Finite Group Databasebeta

← #11675349

Subgroups — #11675349

Order 768.

1852 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86282C211yesmaximal, characteristic, complemented
2#86261C211yesmaximal, characteristic, complemented
3C3×#2961C211yesmaximal, characteristic, complemented
4#79982C214yesmaximal, complemented
5#79980C214yesmaximal, complemented
6#79971C214yesmaximal, complemented
7#79969C214yesmaximal, complemented
8#79090C212yesmaximal, complemented
9#79088C212yesmaximal, complemented
10#79061C211yesmaximal, characteristic, complemented
11#79056C211yesmaximal, characteristic, complemented
12#78563C212yesmaximal, complemented
13#78560C212yesmaximal, complemented
14C2×#4713C211yesmaximal, characteristic, complemented
15#78494C211yesmaximal, characteristic, complemented
16#61337—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×#908C2212yes—
19C2×#904C2211yescharacteristic
20#5729—22no—
21#5725—22no—
22#5697—21no—
23#5697—21no—
24#5688—21no—
25D3×#125—21no—
26#5636—24no—
27#5630—24no—
28#5585C2214yes—
29#5579C2214yescomplemented
30#5577C2214yes—
31#5567C2214yescomplemented
32#5525C2212yes—
33#5523C2212yes—
34#5521C2212yes—
35#5519C2212yes—
36C2×#874C2211yescharacteristic
37C4×#235C2211yescharacteristic
38C4×#235C2211yescharacteristic
39C2×#870C2211yescharacteristic
40C3×#446C2214yescomplemented
41C3×#446C2214yescomplemented
42C3×#424C2212yes—
43C3×#423C2211yescharacteristic
44C3×#405C2212yes—
45C2×C3×#97C2211yescharacteristic
46#5133C2212yes—
47#5131C2212yes—
48#5121C2212yes—
49#5113C2211yescharacteristic
50#5113C2211yescharacteristic
51C2×#821C2212yes—
52C2×#821C2212yes—
53#5106C2211yescharacteristic
54#4848C2214yes—
55#4848C2214yescomplemented
56#4847C2214yescomplemented
57#4847C2214yescomplemented
58#4847C2214yescomplemented
59#4846C2214yescomplemented
60#4845C2214yescomplemented
61#4844C2214yescomplemented
62#4843C2214yes—
63#4843C2214yescomplemented
64#4842C2214yescomplemented
65#4829C2214yescomplemented
66#4716C2212yescomplemented
67#4714C2214yescomplemented
68#4713C2214yes—
69#4713C2214yes—
70#4713C2214yescomplemented
71#4711C2212yescomplemented
72#4711C2214yescomplemented
73#4710C2212yescomplemented
74#4710C2212yescomplemented
75#4710C2214yescomplemented
76#4697C2214yescomplemented
77#4694C2214yescomplemented
78C4×D24C2214yescomplemented
79C4×#204C2214yescomplemented
80#3508—31no—
81#3504—31no—
82#3371—34no—
83#3370—34no—
84#3368—34no—
85#3366—34no—
86#3136—32no—
87C2×#490—31no—
88#3081—32no—
89#3080—31no—
90#3045—32no—
91#3038—31no—
92#2961D311yescharacteristic, complemented
93#2946—32no—
94#2943—31no—
95C2×#239—24no—
96C2×#235C2311yescharacteristic
97C2×#235C2311yescharacteristic
98C22×D12—22no—
99C22×D12C2312yes—
100C2×C3×M16C2312yes—
101C22×C3×C8C2311yescharacteristic
102C22×C3×C8C2311yescharacteristic
103C3×C4⋊Q8D412yes—
104C3×#125D411yescharacteristic
105C3×#125D411yescharacteristic
106C3×C22⋊Q8D414yes—
107C3×C4×Q8C2314yes—
108C3×#117C2312yes—
109C2×C3×C42C2311yescharacteristic
110#947—24no—
111#946—24no—
112#944—24no—
113C4×#67—22no—
114C4×#67—22no—
115#926C2311yescharacteristic
116#926C2311yescharacteristic
117C2×C4⋊Dic3C2312yes—
118C2×Dic3⋊C4—22no—
119#908—28no—
120C2×Dic12C2311yescharacteristic
121#904—24no—
122C2×D24C2314yes—
123C2×D24C2311yescharacteristic
