Finite Group Databasebeta

← #11670579

Subgroups — #11670579

Order 768.

2710 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5771C211yesmaximal, characteristic, complemented
2C22×#912C211yesmaximal, characteristic, complemented
3#86315C212yesmaximal, complemented
4C3×#3155C211yesmaximal, characteristic, complemented
5#83948C212yesmaximal, complemented
6#83945C212yesmaximal, complemented
7#83914C211yesmaximal, characteristic, complemented
8#83836C211yesmaximal, characteristic, complemented
9#83391C212yesmaximal, complemented
10#81859C218yesmaximal, complemented
11#81834C214yesmaximal, complemented
12#81790C214yesmaximal, complemented
13#78573C211yesmaximal, characteristic, complemented
14#78553C211yesmaximal, characteristic, complemented
15#61054—31nomaximal, Sylow
16C2×#1010C2212yes—
17C22×#237C2212yes—
18C2×C3×AGL(1,ℤ/8)C2212yes—
19C2×C3×#135C2211yescharacteristic
20C22×C3×D8C2211yescharacteristic
21C3×#535C2212yes—
22#5828C2212yes—
23C2×#951C2212yes—
24#5805C2212yes—
25C22×#214C2212yes—
26#5771—28no—
27C2×D3×Q16C2212yescomplemented
28C2×#915C2214yescomplemented
29C2×#912—28no—
30C2×#912C2214yes—
31C2×#912C2214yes—
32C2×#912C2212yescomplemented
33C22×Dic12C2211yescharacteristic
34C2×#901C2211yescharacteristic
35C22×D3×C8C2211yescharacteristic
36#5718C2218yescomplemented
37#5677C2218yescomplemented
38#5599—24no—
39#5598—24no—
40#5592—24no—
41#5586—24no—
42#5524C2211yescharacteristic
43#5516C2212yes—
44C3×#496C2214yescomplemented
45C3×#495C2212yescomplemented
46C3×#493C2212yescomplemented
47C3×#405C2211yescharacteristic
48#5233C2212yes—
49#5215C2212yes—
50#5188C2212yescomplemented
51#5173C2218yescomplemented
52#5171C2214yescomplemented
53#5170C2218yescomplemented
54#5159C2214yescomplemented
55#5158C2218yescomplemented
56#5158C2218yescomplemented
57#5157C2214yescomplemented
58#5157C2212yescomplemented
59#5129C2211yescharacteristic
60#5126C2211yescharacteristic
61#5120C2212yes—
62#5106C2211yescharacteristic
63#5075C2218yescomplemented
64#5054C2218yescomplemented
65#4999C2211yescharacteristic
66#4716C2214yescomplemented
67#4706C2214yescomplemented
68C4×Dic12C2214yescomplemented
69C8×D12C2214yescomplemented
70C2×#578—31no—
71C22×#135—31no—
72#3487—32no—
73#3287—34no—
74#3285—34no—
75#3271—38no—
76#3155D311yescharacteristic, complemented
77#3058—31no—
78#3050—31no—
79#3043—32no—
80#3010—32no—
81#3005—32no—
82#2950—31no—
83#2929—31no—
84C2×C3×#41C2312yes—
85C22×C3×D4C2312yes—
86C22×#67—22no—
87C23×Dic3—22no—
88#1010—28no—
89#1010—44no—
90C2×D3×Q8C2314yes—
91C2×#237—24no—
92C2×#237—24no—
93C2×#237—28no—
94C2×#237—44no—
95C2×#237C2314yes—
96C2×#237C2318yes—
97C2×#235—22no—
98C2×#235C2311yescharacteristic
99C22×C4×D3C2311yescharacteristic
100C22×Dic6—22no—
101C22×Dic6C2312yes—
102C3×AGL(1,ℤ/8)—44no—
103C3×#135—24no—
104C2×C3×Q16C2311yescharacteristic
105C2×C3×SD16C2312yes—
106C2×C3×SD16C2314yes—
107C2×C3×D8—24no—
108C2×C3×D8C2312yes—
109C2×C3×D8C2312yes—
110C2×C3×D8C2311yescharacteristic
111C2×C3×D8C2314yes—
112C2×C3×M16C2311yescharacteristic
113C22×C3×C8C2311yescharacteristic
114C3×#127C2314yes—
115C3×#124C2314yes—
116C3×#121—24no—
117C3×#120—24no—
118C3×#117C2311yescharacteristic
119#953—24no—
120#951—28no—
121#946C2314yes—
