Finite Group Databasebeta

← #10786013

Subgroups — #10786013

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3397C211yesmaximal, characteristic, complemented
2#83658C211yesmaximal, characteristic, complemented
3#83571C211yesmaximal, characteristic, complemented
4#79616C211yesmaximal, characteristic, complemented
5#71932C211yesmaximal, characteristic, complemented
6#71919C211yesmaximal, characteristic, complemented
7#71267C211yesmaximal, characteristic, complemented
8#71258C211yesmaximal, characteristic, complemented
9#71047C211yesmaximal, characteristic, complemented
10#71036C211yesmaximal, characteristic, complemented
11#71017C211yesmaximal, characteristic, complemented
12#71006C211yesmaximal, characteristic, complemented
13#70965C211yesmaximal, characteristic, complemented
14#70953C211yesmaximal, characteristic, complemented
15#70878C211yesmaximal, characteristic, complemented
16#19854—31nomaximal, Sylow
17C3×#556C2211yescharacteristic
18C3×#550C2211yescharacteristic
19#5563C2211yescharacteristic
20#5563C2211yescharacteristic
21C3×#496—21no—
22C3×#492C2211yescharacteristic
23C3×#463—21no—
24C3×#461C2211yescharacteristic
25C3×#460—21no—
26C3×#459C2211yescharacteristic
27C3×#446C2211yescharacteristic
28C3×#440—21no—
29C3×C4×D8C2211yescharacteristic
30#5238—21no—
31#5233—21no—
32#5157—21no—
33#5093C2211yescharacteristic
34#5087C2211yescharacteristic
35#5080C2211yescharacteristic
36#5076C2211yescharacteristic
37#5072C2211yescharacteristic
38#5070C2211yescharacteristic
39#5048—21no—
40#5045—21no—
41#5038—21no—
42#5035—21no—
43C4×#215C2211yescharacteristic
44#5021C2211yescharacteristic
45#5021C2211yescharacteristic
46D4×C3⋊C8C2211yescharacteristic
47D4×C3⋊C8C2211yescharacteristic
48#5008C2211yescharacteristic
49#5008C2211yescharacteristic
50#5003—21no—
51#4998—21no—
52#4880—21no—
53#4784—21no—
54#4784—21no—
55C8×D12C2211yescharacteristic
56#4682C2211yescharacteristic
57#4547C2211yescharacteristic
58#4544C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4483C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4457C2211yescharacteristic
67#4455C2211yescharacteristic
68#3397D311yescharacteristic, complemented
69#3341—31no—
70#3340—31no—
71#3305—31no—
72#1714—31no—
73#1696—31no—
74#1677—31no—
75#1671—31no—
76#1665—31no—
77#1646—31no—
78#1638—31no—
79#1631—31no—
80#1624—31no—
81#1582—31no—
82#1580—31no—
83C2×C3×#41—21no—
84C2×C3×SD16—21no—
85C2×C3×SD16—21no—
86C2×C3×SD16—21no—
87C2×C3×D8—21no—
88C2×C3×D8D411yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×#127—21no—
91C3×#127C2311yescharacteristic
92C3×#126—41no—
93C3×#124—21no—
94C3×C22⋊Q8—21no—
95C3×#121—21no—
96C3×#121—41no—
97C3×C4×Q8—21no—
98C3×C4×Q8C2311yescharacteristic
99C3×C4×D4—21no—
100C3×C4×D4C2311yescharacteristic
101C3×C4×D4C2311yescharacteristic
102C3×#117—21no—
103C2×#215D411yescharacteristic
104#938—21no—
105#938—21no—
106C2×#214—41no—
107C2×#213—21no—
108#930—21no—
109#930—21no—
110#926—21no—
111#926—21no—
112C22×C3⋊C8—21no—
113C22×C3⋊C8—21no—
114C2×D3×C8—21no—
115#892—21no—
116#892—21no—
117#885—41no—
118#885—41no—
119C4×D12C2311yescharacteristic
120#869C2311yescharacteristic
121#869C2311yescharacteristic
122C3×#107D411yescharacteristic
123C3×C4⋊C8—21no—
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#102—21no—
127C3×#102D411yescharacteristic
128C3×#102D411yescharacteristic
129C3×#97—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#835D411yescharacteristic
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#830—21no—
140#830—21no—
141(C2×D3)⋊C8—21no—
142#818—21no—
143#818—21no—
144#810—41no—
145#809—21no—
146#809D411yescharacteristic
147#808—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#808D411yescharacteristic