124C2×#204C2314yes—
125C2×#204C2311yescharacteristic
126C2×#204C2311yescharacteristic
127#896—24no—
128#896—24no—
129#895—22no—
130#895—22no—
131#895—24no—
132#895—24no—
133#895—24no—
134#894—24no—
135#894—24no—
136#893—22no—
137#893—22no—
138#893—24no—
139#892—22no—
140D3×C4⋊C4—24no—
141D3×C4⋊C4—22no—
142#890—22no—
143#889—24no—
144#889—24no—
145#888—22no—
146#887—24no—
147#882—24no—
148#67⋊C4—24no—
149D3×C22⋊C4—24no—
150#878—24no—
151#875C2314yes—
152#874—21no—
153#874—21no—
154#874C2314yes—
155#874C2314yes—
156C4×D12C2314yes—
157C4×D12C2314yes—
158C4×D12C2314yes—
159C4×D12C2312yes—
160C4×D12C2312yes—
161#872—21no—
162#872—21no—
163C42×D3—21no—
164C42×D3—21no—
165#870—21no—
166#870—21no—
167#870C2314yes—
168#870C2314yes—
169#869C2314yes—
170C4×Dic6C2314yes—
171C4×Dic6C2312yes—
172C4×Dic6C2312yes—
173C3×C4⋊C8C2314yes—
174C3×C4⋊C8C2314yes—
175C3×C4⋊C8C2314yes—
176C3×#97C2312yes—
177C3×#97C2314yes—
178C3×#97C2312yes—
179C3×C4×C8C2314yes—
180#821C2314yes—
181#821C2314yes—
182#821C2314yes—
183#821C2314yes—
184#821C2312yes—
185#821C2312yes—
186#821C2314yes—
187#821C2314yes—
188#821C2314yes—
189#818C2314yes—
190#818C2312yes—
191#817C2314yes—
192#817C2312yes—
193#816C2314yes—
194#816C2312yes—
195#816C2312yes—
196C2×AGL(1,ℤ/8)—32no—
197C2×#135—31no—
198#559—34no—
199#557—34no—
200#554—34no—
201#554—64no—
202#550—34no—
203#546—64no—
204#544—32no—
205#544—62no—
206#540—61no—
207#540—61no—
208#535—32no—
209#533—61no—
210#533C2×D311yescharacteristic
211#532—62no—
212C2×#127—31no—
213C2×#125—31no—
214C2×#125—61no—
215#520—32no—
216#518—32no—
217C4×#41—31no—
218C4×#41—31no—
219#490—34no—
220#490—34no—
221#490—34no—
222#489—34no—
223#488—34no—
224#486—34no—
225#479—34no—
226#478—34no—
227#478—34no—
228#478—34no—
229#476—34no—
230#474—34no—
231#464—34no—
232#463—34no—
233#460—34no—
234#460—34no—
235#459—34no—
236#459—34no—
237#446C2×D314yescomplemented
238#446C2×D314yescomplemented
239#442—32no—
240#442—34no—
241#441—32no—
242#440—32no—
243#440—34no—
244#440—32no—
245C4×SD16—34no—
246C4×D8—34no—
247#428—32no—
248#427—31no—
249#424C2×D312yes—
250#423C2×D311yescharacteristic
251#418—32no—
252#417—32no—
253#416—31no—
254#416—31no—
255C2×#102—32no—
256C2×#102—32no—
257#405C2×D312yes—
258C2×#97C2×D311yescharacteristic
259C23×D3—22no—
260C2×C3×Q8C2×D414yes—
261C22×C3×C4C2×D412yes—
262C22×C3×C4C2411yescharacteristic
263C2×#67—21no—
264C2×#67—21no—
265C22×Dic3—22no—
266#239—88no—
267#235—22no—
268#235—22no—
269#235—22no—
270#235#4114yescomplemented
271C2×D12—24no—
272C2×D12—24no—
273C2×D12—22no—
274C2×D12—24no—
275C2×D12—24no—
276C2×D12—24no—
277C2×D12—24no—
278C2×D12—84no—
279C2×D12C2411yescharacteristic
280C2×D12C2414yes—
281C2×D12C2411yescharacteristic
282C2×C4×D3—21no—
283C2×C4×D3—21no—
284C2×C4×D3—22no—
285C2×C4×D3—22no—
286C2×C4×D3—24no—
287C2×C4×D3—24no—
288C2×C4×D3—24no—
289C2×Dic6—24no—
290C2×Dic6C2411yescharacteristic
291C2×Dic6C2411yescharacteristic
292C3×M16—24no—
293C2×C3×C8—22no—
294C2×C3×C8—22no—
295C2×C3×C8C2412yes—
296C2×C3×C8C2414yes—
297C2×C3×C8C2411yescharacteristic
298C2×C3×C8C2411yescharacteristic
299C3×C4⋊C4C2×D414yes—
300C3×C4⋊C4C2×D412yes—
301C3×C4⋊C4C2×D414yes—
302C3×C4⋊C4C2×D414yes—
303C3×C4⋊C4C2×D412yes—
304C3×C22⋊C4C2×D414yes—
305C3×C42C2×D411yescharacteristic
306C3×C42C2×D411yescharacteristic
