Finite Group Databasebeta

← #11675345

Subgroups — #11675345

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#86098C211yesmaximal, characteristic, complemented
2#85848C211yesmaximal, characteristic, complemented
3C3×#3080C211yesmaximal, characteristic, complemented
4C2×#5067C211yesmaximal, characteristic, complemented
5#82506C211yesmaximal, characteristic, complemented
6#81218C214yesmaximal, complemented
7#81217C214yesmaximal, complemented
8#81153C214yesmaximal, complemented
9#81152C214yesmaximal, complemented
10#79242C212yesmaximal, complemented
11#79201C212yesmaximal, complemented
12#79056C211yesmaximal, characteristic, complemented
13#79045C211yesmaximal, characteristic, complemented
14#78911C212yesmaximal, complemented
15#78876C212yesmaximal, complemented
16#61337—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5830C2212yes—
20C2×#950C2211yescharacteristic
21C2×#908C2212yes—
22#5690C2214yescomplemented
23#5689—22no—
24#5688—21no—
25D3×#125—21no—
26#5685C2214yescomplemented
27#5585C2214yes—
28#5553C2214yes—
29#5550—24no—
30#5535—21no—
31#5535—21no—
32#5521C2212yes—
33#5511—24no—
34#5504C2212yes—
35#5500C2212yes—
36#5477—22no—
37C2×#874C2211yescharacteristic
38C4×#235C2211yescharacteristic
39#5470C2211yescharacteristic
40C3×#479C2214yescomplemented
41C3×#478C2214yescomplemented
42C3×#428C2212yes—
43C3×#423C2211yescharacteristic
44C3×#417C2212yes—
45#5233C2211yescharacteristic
46#5131C2212yes—
47#5122C2212yes—
48#5119C2212yes—
49#5067C2214yes—
50#5067C2214yes—
51#5030C2212yescomplemented
52#5026C2212yescomplemented
53#5018C2212yescomplemented
54#5014C2212yescomplemented
55#5002C2211yescharacteristic
56#4998C2211yescharacteristic
57C2×#809C2212yes—
58C2×#809C2212yes—
59C2×#803C2211yescharacteristic
60#4891C2214yescomplemented
61#4890C2214yescomplemented
62#4889C2214yescomplemented
63#4872C2214yescomplemented
64#4871C2214yescomplemented
65#4870C2214yescomplemented
66#4869C2214yescomplemented
67#4852C2214yescomplemented
68#4848C2214yes—
69#4843C2214yes—
70#4828C2214yescomplemented
71#4828C2214yescomplemented
72#4819C2214yescomplemented
73#4800C2214yescomplemented
74#4798C2214yescomplemented
75#4794C2214yescomplemented
76#4786C2214yescomplemented
77#4764C2214yescomplemented
78#4762C2214yescomplemented
79#4760C2214yescomplemented
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#3080D311yescharacteristic, complemented
90#3045—32no—
91#3038—31no—
92#2961—31no—
93#2946—32no—
94#2943—31no—
95C2×C3×#41C2311yescharacteristic
96C2×#235—24no—
97C2×#235C2311yescharacteristic
98C22×D12—22no—
99C22×D12C2312yes—
100C2×C3×M16C2312yes—
101C3×#125—21no—
102C3×#125—21no—
103C3×#125C2314yes—
104C3×C4×Q8C2312yes—
105C3×C4×D4—22no—
106C3×C4×D4C2312yes—
107C3×#117—21no—
