Finite Group Databasebeta

← #10786183

Subgroups — #10786183

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81953C211yesmaximal, characteristic
2#81947C211yesmaximal, characteristic, complemented
3#81902C211yesmaximal, characteristic, complemented
4#81621C211yesmaximal, characteristic
5C3×#1723C211yesmaximal, characteristic, complemented
6#71378C211yesmaximal, characteristic, complemented
7#71374C211yesmaximal, characteristic, complemented
8#71344C211yesmaximal, characteristic, complemented
9#71303C211yesmaximal, characteristic, complemented
10#71303C211yesmaximal, characteristic, complemented
11#71241C211yesmaximal, characteristic, complemented
12#71230C211yesmaximal, characteristic, complemented
13#71229C211yesmaximal, characteristic, complemented
14#68084C211yesmaximal, characteristic, complemented
15#68078C211yesmaximal, characteristic
16#19865—31nomaximal, Sylow
17#5732C2211yescharacteristic
18#5730C2211yescharacteristic
19#5730C2211yescharacteristic
20#5730C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#500C2211yescharacteristic
23C3×#498C2211yescharacteristic
24C3×#494C2211yescharacteristic
25C3×#487C2211yescharacteristic
26C3×#487C2211yescharacteristic
27#5094C2211yescharacteristic
28#5091C2211yescharacteristic
29#5091C2211yescharacteristic
30#5091C2211yescharacteristic
31#5084C2211yescharacteristic
32#5084C2211yescharacteristic
33#5084C2211yescharacteristic
34#5083C2211yescharacteristic
35#4891—21no—
36#4891—21no—
37#4887—21no—
38#4884—21no—
39#4884—21no—
40D3×#107—21no—
41#4878—21no—
42#4872—21no—
43#4868—21no—
44#4867—21no—
45#4862—21no—
46#4861—21no—
47#4819—21no—
48#4800—21no—
49#4796—21no—
50#4788—21no—
51C4×Dic12C2211yescharacteristic
52C4×D24C2211yescharacteristic
53#4688C2211yescharacteristic
54C8×D12C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57C3×#322C2211yescharacteristic
58#4547C2211yescharacteristic
59#4546C2211yescharacteristic
60#4546C2211yescharacteristic
61#4546C2211yescharacteristic
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic
64#4458C2211yescharacteristic
65#4457C2211yescharacteristic
66#4456C2211yescharacteristic
67#4456C2211yescharacteristic
68#3380—31no—
69#3360—31no—
70#3356—31no—
71#3338—31no—
72#1723D311yescharacteristic, complemented
73#1695—31no—
74#1684—31no—
75#1676—31no—
76#1671—31no—
77#1643—31no—
78#1638—31no—
79#1632—31no—
80#1632—31no—
81#1581—31no—
82#1579—31no—
83C2×C3×Q16—41no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88#947—21no—
89#946—41no—
90#946—41no—
91#946—41no—
92Q8×Dic3—21no—
93C2×#215—21no—
94C2×Dic12D411yescharacteristic
95C2×D24D411yescharacteristic
96C2×ΓL(1,𝔽25)—21no—
97C2×D3×C8—21no—
98#898—41no—
99#897—21no—
100#897—21no—
101#897—21no—
102#897—21no—
103#893—21no—
104#893—21no—
105#893—21no—
106#892—21no—
107D3×C4⋊C4—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—21no—
110#890—21no—
111#890—21no—
112#890—21no—
113C4×D12C2311yescharacteristic
114C4×D12C2311yescharacteristic
115C4×Dic6C2311yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×#107—21no—
118C3×#107—21no—
119C3×#107—21no—
120C3×#107—21no—
121C3×#106—21no—
122C3×#106—21no—
123C3×#103—21no—
124C3×#103—21no—
125C3×#103—41no—
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#821D411yescharacteristic
131#821D411yescharacteristic
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134#818D411yescharacteristic
135#818D411yescharacteristic
136#816D411yescharacteristic
137#816D411yescharacteristic
