Finite Group Databasebeta

← #10778222

Subgroups — #10778222

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81888C211yesmaximal, characteristic, complemented
2#81093C211yesmaximal, characteristic, complemented
3#80475C211yesmaximal, characteristic, complemented
4#79905C211yesmaximal, characteristic, complemented
5C3×#1672C211yesmaximal, characteristic, complemented
6#71345C211yesmaximal, characteristic, complemented
7#71150C211yesmaximal, characteristic, complemented
8#71055C211yesmaximal, characteristic, complemented
9#70939C211yesmaximal, characteristic, complemented
10#68540C211yesmaximal, characteristic, complemented
11#68528C211yesmaximal, characteristic, complemented
12#68502C211yesmaximal, characteristic, complemented
13#68485C211yesmaximal, characteristic, complemented
14#68454C211yesmaximal, characteristic, complemented
15#68426C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C2×#919—21no—
18#5725C2211yescharacteristic
19#5725C2211yescharacteristic
20C4×#239C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#477C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×C8×Q8C2211yescharacteristic
25#5185—21no—
26#5088C2211yescharacteristic
27#5088C2211yescharacteristic
28#5084C2211yescharacteristic
29#5083C2211yescharacteristic
30#5027C2211yescharacteristic
31#5027C2211yescharacteristic
32Q8×C3⋊C8C2211yescharacteristic
33#4885—21no—
34#4872—21no—
35#4869—21no—
36#4869—21no—
37#4865—21no—
38#4865—21no—
39#4843C2211yescharacteristic
40#4843C2211yescharacteristic
41#4838—21no—
42#4838C2211yescharacteristic
43#4835—21no—
44#4835C2211yescharacteristic
45#4833C2211yescharacteristic
46D3×C4⋊C8C2211yescharacteristic
47#4819—21no—
48#4810—21no—
49#4804—21no—
50#4788—21no—
51#4695C2211yescharacteristic
52#4688—21no—
53#4688C2211yescharacteristic
54C4×ΓL(1,𝔽25)—21no—
55C4×ΓL(1,𝔽25)C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4546C2211yescharacteristic
61#4485C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4485C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3184—31no—
69#3106—31no—
70#3035—31no—
71#2971—31no—
72#1701—31no—
73#1697—31no—
74#1691—31no—
75#1687—31no—
76#1682—31no—
77#1678—31no—
78#1672D311yescharacteristic, complemented
79#1668—31no—
80#1565—31no—
81#1553—31no—
82#1551—31no—
83C2×#239—21no—
84C2×#239—21no—
85C2×#239D411yescharacteristic
86C2×C3×Q16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4×Q8C2311yescharacteristic
90#947—21no—
91#947—21no—
92Q8×Dic3D411yescharacteristic
93C2×#215—21no—
94C2×#215—21no—
95C2×#215—21no—
96#919—44no—
97C2×D24—21no—
98C2×D24—21no—
99C2×D24—21no—
100C2×D24—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)D411yescharacteristic
104C2×ΓL(1,𝔽25)D411yescharacteristic
105C2×D3×C8—21no—
106#895—21no—
107#895—21no—
108#895—21no—
109#895—21no—
110#893—21no—
111#893—21no—
112#893—21no—
113#893—21no—
114#893—21no—
115#893D411yescharacteristic
116#892—21no—
117#892D411yescharacteristic
118#874C2311yescharacteristic
119#874C2311yescharacteristic
120C4×D12—21no—
121C4×D12C2311yescharacteristic
122C42×D3—21no—
123C42×D3C2311yescharacteristic
124C3×#107—21no—
125C3×#106—21no—
126C3×#106—21no—
127C3×C4⋊C8—21no—
128C3×C4⋊C8C2311yescharacteristic
129C3×C4⋊C8C2311yescharacteristic
130C3×C4⋊C8C2311yescharacteristic
131C3×#103—21no—
132C3×#103—21no—
133C3×C4×C8—21no—
134C3×C4×C8C2311yescharacteristic
135#835—21no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#815—21no—
143#815D411yescharacteristic
144#815D411yescharacteristic
145Dic3⋊C8—21no—
146C8×Dic3—21no—
147#809—21no—
148#809—21no—
149#809—21no—
150#809—21no—
151#809—21no—
152#809—21no—
153#809—21no—
154#808—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×#135—61no—
162#532—31no—
163#532—31no—
164C4×#41—31no—
165#498C2×D311yescharacteristic
166#493—31no—
167#486—31no—
168#486—31no—
169#482—61no—
170#477C2×D311yescharacteristic
171#476—31no—
172#469—61no—
173#469—61no—
174#464—61no—
175#463—61no—
176#462C2×D311yescharacteristic
177#461—31no—
178#460—31no—
179#460—31no—
180#459—31no—
181#459—31no—
182#449—61no—
183C8×Q8—31no—
184C8×Q8C2×D311yescharacteristic
185#442—61no—
186#442—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431—31no—
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334C2×D311yescharacteristic
204#333—31no—
205#333C2×D311yescharacteristic
206#332—31no—
207#332C2×D311yescharacteristic
208#331—31no—
209#331—31no—
210#331—31no—
211#331—31no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215#239—22no—
216#239—41no—
217#239—42no—
218#239—42no—
219#239—82no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—41no—
228C2×D12—41no—
229C2×D12C2×D411yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—41no—
234C2×C4×D3—41no—
235C2×C4×D3C2×D411yescharacteristic
236C2×C4×D3C2×D411yescharacteristic
237C2×C4×D3#4111yescharacteristic
238C3×Q16—42no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8C2×D411yescharacteristic
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C42—21no—
251C3×C42C2411yescharacteristic
252#215—42no—
253#215—42no—
254#215—42no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260(C2×D3)⋊C4—41no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D411yescharacteristic
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
274D24—42no—
275D24—42no—
276D24—42no—
277D24—42no—
278ΓL(1,𝔽25)—22no—