108C3×#117C2311yescharacteristic
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113#950—24no—
114#947C2314yes—
115C2×#216C2311yescharacteristic
116C2×#215C2311yescharacteristic
117#940C2314yes—
118#937—24no—
119C2×#214C2311yescharacteristic
120C2×#213C2311yescharacteristic
121#930—22no—
122#930—22no—
123#930—24no—
124C4×#67—22no—
125C4×#67—24no—
126#926C2311yescharacteristic
127#926C2312yes—
128#924—24no—
129#924D414yes—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132#908—28no—
133C2×D24C2314yes—
134C2×#204C2314yes—
135#897—24no—
136#896—24no—
137#896—24no—
138#894—24no—
139#894—24no—
140#893C2314yes—
141#893C2314yes—
142#893C2314yes—
143#892—22no—
144D3×C4⋊C4—22no—
145D3×C4⋊C4—24no—
146#890—22no—
147#890C2314yes—
148#889—24no—
149#889—24no—
150#888—24no—
151#888—22no—
152#888C2314yes—
153#887—24no—
154#887C2314yes—
155#884—24no—
156#884—24no—
157D3×C22⋊C4—24no—
158#875—24no—
159#874—21no—
160#874—21no—
161#874C2314yes—
162C4×D12—22no—
163C4×D12—22no—
164C4×D12—24no—
165C4×D12C2312yes—
166#872—21no—
167C42×D3—21no—
168#870D411yescharacteristic
169#870D411yescharacteristic
170#869D412yes—
171C4×Dic6C2312yes—
172C4×Dic6C2314yes—
173C3×#107C2314yes—
174C3×#106C2314yes—
175C3×C4⋊C8C2314yes—
176C3×#103C2314yes—
177C3×#102C2314yes—
178#835C2312yes—
179#832C2312yes—
180#821C2314yes—
181#821C2314yes—
182#815C2314yes—
183Dic3⋊C8C2314yes—
184Dic3⋊C8C2314yes—
185C8×Dic3C2314yes—
186#810C2312yes—
187#809C2312yes—
188#809C2314yes—
189#809C2314yes—
190#809C2314yes—
191#809C2314yes—
192#808C2312yes—
193#807C2312yes—
194#803C2312yes—
195#803C2312yes—
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—31no—
210#533—61no—
211#532—62no—
212C2×#127—31no—
213C2×#125—61no—
214C2×#125C2×D311yescharacteristic
215#520—32no—
216#518—32no—
217C4×#41—31no—
218C4×#41C2×D311yescharacteristic
219#490—34no—
220#490—34no—
221#490—34no—
222#489—34no—
223#488—34no—
224#486—34no—
225#479C2×D314yescomplemented
226#478—34no—
227#478—34no—
228#478C2×D314yescomplemented
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#446—34no—
238#446—34no—
239#442—32no—
240#442—34no—
241#441—32no—
242#440—32no—
243#440—32no—
244#440—34no—
245C4×SD16—34no—
246C4×D8—34no—
247#428C2×D312yes—
248#427—31no—
249#424—32no—
250#423C2×D311yescharacteristic
251#418—32no—
252#417C2×D312yes—
253#416—31no—
254#416—31no—
255C2×#102—32no—
256C2×#102—32no—
257#405—32no—
258C2×#97—31no—
259C23×D3—22no—
260C3×#41—22no—
261C3×#41—22no—
262C2×C3×Q8C2411yescharacteristic
263C2×C3×D4—21no—
264C2×C3×D4C2411yescharacteristic
265C22×C3×C4—22no—
266C22×C3×C4—21no—
267C22×C3×C4—22no—
268C22×C3×C4C2×D412yes—
269C22×C3×C4C2411yescharacteristic
270C2×#67—21no—
271C2×#67—24no—
272C2×#67—24no—
273#235—22no—
274#235—88no—
275#235#4114yescomplemented