307C3×C42C2414yes—
308C3×C42C2412yes—
309#217—21no—
310#217—21no—
311(C2×D3)⋊C4—22no—
312(C2×D3)⋊C4—22no—
313(C2×D3)⋊C4—24no—
314(C2×D3)⋊C4—24no—
315(C2×D3)⋊C4—24no—
316(C2×D3)⋊C4—24no—
317(C2×D3)⋊C4—24no—
318C4⋊Dic3—24no—
319C4⋊Dic3—22no—
320C4⋊Dic3C2412yes—
321C4⋊Dic3C2412yes—
322C4⋊Dic3C2414yes—
323Dic3⋊C4—22no—
324Dic3⋊C4—22no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330C4×Dic3—22no—
331C4×Dic3—22no—
332C4×Dic3—24no—
333C4×Dic3—21no—
334C4×Dic3—21no—
335Dic12—22no—
336D24—22no—
337D24—28no—
338#204—22no—
339#204—22no—
340#204—28no—
341C2×#41—31no—
342C2×#41—31no—
343C2×#41—64no—
344C22×D4—32no—
345C22×D4—62no—
346AGL(1,ℤ/8)—68no—
347#135—64no—
348C2×Q16—31no—
349C2×SD16—31no—
350C2×SD16—31no—
351C2×SD16—34no—
352C2×D8—31no—
353C2×D8—34no—
354C2×M16C22×D312yes—
355C22×C8C22×D311yescharacteristic
356C22×C8C22×D311yescharacteristic
357C4⋊Q8—34no—
358C4⋊Q8—64no—
359C4⋊Q8—62no—
360C4⋊Q8D1212yes—
361#127—34no—
362#127—34no—
363#127—61no—
364#127—61no—
365#125—34no—
366#125—34no—
367#125—62no—
368#125—64no—
369#125—64no—
370#125—61no—
371#125—61no—
372#125D1211yescharacteristic
373#125D1211yescharacteristic
374#124—34no—
375#124—64no—
376#123—64no—
377#123—64no—
378C22⋊Q8—64no—
379C22⋊Q8—64no—
380C22⋊Q8—64no—
381C22⋊Q8—64no—
382C22⋊Q8D1214yes—
383#121—62no—
384#121—62no—
385#121—64no—
386#121—64no—
387#121—64no—
388#120—64no—
389C4×Q8—32no—
390C4×Q8—32no—
391C4×Q8—34no—
392C4×Q8—64no—
393C4×Q8C22×D314yes—
394C4×D4—32no—
395C4×D4—32no—
396C4×D4—34no—
397C4×D4—34no—
398C4×D4—34no—
399C4×D4—62no—
400C4×D4—62no—
401C4×D4—62no—
402C4×D4—62no—
403C4×D4—64no—
404C4×D4—64no—
405#117—31no—
406#117—31no—
407#117—62no—
408#117—61no—
409#117—61no—
410#117C22×D312yes—
411C2×C4⋊C4—32no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—62no—
414C2×C4⋊C4—64no—
415C2×C22⋊C4—64no—
416C2×C42—61no—
417C2×C42—61no—
418C2×C42C22×D311yescharacteristic
419#107—32no—
420#107—34no—
421#106—32no—
422#106—34no—
423C4⋊C8C22×D314yes—
424C4⋊C8C22×D314yes—
425C4⋊C8C22×D314yes—
426#103—32no—
427#103—32no—
428#103—34no—
429#102—34no—
430#102—34no—
431#102—34no—
432#102—34no—
433#102—32no—
434#102—32no—
435#102—34no—
436#102—34no—
437#102—34no—
438#97C22×D312yes—
439#97C22×D312yes—
440#97C22×D314yes—
441C4×C8C22×D314yes—
442C23×C3C22×D411yescharacteristic
443C22×D3—21no—
444C22×D3—24no—
445C22×D3—21no—
446C22×D3—44no—
447C22×D3—82no—
448C22×D3—82no—
449C3×Q8—44no—
450C2×C3×C4—42no—
451C2×C3×C4C22×D412yes—
452C2×C3×C4C22×D412yes—
453C2×C3×C4C22×D414yes—
454C2×C3×C4C22×D414yes—
455C2×C3×C4C22×D411yescharacteristic
456C2×C3×C4C2×#4112yes—
457C2×C3×C42−1+412yes—
458C2×C3×C4C2511yescharacteristic
459#67—24no—
460#67—42no—
461C2×Dic3—21no—
462C2×Dic3—22no—
463C2×Dic3—21no—
464C2×Dic3—22no—
465C2×Dic3—24no—
466C2×Dic3—82no—
467D12—24no—
468D12—21no—
469D12—21no—
470D12—24no—
471D12—21no—
472D12—48no—
473D12—44no—
474D12—84no—
475D12—84no—
476D12—84no—
477D12C2×#4112yes—
478C4×D3—24no—
479C4×D3—44no—
480C4×D3—42no—
481C4×D3—82no—
482C4×D3—84no—
483C4×D3—84no—
484C4×D3—84no—
485Dic6—21no—
486Dic6—21no—
487Dic6—21no—
488Dic6—84no—
489Dic6C2×#4112yes—
490C3×C8—21no—
491C3×C8—24no—
492C3×C8—21no—
493C3×C8—22no—
494C24—62no—
495#41—34no—
496#41—62no—
497#41—62no—
498#41—62no—
499#41—248no—
500C2×Q8—31no—
501C2×Q8—31no—
502C2×Q8—64no—
