D311yescharacteristic
299#126—61no—
300#125—31no—
301#124—31no—
302#124—61no—
303#123—61no—
304C22⋊Q8—31no—
305#121—31no—
306#121—61no—
307#121—61no—
308C4×Q8C22×D311yescharacteristic
309C4×D4—31no—
310C4×D4—61no—
311C4×D4—61no—
312C4×D4—61no—
313#117—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C42—31no—
317#107—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—21no—
339#102—21no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—21no—
349#97—31no—
350#97—31no—
351#97—61no—
352C4×C8—31no—
353C4×C8—61no—
354C4×C8—61no—
355C4×C8C22×D311yescharacteristic
356C23×C3—41no—
357C22×D3—21no—
358C22×D3#12111yescharacteristic
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—81no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—82no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—82no—
379C2×Dic3#12111yescharacteristic
380C2×Dic3#12111yescharacteristic
381C2×Dic3#12111yescharacteristic
382D12—21no—
383D12—21no—
384D12—81no—
385C4×D3—21no—
386C4×D3—21no—
387C4×D3—21no—
388C4×D3—21no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—82no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—82no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—122no—
409#41—122no—
410C2×Q8—21no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—121no—
414C2×Q8—241no—
415C2×Q8—241no—
416C2×Q8D3×D411yescharacteristic
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—31no—
421C2×D4—61no—
422C2×D4—121no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
427C22×C4—121no—
428C22×C4—121no—
429Q16—81no—
430Q16—241no—
431SD16—81no—
432SD16—241no—
433SD16—241no—
434D8—241no—
435M16—121no—
436M16—121no—
437M16—122no—
438M16—241no—
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—122no—
448C2×C8D3×D411yescharacteristic
449C2×C8D3×D411yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—31no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—121no—
463C4⋊C4D3×D411yescharacteristic
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C42—21no—
472C42—31no—
473C42—31no—
474C42—31no—
475C42C23×D311yescharacteristic
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3C2×#12111yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—21no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C3×C4—21no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4#57511yescharacteristic
491C3×C4#57511yescharacteristic
492C3×C4#57611yescharacteristic
493C3×C4#57711yescharacteristic
494Dic3—21no—
495Dic3—21no—
496Dic3—21no—
497Dic3—41no—
498Dic3—81no—
499Dic3—81no—
500C23—31no—
501C23—41no—
502C23—61no—
503Q8—21no—
504Q8—21no—
505Q8—41no—
506Q8—41no—
507Q8—121no—
508Q8—121no—
509Q8—121no—
510Q8—121no—
511Q8—242no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—61no—
517D4—61no—
518D4—81no—
519D4—241no—
520D4—242no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—31no—
524C2×C4—31no—
525C2×C4—31no—
526C2×C4—41no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—121no—
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—161no—
553C2×C3#304011yescharacteristic
554C2×C3#306211yescharacteristic
555C2×C3#306911yescharacteristic
556D3—41no—
557D3—81no—
558C22—61no—
559C22—61no—
560C22—61no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—121no—
565C22—121no—
566C22—121no—
567C22D3×#12111yescharacteristic
568C4—21no—
569C4—41no—
570C4—41no—
571C4—61no—
572C4—61no—
573C4—61no—
574C4—81no—
575C4—121no—
576C4—241no—
577C4—241no—
578C4#576811yescharacteristic
579C4#577411yescharacteristic
580C4#577511yescharacteristic
581C4#577711yescharacteristic
582C3#1938311yescharacteristic, complemented, Sylow
583C2—121no—
584C2—161no—
585C2—241no—
586C2#8059911yescharacteristic
587C2#8086711yescharacteristic
588C2#8093111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1702C2
socleC22×C3C2×#121
prime radical#80402C2
hypercenter#1702D3
hypocenterC3#19383
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1702

Complements

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

NH ≅ G/Ntype
1. #81565C2semidirect
2. #80483C2semidirect
4. #79942C2semidirect
5. C3×#1702C2semidirect
6. #71201C2semidirect
7. #71054C2semidirect
8. #71023C2semidirect
9. #70966C2semidirect
10. #68539C2semidirect
11. #68531C2semidirect
12. #68453C2semidirect
13. #68420C2semidirect
14. #68401C2semidirect
15. #68386C2semidirect
72. #1702D3semidirect
582. C3#19383semidirect

Maximal subgroups

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

subgroupindex
#815652
#804832
#804022
#799422
C3×#17022
#712012
#710542
#710232
#709662
#685392
#685312
#684532
#684202
#684012
#683862
#193833