276C2×D12—24no—
277C2×D12—24no—
278C2×D12—22no—
279C2×D12—24no—
280C2×D12—24no—
281C2×D12—84no—
282C2×D12C2414yes—
283C2×D12C2411yescharacteristic
284C2×C4×D3—21no—
285C2×C4×D3—22no—
286C2×C4×D3—24no—
287C2×C4×D3—24no—
288C2×C4×D3—24no—
289C2×Dic6—24no—
290C2×Dic6C2×D414yes—
291C2×Dic6C2411yescharacteristic
292C3×M16—24no—
293C2×C3×C8C2414yes—
294C3×C4⋊C4—24no—
295C3×C4⋊C4—24no—
296C3×C4⋊C4—22no—
297C3×C4⋊C4—22no—
298C3×C4⋊C4C2412yes—
299C3×C4⋊C4C2414yes—
300C3×C22⋊C4—21no—
301C3×C22⋊C4—22no—
302C3×C42—21no—
303C3×C42—22no—
304C3×C42C2×D411yescharacteristic
305C3×C42C2×D411yescharacteristic
306C3×C42C2412yes—
307#217—21no—
308#217C2×D414yes—
309#216—22no—
310#215—22no—
311#214—22no—
312#213—22no—
313(C2×D3)⋊C4—22no—
314(C2×D3)⋊C4—24no—
315(C2×D3)⋊C4—24no—
316(C2×D3)⋊C4—24no—
317(C2×D3)⋊C4—24no—
318(C2×D3)⋊C4—24no—
319C4⋊Dic3—24no—
320C4⋊Dic3C2×D414yes—
321C4⋊Dic3C2×D412yes—
322C4⋊Dic3C2×D412yes—
323C4⋊Dic3C2412yes—
324Dic3⋊C4—22no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4C2×D414yes—
329Dic3⋊C4C2×D414yes—
330C4×Dic3—22no—
331C4×Dic3—21no—
332C4×Dic3—24no—
333C4×Dic3C2414yes—
334C2×C3⋊C8—22no—
335C2×C3⋊C8—22no—
336C2×C3⋊C8C2411yescharacteristic
337C2×C3⋊C8C2411yescharacteristic
338C2×C3⋊C8C2412yes—
339D24—28no—
340#204—28no—
341C2×#41—31no—
342C2×#41—64no—
343C2×#41C22×D311yescharacteristic
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×C8—31no—
356C22×C8—31no—
357C4⋊Q8—34no—
358C4⋊Q8—32no—
359C4⋊Q8—62no—
360C4⋊Q8—64no—
361#127—34no—
362#127—34no—
363#127—61no—
364#127—61no—
365#125—21no—
366#125—21no—
367#125—34no—
368#125—31no—
369#125—31no—
370#125—62no—
371#125—64no—
372#125—64no—
373#125C22×D314yes—
374#124—34no—
375#124—64no—
376#123—64no—
377#123—64no—
378C22⋊Q8—34no—
379C22⋊Q8—64no—
380C22⋊Q8—64no—
381C22⋊Q8—64no—
382C22⋊Q8—64no—
383#121—64no—
384#121—64no—
385#121—62no—
386#121—62no—
387#121—64no—
388#120—64no—
389C4×Q8—34no—
390C4×Q8—34no—
391C4×Q8—32no—
392C4×Q8—64no—
393C4×Q8C22×D312yes—
394C4×D4—22no—
395C4×D4—34no—
396C4×D4—34no—
397C4×D4—34no—
398C4×D4—32no—
399C4×D4—62no—
400C4×D4—62no—
401C4×D4—64no—
402C4×D4—64no—
403C4×D4—62no—
404C4×D4C22×D312yes—
405#117—21no—
406#117—32no—
407#117—31no—
408#117—62no—
409#117—61no—
410#117C22×D311yescharacteristic
411C2×C4⋊C4—22no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—64no—
414C2×C4⋊C4C22×D312yes—
415C2×C22⋊C4—64no—
416C2×C42—21no—
417C2×C42—61no—
418C2×C42C22×D311yescharacteristic
419#107—32no—
420#107C22×D314yes—
421#106—32no—
422#106C22×D314yes—
423C4⋊C8—34no—
424C4⋊C8—34no—
425C4⋊C8C22×D314yes—
426#103—32no—
427#103—32no—
428#103C22×D314yes—
429#102—32no—
430#102—34no—
431#102—34no—
432#102—34no—
433#102—34no—
434#102—34no—
435#102—34no—
436#102—32no—
437#102C22×D314yes—
438#97—34no—
439#97—32no—
