Finite Group Databasebeta

← #11675353

Subgroups — #11675353

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#86190C211yesmaximal, characteristic, complemented
2#86098C211yesmaximal, characteristic, complemented
3C2×C3×#490C211yesmaximal, characteristic, complemented
4#83165C211yesmaximal, characteristic, complemented
5#83017C211yesmaximal, characteristic, complemented
6#81711C214yesmaximal, complemented
7#81710C214yesmaximal, complemented
8#81709C214yesmaximal, complemented
9#81708C214yesmaximal, complemented
10#78767C212yesmaximal, complemented
11#78737C212yesmaximal, complemented
12#78731C212yesmaximal, complemented
13#78726C212yesmaximal, complemented
14#78494C211yesmaximal, characteristic, complemented
15#78480C211yesmaximal, characteristic, complemented
16#61337—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19#5806—24no—
20#5805C2212yes—
21#5798C2212yes—
22C2×#932C2212yes—
23C2×#904C2211yescharacteristic
24#5719—21no—
25#5719—21no—
26#5718C2214yes—
27#5716—22no—
28#5713C2214yes—
29#5690C2214yescomplemented
30#5689—22no—
31#5688—21no—
32D3×#125—21no—
33#5685C2214yescomplemented
34#5511—24no—
35#5504C2212yes—
36#5500C2212yes—
37C4×#235C2211yescharacteristic
38C4×#235C2211yescharacteristic
39#5470C2211yescharacteristic
40C3×#490C2214yes—
41C3×#490C2214yes—
42C2×C3×#102C2212yes—
43C2×C3×#102C2212yes—
44C2×C3×#97C2211yescharacteristic
45#5215C2212yes—
46#5113C2211yescharacteristic
47#5113C2211yescharacteristic
48#5106C2211yescharacteristic
49#5102C2212yes—
50#5101C2212yes—
51#5075C2214yes—
52#5072C2214yes—
53#5066C2214yescomplemented
54#5063C2214yescomplemented
55#4966C2212yes—
56#4964C2212yes—
57#4924C2211yescharacteristic
58#4787C2214yescomplemented
59#4786C2214yescomplemented
60#4785C2214yescomplemented
61#4768C2214yescomplemented
62#4767C2214yescomplemented
63#4766C2214yescomplemented
64#4765C2214yescomplemented
65#4764C2214yescomplemented
66#4763C2214yescomplemented
67#4762C2214yescomplemented
68#4761C2214yescomplemented
69#4760C2214yescomplemented
70#4759C2214yescomplemented
71#4758C2214yescomplemented
72#4757C2214yescomplemented
73#4756C2214yescomplemented
74#4716C2212yescomplemented
75#4711C2212yescomplemented
76#4710C2212yescomplemented
77#4710C2212yescomplemented
78#4701C2214yescomplemented
79#4701C2214yescomplemented
80#3508—31no—
81#3504—31no—
82#3371—34no—
83#3370—34no—
84#3368—34no—
85#3366—34no—
86#3136—32no—
87C2×#490D311yescharacteristic, complemented
88#3081—32no—
89#3080—31no—
90#3045—32no—
91#3038—31no—
92#2961—31no—
93#2946—32no—
94#2943—31no—
95C22×C3×D4—22no—
96C22×C3×D4C2312yes—
97C2×#237—24no—
98C2×#235C2311yescharacteristic
99C2×#235C2311yescharacteristic
100C22×C3×C8C2311yescharacteristic
101C22×C3×C8C2311yescharacteristic
102C3×#127—21no—
103C3×#127—21no—
104C3×#127C2314yes—
105C3×#125—21no—
106C3×#125—21no—
107C3×#125C2314yes—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C42C2311yescharacteristic
111#953—24no—
112C2×#217—24no—
113#940C2314yes—
114#939—24no—
115#939—24no—
116#938—24no—
117#938—22no—
118#938—22no—
119#936—24no—
120#936C2314yes—
121#935—24no—
122#935—24no—
123D4×Dic3—24no—
124D4×Dic3—22no—
125D4×Dic3—22no—
126D4×Dic3C2314yes—
127D4×Dic3C2314yes—
128C2×#214C2314yes—
129#932—28no—
130C2×#213C2314yes—
131C4×#67—22no—
132C4×#67—22no—
133C4×#67—24no—
134#926C2311yescharacteristic
135#926C2311yescharacteristic