503C2×Q8C2×D1214yes—
504C2×D4—31no—
505C2×D4—34no—
506C2×D4—31no—
507C2×D4—61no—
508C2×D4—61no—
509C2×D4—62no—
510C2×D4—64no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—64no—
515C2×D4—64no—
516C2×D4—244no—
517C22×C4—61no—
518C22×C4—62no—
519C22×C4—61no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4—64no—
523C22×C4—64no—
524C22×C4—64no—
525C22×C4C2×D1212yes—
526C22×C4C23×D311yescharacteristic
527Q16—62no—
528SD16—62no—
529SD16—62no—
530SD16—68no—
531D8—62no—
532D8—68no—
533M16—24no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8C23×D312yes—
537C2×C8C23×D314yes—
538C2×C8C23×D311yescharacteristic
539C2×C8C23×D311yescharacteristic
540C4⋊C4—34no—
541C4⋊C4—32no—
542C4⋊C4—32no—
543C4⋊C4—62no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—64no—
549C4⋊C4—62no—
550C4⋊C4—62no—
551C4⋊C4—64no—
552C4⋊C4C2×D1212yes—
553C4⋊C4C2×D1214yes—
554C4⋊C4C2×D1212yes—
555C4⋊C4C2×D1214yes—
556C4⋊C4C2×D1214yes—
557C22⋊C4—61no—
558C22⋊C4—61no—
559C22⋊C4—64no—
560C22⋊C4—64no—
561C22⋊C4—62no—
562C22⋊C4—62no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C22⋊C4—64no—
566C22⋊C4C2×D1214yes—
567C42—61no—
568C42—61no—
569C42—62no—
570C42—62no—
571C42—64no—
572C42C2×D1211yescharacteristic
573C42C2×D1211yescharacteristic
574C42C23×D314yes—
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3#54812yes—
578C22×C3C23×D411yescharacteristic
579C2×D3—22no—
580C2×D3—42no—
581C2×D3—42no—
582C2×D3—44no—
583C2×D3—44no—
584C2×D3—41no—
585C2×D3—44no—
586C2×D3—164no—
587C3×C4—24no—
588C3×C4—22no—
589C3×C4—24no—
590C3×C4—22no—
591C3×C4#54812yes—
592C3×C4C22×#4111yescharacteristic
593C3×C4C2×2−1+411yescharacteristic
594Dic3—22no—
595Dic3—42no—
596Dic3—44no—
597Dic3—42no—
598Dic3—41no—
599C23—61no—
600C23—64no—
601C23—61no—
602C23—124no—
603C23—242no—
604C23—242no—
605C23C22×D1211yescharacteristic
606Q8—32no—
607Q8—44no—
608Q8—61no—
609Q8—61no—
610Q8—61no—
611Q8—244no—
612D4—32no—
613D4—64no—
614D4—61no—
615D4—64no—
616D4—61no—
617D4—64no—
618D4—61no—
619D4—128no—
620D4—122no—
621D4—124no—
622D4—244no—
623D4—244no—
624D4—244no—
625C2×C4—42no—
626C2×C4—61no—
627C2×C4—64no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—64no—
632C2×C4—124no—
633C2×C4—122no—
634C2×C4—244no—
635C2×C4—244no—
636C2×C4—242no—
637C2×C4—244no—
638C2×C4—242no—
639C2×C4C22×D1212yes—
640C2×C4C22×D1212yes—
641C2×C4C22×D1214yes—
642C2×C4C22×D1214yes—
643C2×C4C22×D1211yescharacteristic
644C2×C4#100712yes—
645C2×C4D3×#4112yes—
646C2×C4C24×D311yescharacteristic
647C8—22no—
648C8—21no—
649C8—21no—
650C8—24no—
651C2×C3—22no—
652C2×C3C2×#54811yescharacteristic
653C2×C3#358512yes—
654D3—42no—
655D3—81no—
656D3—84no—
657C22—22no—
658C22—62no—
659C22—122no—
660C22—121no—
661C22—124no—
662C22—122no—
663C22—124no—
664C22—124no—
665C22—484no—
666C22#557612yes—
667C22C23×D1211yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—24no—
672C4—62no—
673C4—121no—
674C4—122no—
675C4—124no—
676C4—122no—
677C4#557612yes—
678C4C2×#100711yescharacteristic
679C4C2×D3×#4111yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—122no—
683C2—244no—
684C2—241no—
685C2C2×#557611yescharacteristic
686C2#8716712yes—
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2961C2
socleC22×C3C23×D4
hypercenter#2961D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #2961