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140C8×Dic3—21no—
141C8×Dic3—21no—
142#810—21no—
143#810—21no—
144#810—21no—
145#810—21no—
146#809—21no—
147#809—21no—
148#809—21no—
149#808—21no—
150#808—21no—
151#808—21no—
152#808—21no—
153#807—21no—
154#807—21no—
155#807—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#556—31no—
161#554—31no—
162#554—31no—
163#554—31no—
164#500C2×D311yescharacteristic
165#500C2×D311yescharacteristic
166#499—61no—
167#498C2×D311yescharacteristic
168#494C2×D311yescharacteristic
169#488—61no—
170#487C2×D311yescharacteristic
171#487C2×D311yescharacteristic
172#485—61no—
173#484—61no—
174#484—61no—
175#483—61no—
176#479—61no—
177#478—61no—
178#478—61no—
179#477—31no—
180#477—31no—
181#477—31no—
182#476—31no—
183#476—31no—
184#476—31no—
185#475—31no—
186#470—61no—
187#467—61no—
188#463—61no—
189#461—31no—
190C8×Q8—31no—
191C8×Q8—31no—
192C4×Q16—31no—
193C4×SD16—61no—
194C4×D8—31no—
195C4×D8—61no—
196#436—31no—
197C8×D4—31no—
198#428—61no—
199C2×#107—61no—
200#335—31no—
201#334—31no—
202#333—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×D12C2×D411yescharacteristic
216C2×D12C2×D411yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×C4×D3—41no—
222C2×Dic6C2×D411yescharacteristic
223C2×Dic6C2×D411yescharacteristic
224C3×Q16—82no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D411yescharacteristic
227C2×C3×C8C2×D411yescharacteristic
228C2×C3×C8C2×D411yescharacteristic
229C2×C3×C8C2×D411yescharacteristic
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C42C2411yescharacteristic
239#215—42no—
240(C2×D3)⋊C4—21no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—41no—
243(C2×D3)⋊C4—41no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246(C2×D3)⋊C4—41no—
247C4⋊Dic3—41no—
248C4⋊Dic3—41no—
249C4⋊Dic3—41no—
250C4⋊Dic3—41no—
251C4⋊Dic3C2×D411yescharacteristic
252C4⋊Dic3C2×D411yescharacteristic
253C4⋊Dic3C2×D411yescharacteristic
254C4⋊Dic3C2×D411yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—41no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267Dic12—22no—
268D24—22no—
269ΓL(1,𝔽25)—42no—
270D3×C8—42no—
271C2×Q16—31no—
272C2×Q16—41no—
273C2×SD16—61no—
274C2×D8—31no—
275C2×M16—61no—
276C22×C8—61no—
277C4⋊Q8C22×D311yescharacteristic
278C4⋊Q8C22×D311yescharacteristic
279C4⋊Q8C22×D311yescharacteristic
280C4⋊Q8C22×D311yescharacteristic
281#126—121no—
282#125—61no—
283#125—61no—
284#125—61no—
285#124—61no—
286C22⋊Q8—61no—
287C22⋊Q8—61no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290C22⋊Q8—121no—
291C22⋊Q8—121no—
292C22⋊Q8—121no—
293C4×Q8—31no—
294C4×Q8—31no—
295C4×Q8—61no—
296C4×D4—31no—
297C4×D4—31no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4—61no—
301#117—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305#107—21no—
306#107—21no—
307#107—21no—
308#107—21no—
309#107—31no—
310#107—31no—
311#107—61no—
312#107—61no—
313#107—61no—
314#106—21no—
315#106—21no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326#103—21no—
327#103—21no—
328#103—31no—
329#103—31no—
330#103—41no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#102—31no—
337#102—31no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C4×C8—61no—
344C4×C8—61no—
345C4×C8C22×D311yescharacteristic
346C4×C8C22×D311yescharacteristic
347C4×C8C22×D311yescharacteristic
348C22×D3—21no—
349C22×D3—21no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—81no—
353C3×Q8—81no—
354C3×Q8—81no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368D12—21no—
369D12—21no—