136#926C2312yes—
137#924—24no—
138#924D414yes—
139C2×#208C2312yes—
140C2×Dic12C2311yescharacteristic
141#904—24no—
142C2×D24C2311yescharacteristic
143C2×#204C2311yescharacteristic
144C2×#204C2311yescharacteristic
145#897—24no—
146#896—24no—
147#896—24no—
148#893C2314yes—
149#892—22no—
150D3×C4⋊C4—22no—
151D3×C4⋊C4—24no—
152#890—22no—
153#890C2314yes—
154#889—24no—
155#889—24no—
156#888—24no—
157#888—22no—
158#888C2314yes—
159#887—24no—
160#887C2314yes—
161C4×D12C2312yes—
162C4×D12C2312yes—
163#872—21no—
164#872—21no—
165C42×D3—21no—
166C42×D3—21no—
167#870D411yescharacteristic
168#870D411yescharacteristic
169#869D412yes—
170C4×Dic6C2312yes—
171C4×Dic6C2312yes—
172C4×Dic6C2314yes—
173C3×#102C2314yes—
174C3×#102C2314yes—
175C3×#102C2314yes—
176C3×#102C2314yes—
177C3×#97C2312yes—
178C3×#97C2312yes—
179#832C2314yes—
180#832C2314yes—
181#821C2312yes—
182#821C2312yes—
183#818C2312yes—
184#817C2312yes—
185#816C2312yes—
186#816C2312yes—
187#815C2314yes—
188Dic3⋊C8C2314yes—
189Dic3⋊C8C2314yes—
190C8×Dic3C2314yes—
191#810C2314yes—
192#809C2314yes—
193#808C2314yes—
194#807C2314yes—
195C4⋊(C3⋊C8)C2314yes—
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×#127C2×D311yescharacteristic
213C2×#125—61no—
214C2×#125C2×D311yescharacteristic
215#520—32no—
216#518—32no—
217C4×#41—31no—
218C4×#41—31no—
219#490—34no—
220#490C2×D314yes—
221#490C2×D314yes—
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#446—34no—
238#446—34no—
239#442—34no—
240#442—32no—
241#441—32no—
242#440—34no—
243#440—32no—
244#440—32no—
245C4×SD16—34no—
246C4×D8—34no—
247#428—32no—
248#427—31no—
249#424—32no—
250#423—31no—
251#418—32no—
252#417—32no—
253#416—31no—
254#416—31no—
255C2×#102C2×D312yes—
256C2×#102C2×D312yes—
257#405—32no—
258C2×#97C2×D311yescharacteristic
259C24×C3—22no—
260C2×C3×D4—22no—
261C2×C3×D4—24no—
262C2×C3×D4—24no—
263C2×C3×D4—24no—
264C2×C3×D4—24no—
265C2×C3×D4—84no—
266C2×C3×D4C2414yes—
267C22×C3×C4—22no—
268C22×C3×C4C2×D412yes—
269C22×C3×C4C2411yescharacteristic
270C2×#67—21no—
271C2×#67—21no—
272C2×#67—24no—
273C2×#67—24no—
274C22×Dic3—24no—
275C22×Dic3—24no—
276#237—88no—
277#235—22no—
278#235—22no—
279#235—22no—
280#235#4114yescomplemented
281C2×D12C2411yescharacteristic
282C2×D12C2411yescharacteristic
283C2×C4×D3—21no—
284C2×C4×D3—21no—
285C2×C4×D3—22no—
286C2×C4×D3—22no—
287C2×C4×D3—24no—
288C2×Dic6—24no—
289C2×Dic6C2×D414yes—
290C2×Dic6C2411yescharacteristic
291C2×Dic6C2411yescharacteristic
292C2×C3×C8—22no—
293C2×C3×C8—22no—
294C2×C3×C8C2412yes—
295C2×C3×C8C2411yescharacteristic
296C2×C3×C8C2411yescharacteristic
297C3×C4⋊C4—24no—
298C3×C4⋊C4—24no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4C2414yes—
301C3×C42C2×D411yescharacteristic
302C3×C42C2×D411yescharacteristic
303C3×C42C2412yes—
304#217—21no—
305#217—21no—
306#217—24no—
307#217—24no—
308#217—24no—
309#217—24no—
310#217C2×D414yes—
311#214—28no—
312#213—28no—
313(C2×D3)⋊C4—22no—
314(C2×D3)⋊C4—22no—
315(C2×D3)⋊C4—24no—
316C4⋊Dic3—24no—
317C4⋊Dic3C2×D414yes—
318C4⋊Dic3C2×D412yes—
319C4⋊Dic3C2×D412yes—
320C4⋊Dic3C2412yes—
321C4⋊Dic3C2412yes—
322Dic3⋊C4—22no—
323Dic3⋊C4—22no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4C2×D414yes—
328Dic3⋊C4C2×D414yes—
329C4×Dic3—22no—
330C4×Dic3—22no—
331C4×Dic3—21no—
332C4×Dic3—21no—
333C4×Dic3—24no—
334C4×Dic3C2414yes—
335#208—24no—
336C2×C3⋊C8C2414yes—
337Dic12—22no—
338D24—22no—