152#807—21no—
153#807—21no—
154#807—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556C2×D311yescharacteristic
161#555—31no—
162#555—31no—
163#550C2×D311yescharacteristic
164#500—31no—
165#498—31no—
166#496—21no—
167#492—31no—
168#492C2×D311yescharacteristic
169#487—31no—
170#486—31no—
171#485—61no—
172#482—61no—
173#482—61no—
174#482—61no—
175#482—61no—
176#477—31no—
177#477—31no—
178#476—31no—
179#476—31no—
180#474—31no—
181#466—61no—
182#466—61no—
183#465—61no—
184#463—21no—
185#461C2×D311yescharacteristic
186#460—21no—
187#459—31no—
188#459C2×D311yescharacteristic
189#449—61no—
190#446C2×D311yescharacteristic
191#440—21no—
192C4×SD16—31no—
193C4×D8C2×D311yescharacteristic
194C8×D4—31no—
195C8×D4—31no—
196C8×D4—31no—
197#427—61no—
198#427—61no—
199#416—61no—
200#334—31no—
201#334—31no—
202#333—31no—
203#333—31no—
204#332—31no—
205#331—31no—
206Q8⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C3×#41—42no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C2×C3×D4—41no—
220C2×C3×D4C2×D411yescharacteristic
221C2×C3×D4C2×D411yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C2×#67—41no—
226C2×#67—41no—
227C2×D12C2×D411yescharacteristic
228C2×C4×D3—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×SD16—42no—
232C3×SD16—81no—
233C3×SD16—81no—
234C3×D8—22no—
235C3×D8—81no—
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⋊C4—41no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C42—21no—
252C3×C42—41no—
253C3×C42C2411yescharacteristic
254#217—41no—
255#217—41no—
256#215—41no—
257#214—82no—
258#213—81no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—41no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D411yescharacteristic
267C4×Dic3—41no—
268C4×Dic3—41no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274C2×C3⋊C8—81no—
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277D3×C8—42no—
278C2×#41—21no—
279C2×SD16—21no—
280C2×SD16—21no—
281C2×SD16—21no—
282C2×SD16—31no—
283C2×SD16—121no—
284C2×D8—21no—
285C2×D8—61no—
286C2×D8#6711yescharacteristic
287C22×C8—61no—
288C22×C8—61no—
289C22×C8—61no—
290C4⋊Q8—31no—
291C4⋊Q8—31no—
292C4⋊Q8C22×D311yescharacteristic
293#127—21no—
294#127C22×D311yescharacteristic
295#126—41no—
296#124—21no—
297#124—121no—
298#124—121no—
299C22⋊Q8—21no—
300#121—21no—
301#121—41no—
302#121—61no—
303#121—61no—
304#121—61no—
305#121—61no—
306C4×Q8—21no—
307C4×Q8C22×D311yescharacteristic
308C4×D4—21no—
309C4×D4—31no—
310C4×D4C22×D311yescharacteristic
311C4×D4C22×D311yescharacteristic
312#117—21no—
313#117—61no—
314#117—61no—
315#117—61no—
316#117—61no—
317#107—61no—
318#107—61no—
319#107—61no—
320#107—61no—
321#107—61no—
322#107#6711yescharacteristic
323#106—31no—
324#106—61no—
325#106—61no—
326#106—61no—
327#106—61no—
328C4⋊C8—21no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8C22×D311yescharacteristic
333#103—21no—
334#103—31no—
335#103—61no—
336#103—61no—
337#103—121no—
338#102—21no—
339#102—31no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102#6711yescharacteristic
347#102#6711yescharacteristic
348C22⋊C8—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351#97—21no—
352C4×C8—31no—
353C4×C8—31no—
354C4×C8C22×D311yescharacteristic
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—41no—
358C22×D3—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4—21no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—81no—
372C3×D4—82no—
373C3×D4—161no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—82no—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387#67—161no—
388#67—161no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392D12—21no—
393D12—21no—
394C4×D3—41no—
395C4×D3—41no—
