Finite Group Databasebeta

← #10780307

Subgroups — #10780307

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81600C211yesmaximal, characteristic, complemented
2#81132C211yesmaximal, characteristic
3#81018C211yesmaximal, characteristic
4#79942C211yesmaximal, characteristic
5C3×#1704C211yesmaximal, characteristic, complemented
6#71233C211yesmaximal, characteristic, complemented
7#71151C211yesmaximal, characteristic, complemented
8#71118C211yesmaximal, characteristic, complemented
9#70966C211yesmaximal, characteristic, complemented
10#68539C211yesmaximal, characteristic, complemented
11#68531C211yesmaximal, characteristic, complemented
12#68503C211yesmaximal, characteristic, complemented
13#68486C211yesmaximal, characteristic, complemented
14#68452C211yesmaximal, characteristic, complemented
15#68422C211yesmaximal, characteristic, complemented
16#19381—31nomaximal, Sylow
17#5728C2211yescharacteristic
18D3×C4⋊Q8C2211yescharacteristic
19#5580C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×#446C2211yescharacteristic
24#5093C2211yescharacteristic
25#5087C2211yescharacteristic
26#5084C2211yescharacteristic
27#5079C2211yescharacteristic
28#5031C2211yescharacteristic
29Q8×C3⋊C8C2211yescharacteristic
30#5020C2211yescharacteristic
31#4886—21no—
32#4877—21no—
33#4863—21no—
34#4856—21no—
35#4846C2211yescharacteristic
36#4841C2211yescharacteristic
37#4840—21no—
38#4838—21no—
39#4838C2211yescharacteristic
40#4837C2211yescharacteristic
41#4835C2211yescharacteristic
42#4833C2211yescharacteristic
43#4816—21no—
44#4811—21no—
45#4802—21no—
46#4802—21no—
47#4798—21no—
48#4797—21no—
49#4797—21no—
50#4794—21no—
51#4708—21no—
52#4706—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54C8×D12C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4546C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3252—31no—
69#3239—31no—
70#3139—31no—
71#2969—31no—
72#1704D311yescharacteristic, complemented
73#1703—31no—
74#1700—31no—
75#1699—31no—
76#1684—31no—
77#1683—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1554—31no—
82#1549—31no—
83C2×D3×Q8—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4×Q8C2311yescharacteristic
87#947D411yescharacteristic
88#946—21no—
89#946D411yescharacteristic
90Q8×Dic3—21no—
91#944—21no—
92C2×#216—21no—
93C2×#204—21no—
94C2×ΓL(1,𝔽25)—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×ΓL(1,𝔽25)D411yescharacteristic
97C2×ΓL(1,𝔽25)D411yescharacteristic
98C2×D3×C8—21no—
99#898—21no—
100#897—21no—
101#895D411yescharacteristic
102#894—21no—
103#892—21no—
104D3×C4⋊C4—21no—
105D3×C4⋊C4—21no—
106#890—21no—
107#890D411yescharacteristic
108#888—21no—
109#888—21no—
110#887—21no—
111C4×D12—21no—
112C4×D12C2311yescharacteristic
113#872—21no—
114C42×D3C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×#106—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#97—21no—
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#835—21no—
131#835—21no—
132#821—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137#818—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815D411yescharacteristic
143#815D411yescharacteristic
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146#810—21no—
147#810—21no—
148#810—21no—
149#809—21no—
150#809—21no—
151#808—21no—
152#808—21no—
153#807—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#533—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#491—61no—
165#491—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487C2×D311yescharacteristic
171#484—61no—
172#481—61no—
173#479—61no—
174#477C2×D311yescharacteristic
175#476—31no—
176#475—31no—
177#475C2×D311yescharacteristic
178#474—31no—
179#474—31no—
180#470—61no—
181#468—61no—
182#464—61no—
183#462—31no—
184#461—31no—
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#419—61no—
196#419—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#333—31no—
204#333C2×D311yescharacteristic
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214D3×Q8—42no—
215D3×Q8—42no—
216C2×D12—21no—
217C2×D12C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—41no—
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3#4111yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—41no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C2×D411yescharacteristic
230C2×C3×C8C2×D411yescharacteristic
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2×D411yescharacteristic
237C3×C42—21no—
238C3×C42C2411yescharacteristic
239#216—81no—
240(C2×D3)⋊C4—21no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—41no—
252C4⋊Dic3C2×D411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—41no—
256Dic3⋊C4—41no—
257Dic3⋊C4—41no—
258C4×Dic3—41no—
259C4×Dic3C2×D411yescharacteristic