339#204—22no—
340#204—22no—
341C2×#41—31no—
342C2×#41—31no—
343C2×#41—64no—
344C22×D4—22no—
345C22×D4C22×D312yes—
346AGL(1,ℤ/8)—68no—
347#135—64no—
348C2×Q16—31no—
349C2×SD16—34no—
350C2×SD16—31no—
351C2×SD16—31no—
352C2×D8—34no—
353C2×D8—31no—
354C2×M16—32no—
355C22×C8C22×D311yescharacteristic
356C22×C8C22×D311yescharacteristic
357C4⋊Q8—34no—
358C4⋊Q8—32no—
359C4⋊Q8—62no—
360C4⋊Q8—64no—
361#127—21no—
362#127—21no—
363#127—34no—
364#127C22×D314yes—
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—62no—
384#121—62no—
385#121—64no—
386#121—64no—
387#121—64no—
388#120—64no—
389C4×Q8—34no—
390C4×Q8—34no—
391C4×Q8—32no—
392C4×Q8—32no—
393C4×Q8—64no—
394C4×D4—34no—
395C4×D4—34no—
396C4×D4—34no—
397C4×D4—32no—
398C4×D4—32no—
399C4×D4—64no—
400C4×D4—62no—
401C4×D4—62no—
402C4×D4—64no—
403C4×D4—62no—
404C4×D4—62no—
405#117—32no—
406#117—31no—
407#117—31no—
408#117—62no—
409#117—61no—
410#117—61no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—64no—
414C2×C4⋊C4C22×D312yes—
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⋊C8—34no—
424C4⋊C8—34no—
425C4⋊C8—34no—
426#103—32no—
427#103—32no—
428#103—34no—
429#102—34no—
430#102—34no—
431#102—32no—
432#102—32no—
433#102—34no—
434#102C22×D314yes—
435#102C22×D314yes—
436#102C22×D314yes—
437#102C22×D314yes—
438#97—34no—
439#97C22×D312yes—
440#97C22×D312yes—
441C4×C8—34no—
442C23×C3—24no—
443C23×C3—44no—
444C23×C3—82no—
445C23×C3—82no—
446C23×C3C22×D411yescharacteristic
447C22×D3—21no—
448C22×D3—21no—
449C3×D4—24no—
450C3×D4—24no—
451C3×D4—48no—
452C3×D4—44no—
453C3×D4—84no—
454C2×C3×C4—24no—
455C2×C3×C4—42no—
456C2×C3×C4—82no—
457C2×C3×C4C22×D412yes—
458C2×C3×C4C22×D412yes—
459C2×C3×C4C22×D411yescharacteristic
460C2×C3×C4C2×#4112yes—
461C2×C3×C42−1+412yes—
462C2×C3×C4C2511yescharacteristic
463#67—24no—
464#67—42no—
465#67—84no—
466#67—84no—
467C2×Dic3—21no—
468C2×Dic3—22no—
469C2×Dic3—21no—
470C2×Dic3—22no—
471C2×Dic3—84no—
472C2×Dic3—84no—
473C2×Dic3C22×D414yes—
474C2×Dic3C22×D414yes—
475D12—21no—
476D12—21no—
477D12—21no—
478D12C2×#4112yes—
479C4×D3—24no—
480C4×D3—44no—
481C4×D3—42no—
482C4×D3—84no—
483C4×D3—82no—
484Dic6—21no—
485Dic6—21no—
486Dic6—21no—
487Dic6—44no—
488Dic6—84no—
489Dic6C2×#4112yes—
490C3×C8—21no—
491C3×C8—21no—
492C3×C8—22no—
493C3⋊C8—24no—
494C24—22no—
495#41—34no—
496#41—62no—
497#41—62no—
498#41—62no—
499#41—248no—
500C2×Q8—34no—
501C2×Q8—31no—
502C2×Q8—31no—
503C2×Q8—64no—
504C2×D4—22no—
505C2×D4—24no—
506C2×D4—24no—
507C2×D4—24no—
508C2×D4—24no—
509C2×D4—31no—
510C2×D4—31no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—61no—
514C2×D4—61no—
515C2×D4—84no—
516C2×D4C23×D314yes—
517C22×C4—22no—
518C22×C4—64no—
519C22×C4—64no—
520C22×C4—64no—
521C22×C4—61no—
522C22×C4—61no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4D3×D412yes—
526C22×C4C23×D311yescharacteristic
527Q16—62no—
528SD16—62no—
529SD16—62no—
530SD16—68no—
531D8—62no—
532D8—68no—
533M16—64no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—34no—
537C2×C8C23×D312yes—
538C2×C8C23×D311yescharacteristic
539C2×C8C23×D311yescharacteristic
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—32no—
544C4⋊C4—34no—
545C4⋊C4—32no—
546C4⋊C4—34no—
547C4⋊C4—34no—
548C4⋊C4—32no—
549C4⋊C4—32no—
550C4⋊C4—64no—
551C4⋊C4—64no—
552C4⋊C4—64no—
553C4⋊C4—64no—
554C4⋊C4—62no—
555C4⋊C4—62no—
556C4⋊C4C23×D314yes—
557C22⋊C4—34no—
558C22⋊C4—64no—