440#97—32no—
441C4×C8—34no—
442C23×C3—21no—
443C23×C3C22×D411yescharacteristic
444C22×D3—21no—
445C22×D3—24no—
446C22×D3—44no—
447C22×D3—82no—
448C22×D3—82no—
449C3×Q8—21no—
450C3×Q8—21no—
451C3×D4—21no—
452C3×D4—21no—
453C3×D4—42no—
454C2×C3×C4—24no—
455C2×C3×C4—21no—
456C2×C3×C4—22no—
457C2×C3×C4—42no—
458C2×C3×C4—44no—
459C2×C3×C4—42no—
460C2×C3×C4—82no—
461C2×C3×C4C22×D412yes—
462C2×C3×C4C22×D412yes—
463C2×C3×C4C22×D411yescharacteristic
464C2×C3×C4C2×#4112yes—
465C2×C3×C42−1+412yes—
466C2×C3×C4C2511yescharacteristic
467#67—24no—
468#67—84no—
469#67—84no—
470C2×Dic3—21no—
471C2×Dic3—22no—
472C2×Dic3C22×D414yes—
473C2×Dic3C22×D414yes—
474D12—24no—
475D12—21no—
476D12—24no—
477D12—48no—
478D12—44no—
479D12—84no—
480D12C2×#4112yes—
481C4×D3—24no—
482C4×D3—84no—
483C4×D3—84no—
484C4×D3—84no—
485C4×D3—82no—
486Dic6—21no—
487Dic6—44no—
488Dic6—84no—
489Dic6C2×#4112yes—
490C3×C8—24no—
491C3⋊C8—21no—
492C3⋊C8—21no—
493C3⋊C8—22no—
494C24—62no—
495#41—22no—
496#41—22no—
497#41—34no—
498#41—62no—
499#41—248no—
500C2×Q8—34no—
501C2×Q8—31no—
502C2×Q8—64no—
503C2×Q8C23×D311yescharacteristic
504C2×D4—21no—
505C2×D4—31no—
506C2×D4—34no—
507C2×D4—64no—
508C2×D4—64no—
509C2×D4—61no—
510C2×D4—62no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—64no—
515C2×D4—244no—
516C2×D4C23×D311yescharacteristic
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—64no—
521C22×C4—64no—
522C22×C4—64no—
523C22×C4—61no—
524C22×C4—62no—
525C22×C4D3×D412yes—
526C22×C4C23×D311yescharacteristic
527Q16—62no—
528SD16—68no—
529SD16—62no—
530SD16—62no—
531D8—68no—
532D8—62no—
533M16—24no—
534C2×C8—32no—
535C2×C8—31no—
536C2×C8—31no—
537C2×C8—62no—
538C2×C8—62no—
539C2×C8C23×D314yes—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—32no—
545C4⋊C4—34no—
546C4⋊C4—32no—
547C4⋊C4—34no—
548C4⋊C4—34no—
549C4⋊C4—32no—
550C4⋊C4—64no—
551C4⋊C4—64no—
552C4⋊C4—64no—
553C4⋊C4—64no—
554C4⋊C4—62no—
555C4⋊C4C23×D314yes—
556C4⋊C4C23×D312yes—
557C22⋊C4—21no—
558C22⋊C4—22no—
559C22⋊C4—34no—
560C22⋊C4—64no—
561C22⋊C4—64no—
562C22⋊C4—64no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C22⋊C4—61no—
566C22⋊C4—62no—
567C42—21no—
568C42—22no—
569C42—34no—
570C42—64no—
571C42—61no—
572C42—62no—
573C42D3×D411yescharacteristic
574C42D3×D411yescharacteristic
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3—42no—
578C22×C3—41no—
579C22×C3#54812yes—
580C22×C3C23×D411yescharacteristic
581C2×D3—22no—
582C2×D3—42no—
583C2×D3—44no—
584C2×D3—44no—
585C2×D3—44no—
586C2×D3—164no—
587C3×C4—22no—
588C3×C4—22no—
589C3×C4—44no—
590C3×C4—42no—
591C3×C4—41no—
592C3×C4#54812yes—
593C3×C4C22×#4111yescharacteristic
594C3×C4C2×2−1+411yescharacteristic
595Dic3—24no—
596Dic3—24no—
597Dic3—22no—
598Dic3—42no—
599C23—21no—
600C23—61no—
601C23—64no—