Complements

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

NH ≅ G/Ntype
1. #86282C2semidirect
2. #86261C2semidirect
3. C3×#2961C2semidirect
4. #79982C2semidirect
5. #79980C2semidirect
6. #79971C2semidirect
7. #79969C2semidirect
8. #79090C2semidirect
9. #79088C2semidirect
10. #79061C2semidirect
11. #79056C2semidirect
12. #78563C2semidirect
13. #78560C2semidirect
14. C2×#4713C2semidirect
15. #78494C2semidirect
29. #5579C22semidirect
31. #5567C22semidirect
40. C3×#446C22semidirect
41. C3×#446C22semidirect
55. #4848C22semidirect
56. #4847C22semidirect
57. #4847C22semidirect
58. #4847C22semidirect
59. #4846C22semidirect
60. #4845C22semidirect
61. #4844C22semidirect
63. #4843C22semidirect
64. #4842C22semidirect
65. #4829C22semidirect
66. #4716C22semidirect
67. #4714C22semidirect
70. #4713C22semidirect
71. #4711C22semidirect
72. #4711C22semidirect
73. #4710C22semidirect
74. #4710C22semidirect
75. #4710C22semidirect
76. #4697C22semidirect
77. #4694C22semidirect
78. C4×D24C22semidirect
79. C4×#204C22semidirect
92. #2961D3semidirect
237. #446C2×D3semidirect
238. #446C2×D3semidirect
270. #235#41semidirect
680. C3#61337semidirect

Maximal subgroups

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

subgroupindex
#862822
#862612
C3×#29612
#799822
#799802
#799712
#799692
#790902
#790882
#790612
#790562
#785632
#785602
C2×#47132
#784942
#613373