559C22⋊C4—64no—
560C22⋊C4—64no—
561C22⋊C4—64no—
562C22⋊C4—64no—
563C22⋊C4—61no—
564C22⋊C4—61no—
565C22⋊C4—62no—
566C22⋊C4—62no—
567C42—34no—
568C42—64no—
569C42—61no—
570C42—61no—
571C42—62no—
572C42—62no—
573C42D3×D411yescharacteristic
574C42D3×D411yescharacteristic
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3—44no—
578C22×C3—44no—
579C22×C3—44no—
580C22×C3—164no—
581C22×C3#54812yes—
582C22×C3C23×D411yescharacteristic
583C2×D3—22no—
584C2×D3—42no—
585C2×D3—42no—
586C2×D3—41no—
587C3×C4—22no—
588C3×C4—22no—
589C3×C4—44no—
590C3×C4#54812yes—
591C3×C4C22×#4111yescharacteristic
592C3×C4C2×2−1+411yescharacteristic
593Dic3—24no—
594Dic3—24no—
595Dic3—22no—
596Dic3—42no—
597Dic3—42no—
598Dic3—41no—
599C23—24no—
600C23—44no—
601C23—61no—
602C23—61no—
603C23—82no—
604C23—82no—
605C23C2×D3×D411yescharacteristic
606Q8—32no—
607Q8—61no—
608Q8—61no—
609Q8—61no—
610Q8—124no—
611Q8—244no—
612D4—24no—
613D4—24no—
614D4—32no—
615D4—48no—
616D4—44no—
617D4—64no—
618D4—61no—
619D4—61no—
620D4—61no—
621D4—84no—
622D4—122no—
623D4—244no—
624D4—244no—
625C2×C4—24no—
626C2×C4—34no—
627C2×C4—34no—
628C2×C4—42no—
629C2×C4—61no—
630C2×C4—64no—
631C2×C4—61no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—82no—
635C2×C4—124no—
636C2×C4—122no—
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×C4D3×#4112yes—
645C2×C4#101012yes—
646C2×C4C24×D311yescharacteristic
647C8—21no—
648C8—21no—
649C8—22no—
650C8—64no—
651C2×C3—22no—
652C2×C3—84no—
653C2×C3C2×#54811yescharacteristic
654C2×C3#358512yes—
655D3—42no—
656D3—81no—
657C22—22no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—62no—
662C22—122no—
663C22—121no—
664C22—122no—
665C22—164no—
666C22#567712yes—
667C22C22×D3×D411yescharacteristic
668C4—22no—
669C4—22no—
670C4—44no—
671C4—62no—
672C4—64no—
673C4—64no—
674C4—121no—
675C4—122no—
676C4—122no—
677C4#567712yes—
678C4C2×D3×#4111yescharacteristic
679C4C2×#101011yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—84no—
683C2—122no—
684C2—241no—
685C2C2×#567711yescharacteristic
686C2#8719212yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#490C2
socleC22×C3C23×D4
hypercenterC2×#490D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
1. #86190C2semidirect
2. #86098C2semidirect
3. C2×C3×#490C2semidirect
4. #83165C2semidirect
5. #83017C2semidirect
6. #81711C2semidirect
7. #81710C2semidirect
8. #81709C2semidirect
9. #81708C2semidirect
10. #78767C2semidirect
11. #78737C2semidirect
12. #78731C2semidirect
13. #78726C2semidirect
14. #78494C2semidirect
15. #78480C2semidirect
29. #5690C22semidirect
33. #5685C22semidirect
53. #5066C22semidirect
54. #5063C22semidirect
58. #4787C22semidirect
59. #4786C22semidirect
60. #4785C22semidirect
61. #4768C22semidirect
62. #4767C22semidirect
63. #4766C22semidirect
64. #4765C22semidirect
65. #4764C22semidirect
66. #4763C22semidirect
67. #4762C22semidirect
68. #4761C22semidirect
69. #4760C22semidirect
70. #4759C22semidirect
71. #4758C22semidirect
72. #4757C22semidirect
73. #4756C22semidirect
74. #4716C22semidirect
75. #4711C22semidirect
76. #4710C22semidirect
77. #4710C22semidirect
78. #4701C22semidirect
79. #4701C22semidirect
87. C2×#490D3semidirect
280. #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
#861902
#860982
C2×C3×#4902
#831652
#830172
#817112
#817102
#817092
#817082
#787672
#787372
#787312
#787262
#784942
#784802
#613373