370D12—21no—
371D12#13712yes—
372C4×D3—41no—
373C4×D3—41no—
374C4×D3—41no—
375C4×D3—41no—
376C4×D3—161no—
377C4×D3—161no—
378C4×D3—161no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—21no—
383C3×C8—21no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—41no—
387C3×C8#13512yes—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—31no—
397C2×Q8—31no—
398C2×D4—31no—
399C2×D4—31no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—121no—
403C22×C4—121no—
404C22×C4—121no—
405Q16—62no—
406Q16—82no—
407SD16—122no—
408D8—62no—
409M16—122no—
410C2×C8—21no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—122no—
416C2×C8D3×D411yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8D3×D411yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—31no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—121no—
435C4⋊C4—121no—
436C4⋊C4—121no—
437C4⋊C4—121no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4—121no—
441C22⋊C4—61no—
442C22⋊C4—61no—
443C22⋊C4—121no—
444C22⋊C4—121no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C42—61no—
449C42—61no—
450C42—121no—
451C42C23×D311yescharacteristic
452C22×C3D4211yescharacteristic
453C2×D3—22no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C3×C4—21no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4C2×#13511yescharacteristic
465C3×C4C2×#13711yescharacteristic
466C3×C4#57811yescharacteristic
467C3×C4#57811yescharacteristic
468Dic3—41no—
469Dic3—41no—
470Dic3—41no—
471Dic3—41no—
472Dic3—41no—
473Dic3—41no—
474C23—61no—
475C23—61no—
476Q8—41no—
477Q8—41no—
478Q8—61no—
479Q8—61no—
480Q8—61no—
481Q8—61no—
482Q8—81no—
483Q8—81no—
484Q8—81no—
485D4—32no—
486D4—61no—
487D4—61no—
488D4—61no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—481no—
504C2×C4—481no—
505C2×C4—481no—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4#100411yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—41no—
513C8—121no—
514C8—121no—
515C8—121no—
516C8—121no—
517C8#91912yes—
518C2×C3#327011yescharacteristic
519C2×C3#328311yescharacteristic
520C2×C3#328811yescharacteristic
521D3—42no—
522D3—81no—
523C22—62no—
524C22—121no—
525C22—121no—
526C22—121no—
527C22—121no—
528C22—121no—
529C22#571711yescharacteristic
530C4—21no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—121no—
536C4—121no—
537C4—121no—
538C4—121no—
539C4—121no—
540C4—121no—
541C4C2×#91911yescharacteristic
542C4#576611yescharacteristic
543C4D3×#13711yescharacteristic
544C4#577911yescharacteristic
545C3#1986511yescharacteristic, complemented, Sylow
546C2—122no—
547C2—241no—
548C2#8180611yescharacteristic
549C2#8186011yescharacteristic
550C2#8188411yescharacteristic
subgroupG/H
centerC22#5717
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1723C2
socleC22×C3D42
prime radicalC2×D24D4
hypercenter#1723D3
hypocenterC3#19865
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1723

Complements

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

NH ≅ G/Ntype
2. #81947C2semidirect
3. #81902C2semidirect
5. C3×#1723C2semidirect
6. #71378C2semidirect
7. #71374C2semidirect
8. #71344C2semidirect
9. #71303C2semidirect
10. #71303C2semidirect
11. #71241C2semidirect
12. #71230C2semidirect
13. #71229C2semidirect
14. #68084C2semidirect
72. #1723D3semidirect
545. C3#19865semidirect

Maximal subgroups

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

subgroupindex
#819532
#819472
#819022
#816212
C3×#17232
#713782
#713742
#713442
#713032
#712412
#712302
#712292
#680842
#680782
#198653