396Dic6—81no—
397Dic6—81no—
398C3×C8—21no—
399C3×C8—41no—
400C3×C8#13512yes—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407#41—42no—
408#41—81no—
409C2×Q8—21no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8D3×D411yescharacteristic
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—31no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—121no—
420C2×D4—121no—
421C2×D4C2×#6711yescharacteristic
422C2×D4C2×#6711yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—61no—
427SD16—42no—
428SD16—81no—
429SD16—81no—
430SD16—121no—
431SD16—242no—
432D8—22no—
433D8—81no—
434D8—241no—
435C2×C8—21no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—122no—
443C2×C8—122no—
444C2×C8—241no—
445C2×C8C2×#6711yescharacteristic
446C2×C8C2×#6711yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—31no—
451C4⋊C4—41no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4D3×D411yescharacteristic
457C4⋊C4C2×#6711yescharacteristic
458C4⋊C4C2×#6711yescharacteristic
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—61no—
465C22⋊C4—121no—
466C22⋊C4—121no—
467C22⋊C4—121no—
468C22⋊C4—121no—
469C42—21no—
470C42—41no—
471C42—121no—
472C42—121no—
473C42C23×D311yescharacteristic
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3D4211yescharacteristic
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C3×C4—21no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—81no—
493C3×C4C2×#13511yescharacteristic
494C3×C4#57511yescharacteristic
495C3×C4#57711yescharacteristic
496C3×C4#57711yescharacteristic
497Dic3—41no—
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—21no—
502C23—41no—
503C23—61no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—241no—
509Q8—241no—
510D4—21no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—61no—
519D4—61no—
520D4—81no—
521D4—82no—
522D4—161no—
523D4—481no—
524D4—481no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—82no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4#100411yescharacteristic
542C2×C4C22×#6711yescharacteristic
543C8—21no—
544C8—41no—
545C8—61no—
546C8—61no—
547C8—121no—
548C8—121no—
549C8—121no—
550C8—121no—
551C8#95012yes—
552C2×C3—81no—
553C2×C3—81no—
554C2×C3—161no—
555C2×C3#326711yescharacteristic
556C2×C3#328411yescharacteristic
557C2×C3#328811yescharacteristic
558D3—81no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—121no—
569C22—121no—
570C22—121no—
571C22D4×#6711yescharacteristic
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—121no—
579C4—241no—
580C4—241no—
581C4#577311yescharacteristic
582C4#577511yescharacteristic
583C4C2×#95011yescharacteristic
584C4#583611yescharacteristic
585C3#1985411yescharacteristic, complemented, Sylow
586C2—81no—
587C2—81no—
588C2—161no—
589C2—241no—
590C2#8357911yescharacteristic
591C2#8395411yescharacteristic
592C2#8401511yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3397C2
socleC22×C3D42
hypercenter#3397D3
hypocenterC3#19854
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3397

Complements

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

NH ≅ G/Ntype
1. C3×#3397C2semidirect
2. #83658C2semidirect
3. #83571C2semidirect
4. #79616C2semidirect
5. #71932C2semidirect
6. #71919C2semidirect
7. #71267C2semidirect
8. #71258C2semidirect
9. #71047C2semidirect
10. #71036C2semidirect
11. #71017C2semidirect
12. #71006C2semidirect
13. #70965C2semidirect
14. #70953C2semidirect
15. #70878C2semidirect
68. #3397D3semidirect
585. C3#19854semidirect

Maximal subgroups

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

subgroupindex
C3×#33972
#836582
#835712
#796162
#719322
#719192
#712672
#712582
#710472
#710362
#710172
#710062
#709652
#709532
#708782
#198543