260C4×Dic3C2×D411yescharacteristic
261C4×Dic3#4111yescharacteristic
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8C2×D411yescharacteristic
266C2×C3⋊C8C2×D411yescharacteristic
267#204—81no—
268ΓL(1,𝔽25)—41no—
269ΓL(1,𝔽25)—41no—
270ΓL(1,𝔽25)—42no—
271ΓL(1,𝔽25)—81no—
272D3×C8—42no—
273C22×Q8—61no—
274C2×Q16—61no—
275C2×SD16—61no—
276C2×M16—31no—
277C2×M16—31no—
278C2×M16—61no—
279C2×M16—61no—
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288#126—61no—
289#125—31no—
290#125—61no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296#121—31no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8C22×D311yescharacteristic
300C4×D4—31no—
301C4×D4—61no—
302#117—61no—
303#117—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C42—31no—
307#107—21no—
308#107—61no—
309#107—61no—
310#106—21no—
311#106—61no—
312#106—61no—
313C4⋊C8—21no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322#103—21no—
323#103—21no—
324#103—21no—
325#103—21no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#97—21no—
343#97—31no—
344#97—31no—
345#97—61no—
346C4×C8—31no—
347C4×C8—61no—
348C4×C8—61no—
349C4×C8C22×D311yescharacteristic
350C22×D3—21no—
351C22×D3#12111yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—81no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3#12111yescharacteristic
368C2×Dic3#12111yescharacteristic
369C2×Dic3#12111yescharacteristic
370D12—21no—
371D12—21no—
372D12—81no—
373C4×D3—21no—
374C4×D3—21no—
375C4×D3—21no—
376C4×D3—21no—
377C4×D3—41no—
378C4×D3—41no—
379C4×D3—41no—
380C4×D3—82no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—82no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—121no—
401C2×Q8—122no—
402C2×Q8—122no—
403C2×Q8D3×D411yescharacteristic
404C2×D4—31no—
405C2×D4—61no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—121no—
411Q16—241no—
412SD16—241no—
413M16—121no—
414M16—121no—
415M16—122no—
416M16—241no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—122no—
426C2×C8D3×D411yescharacteristic
427C2×C8D3×D411yescharacteristic
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—31no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—121no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4D3×D411yescharacteristic
446C22⋊C4—61no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—121no—
452C42—21no—
453C42—31no—
454C42—31no—
455C42—31no—
456C42—121no—
457C42C23×D311yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C2×D3—21no—
460C2×D3—21no—
461C2×D3—21no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4#57511yescharacteristic
471C3×C4#57511yescharacteristic
472C3×C4#57711yescharacteristic
473C3×C4#57811yescharacteristic
474Dic3—21no—
475Dic3—21no—
476Dic3—21no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—31no—
481C23—61no—
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—121no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—242no—
492D4—61no—
493D4—61no—
494D4—241no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—242no—
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C8—21no—
517C8—21no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—61no—
522C8—61no—
523C8—121no—
524C8—121no—
525C8—121no—
526C2×C3#304011yescharacteristic
527C2×C3#306311yescharacteristic
528C2×C3#306911yescharacteristic
529D3—41no—
530D3—81no—
531C22—61no—
532C22—61no—
533C22—61no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22D3×#12111yescharacteristic
538C4—21no—
539C4—41no—
540C4—41no—
541C4—61no—
542C4—61no—
543C4—61no—
544C4—81no—
545C4—81no—
546C4—121no—
547C4—241no—
548C4—241no—
549C4#576811yescharacteristic
550C4#577711yescharacteristic
551C4#577811yescharacteristic
552C4#577911yescharacteristic
553C3#1938111yescharacteristic, complemented, Sylow
554C2—121no—
555C2—241no—
556C2#8059911yescharacteristic
557C2#8090711yescharacteristic
558C2#8093111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1704C2
socleC22×C3C2×#121
prime radical#895D4
hypercenter#1704D3
hypocenterC3#19381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1704

Complements

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

NH ≅ G/Ntype
1. #81600C2semidirect
5. C3×#1704C2semidirect
6. #71233C2semidirect
7. #71151C2semidirect
8. #71118C2semidirect
9. #70966C2semidirect
10. #68539C2semidirect
11. #68531C2semidirect
12. #68503C2semidirect
13. #68486C2semidirect
14. #68452C2semidirect
15. #68422C2semidirect
72. #1704D3semidirect
553. C3#19381semidirect

Maximal subgroups

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

subgroupindex
#816002
#811322
#810182
#799422
C3×#17042
#712332
#711512
#711182
#709662
#685392
#685312
#685032
#684862
#684522
#684222
#193813