279ΓL(1,𝔽25)—41no—
280ΓL(1,𝔽25)—42no—
281ΓL(1,𝔽25)—42no—
282D3×C8—42no—
283C2×#41—31no—
284C2×#41—61no—
285C2×#41—61no—
286#135—124no—
287C2×Q16—21no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×D8—61no—
292C2×D8—61no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—31no—
296C2×M16—31no—
297C2×M16—61no—
298C2×M16—61no—
299C22×C8—61no—
300C4⋊Q8C22×D311yescharacteristic
301C4⋊Q8C22×D311yescharacteristic
302#127—31no—
303#127—31no—
304#124—61no—
305#124—61no—
306#121—61no—
307#121—61no—
308#121—61no—
309#121—61no—
310C4×Q8—31no—
311C4×Q8C22×D311yescharacteristic
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4—61no—
319C4×D4—61no—
320#117—31no—
321#117—61no—
322C2×C42—31no—
323C2×C42—61no—
324#107—21no—
325#106—21no—
326#106—21no—
327#106—61no—
328C4⋊C8—21no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—61no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337#103—21no—
338#103—21no—
339#103—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353#97—31no—
354#97—31no—
355#97—61no—
356C4×C8—21no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C4×C8C22×D311yescharacteristic
361C22×D3—21no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3#12111yescharacteristic
365C3×Q8—21no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—81no—
369C3×Q8AGL(1,ℤ/8)12yes—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×Dic3—21no—
378C2×Dic3#12111yescharacteristic
379C2×Dic3#12111yescharacteristic
380C2×Dic3#12111yescharacteristic
381D12—21no—
382D12—21no—
383D12—42no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391D12—41no—
392D12—81no—
393D12—81no—
394D12—82no—
395D12—82no—
396C4×D3—21no—
397C4×D3—21no—
398C4×D3—21no—
399C4×D3—21no—
400C4×D3—41no—
401C4×D3—42no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—81no—
405C4×D3—82no—
406C4×D3—82no—
407C3×C8—21no—
408C3×C8—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3×C8#13512yescomplemented
412C3⋊C8—21no—
413C3⋊C8—21no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417#41—62no—
418#41—121no—
419#41—122no—
420#41—122no—
421#41—242no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8D3×D411yescharacteristic
425C2×D4—31no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—121no—
434C2×D4—121no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—121no—
442C22×C4—121no—
443Q16—42no—
444SD16—122no—
445SD16—122no—
446SD16—122no—
447D8—122no—
448D8—122no—
449D8—122no—
450D8—122no—
451M16—62no—
452M16—121no—
453M16—122no—
454M16—122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—31no—
459C2×C8—31no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—122no—
464C2×C8D3×D411yescharacteristic
465C2×C8D3×D411yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—31no—
472C4⋊C4—61no—
473C4⋊C4D3×D411yescharacteristic
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—121no—
478C22⋊C4—121no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—21no—
482C42—31no—
483C42—31no—
484C42—31no—
485C42—61no—
486C42—61no—
487C42C23×D311yescharacteristic
488C22×C3C2×#12111yescharacteristic
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4C2×#13511yescharacteristic
507C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C3×C4#57511yescharacteristic
509C3×C4#57511yescharacteristic
510Dic3—21no—
511Dic3—21no—
512Dic3—21no—
513Dic3—41no—
514C23—31no—
515C23—61no—
516C23—121no—
517C23—121no—
518Q8—21no—
519Q8—41no—
520Q8—41no—
521Q8—81no—
522Q8#91412yes—
523D4—61no—
524D4—61no—
525D4—122no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—121no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—241no—
535D4—241no—
536D4—242no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—241no—
555C2×C4—242no—
556C2×C4—242no—
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—61no—
565C8—61no—
566C8—121no—
567C8—121no—
568C8—121no—
569C8#91912yescomplemented
570C2×C3#303511yescharacteristic
571C2×C3#305511yescharacteristic
572C2×C3#306811yescharacteristic
573D3—41no—
574D3—81no—
575D3—161no—
576D3—161no—
577C22—61no—
578C22—61no—
579C22—61no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22D3×#12111yescharacteristic
590C4—22no—
591C4—21no—
592C4—41no—
593C4—61no—
594C4—61no—
595C4—61no—
596C4—81no—
597C4—81no—
598C4—121no—
599C4C2×#91411yescharacteristic
600C4C2×#91911yescharacteristic
601C4#576811yescharacteristic
602C4#577711yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—121no—
605C2—241no—
606C2—481no—
607C2—481no—
608C2#8047511yescharacteristic
609C2#8065711yescharacteristic
610C2#8093011yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1672C2
socleC22×C3C2×#121
hypercenter#1672D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1672

Complements

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

NH ≅ G/Ntype
1. #81888C2semidirect
2. #81093C2semidirect
3. #80475C2semidirect
4. #79905C2semidirect
5. C3×#1672C2semidirect
6. #71345C2semidirect
7. #71150C2semidirect
8. #71055C2semidirect
9. #70939C2semidirect
10. #68540C2semidirect
11. #68528C2semidirect
12. #68502C2semidirect
13. #68485C2semidirect
14. #68454C2semidirect
15. #68426C2semidirect
78. #1672D3semidirect
411. C3×C8#135semidirect
569. C8#919semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
#818882
#810932
#804752
#799052
C3×#16722
#713452
#711502
#710552
#709392
#685402
#685282
#685022
#684852
#684542
#684262
#191843