602C23—124no—
603C23—242no—
604C23—242no—
605C23C2×D3×D411yescharacteristic
606Q8—21no—
607Q8—21no—
608Q8—32no—
609Q8—61no—
610Q8—124no—
611Q8—244no—
612D4—21no—
613D4—21no—
614D4—32no—
615D4—42no—
616D4—64no—
617D4—61no—
618D4—64no—
619D4—64no—
620D4—128no—
621D4—124no—
622D4—244no—
623D4—244no—
624D4—244no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—34no—
629C2×C4—34no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—61no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—82no—
637C2×C4—242no—
638C2×C4—244no—
639C2×C4—244no—
640C2×C4—244no—
641C2×C4C2×D3×D412yes—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D411yescharacteristic
644C2×C4#100712yes—
645C2×C4D3×#4112yes—
646C2×C4C24×D311yescharacteristic
647C8—24no—
648C8—62no—
649C8—61no—
650C8—61no—
651C2×C3—22no—
652C2×C3—81no—
653C2×C3C2×#54811yescharacteristic
654C2×C3#358512yes—
655D3—42no—
656D3—84no—
657C22—22no—
658C22—42no—
659C22—41no—
660C22—62no—
661C22—122no—
662C22—124no—
663C22—124no—
664C22—124no—
665C22—484no—
666C22#563312yes—
667C22C22×D3×D411yescharacteristic
668C4—22no—
669C4—22no—
670C4—42no—
671C4—44no—
672C4—41no—
673C4—64no—
674C4—64no—
675C4—62no—
676C4—122no—
677C4#563312yes—
678C4C2×#100711yescharacteristic
679C4C2×D3×#4111yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—81no—
683C2—122no—
684C2—244no—
685C2C2×#563311yescharacteristic
686C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3080C2
socleC22×C3C23×D4
hypercenter#3080D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
1. #86098C2semidirect
2. #85848C2semidirect
3. C3×#3080C2semidirect
4. C2×#5067C2semidirect
5. #82506C2semidirect
6. #81218C2semidirect
7. #81217C2semidirect
8. #81153C2semidirect
9. #81152C2semidirect
10. #79242C2semidirect
11. #79201C2semidirect
12. #79056C2semidirect
13. #79045C2semidirect
14. #78911C2semidirect
15. #78876C2semidirect
22. #5690C22semidirect
26. #5685C22semidirect
40. C3×#479C22semidirect
41. C3×#478C22semidirect
51. #5030C22semidirect
52. #5026C22semidirect
53. #5018C22semidirect
54. #5014C22semidirect
60. #4891C22semidirect
61. #4890C22semidirect
62. #4889C22semidirect
63. #4872C22semidirect
64. #4871C22semidirect
65. #4870C22semidirect
66. #4869C22semidirect
67. #4852C22semidirect
70. #4828C22semidirect
71. #4828C22semidirect
72. #4819C22semidirect
73. #4800C22semidirect
74. #4798C22semidirect
75. #4794C22semidirect
76. #4786C22semidirect
77. #4764C22semidirect
78. #4762C22semidirect
79. #4760C22semidirect
89. #3080D3semidirect
225. #479C2×D3semidirect
228. #478C2×D3semidirect
275. #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
#860982
#858482
C3×#30802
C2×#50672
#825062
#812182
#812172
#811532
#811522
#792422
#792012
#790562
#790452
#789112
#788762
#613373