122C2×#216C2314yes—
123#940—28no—
124#940—22no—
125#939—28no—
126#939—28no—
127#938C2314yes—
128#938C2318yes—
129#937—22no—
130#936—22no—
131#935—22no—
132D4×Dic3—28no—
133C2×#214—28no—
134C2×#214C2314yes—
135C2×#214C2318yes—
136C4×#67—24no—
137#926C2312yes—
138C2×#208C2311yescharacteristic
139C22×C3⋊C8C2311yescharacteristic
140D3×Q16—28no—
141#915—216no—
142#912—216no—
143#912—216no—
144#912—28no—
145C2×Dic12D418yescomplemented
146C2×Dic12C2312yes—
147C2×Dic12C2312yes—
148#901—24no—
149C2×ΓL(1,𝔽25)C2312yes—
150C2×D3×C8—24no—
151C2×D3×C8C2312yes—
152C2×D3×C8C2312yes—
153C2×D3×C8C2312yes—
154#897D412yes—
155#895D412yes—
156#887—24no—
157#886D412yes—
158#885—28no—
159#884—28no—
160#882D412yes—
161#67⋊C4—28no—
162#878—28no—
163#877—24no—
164C4×D12C2314yes—
165C4×Dic6C2318yes—
166C3×#97C2312yes—
167C3×C4×C8C2312yes—
168#835C2314yes—
169#832C2314yes—
170#832C2318yes—
171(C2×D3)⋊C8C2314yes—
172(C2×D3)⋊C8C2314yes—
173#818C2312yes—
174#817C2312yes—
175#816C2318yes—
176C4⋊(C3⋊C8)C2314yes—
177C2×2−1+4—32no—
178C22×#41—32no—
179#578—68no—
180C2×#137—32no—
181C2×#137—34no—
182C2×AGL(1,ℤ/8)C2×D312yes—
183C2×#135—34no—
184C2×#135—34no—
185C2×#135—32no—
186C2×#135—68no—
187C2×#135C2×D311yescharacteristic
188C22×Q16—32no—
189C22×Q16—31no—
190C22×SD16—32no—
191C22×D8C2×D311yescharacteristic
192C2×#131—31no—
193C23×C8—31no—
194#550—38no—
195#550—64no—
196#548—38no—
197#548—64no—
198#535—32no—
199#535—64no—
200#535C2×D312yes—
201#525—32no—
202#525—31no—
203#525—64no—
204#519—32no—
205#496C2×D314yescomplemented
206#495C2×D312yescomplemented
207#493C2×D312yescomplemented
208#469—34no—
209#467—32no—
210#466—34no—
211#466—32no—
212#465—34no—
213#464—38no—
214#460—38no—
215#452—38no—
216#449—38no—
217#449—38no—
218#448—38no—
219#441—34no—
220C4×Q16—34no—
221#435—34no—
222C8×D4—34no—
223#427—31no—
224#424—31no—
225#419—32no—
226#418—32no—
227#416—32no—
228#408—31no—
229#407—31no—
230#405C2×D311yescharacteristic
231C24×C3—22no—
232C23×D3C2×D411yescharacteristic
233C3×#41—24no—
234C3×#41—24no—
235C2×C3×Q8C2412yes—
236C2×C3×D4—22no—
237C2×C3×D4—24no—
238C2×C3×D4—22no—
239C2×C3×D4—28no—
240C2×C3×D4—24no—
241C2×C3×D4C2414yes—
242C2×C3×D4C2412yes—
243C22×C3×C4—22no—
244C22×C3×C4C2411yescharacteristic
245C2×#67—28no—
246C2×#67—24no—
247C2×#67—28no—
248C2×#67—24no—
249C2×#67—48no—
250C2×#67C2×D412yes—
251C22×Dic3—22no—
252C22×Dic3—28no—
253C22×Dic3—24no—
254C22×Dic3—48no—
255C22×Dic3C2×D411yescharacteristic
256D3×Q8—28no—
257D3×Q8—44no—
258#237—216no—
259#237—28no—
260#237—48no—
261#237—44no—
262#237—416no—
263#237—416no—
264#235—22no—
265#235—22no—
266#235—48no—
267C2×D12C2×D411yescharacteristic
268C2×D12C2×D414yes—
269C2×C4×D3—22no—
270C2×C4×D3—22no—
271C2×C4×D3—24no—
272C2×C4×D3C2×D412yes—
273C2×C4×D3C2412yes—
274C2×C4×D3C2412yes—
275C2×Dic6—22no—
276C2×Dic6—24no—
277C2×Dic6—24no—
278C2×Dic6—48no—
279C2×Dic6C2×D411yescharacteristic
280C2×Dic6C2×D418yes—
281C2×Dic6C2414yes—
282C3×Q16—22no—
283C3×SD16—24no—
284C3×SD16—44no—
285C3×D8—24no—
286C3×D8—24no—
287C3×D8—22no—
288C3×D8—44no—
289C3×M16—22no—
290C2×C3×C8C2×D414yes—
291C2×C3×C8C2412yes—
292C2×C3×C8C2411yescharacteristic
293C2×C3×C8C2411yescharacteristic
294C3×C4⋊C4C2×D412yes—
295C3×C22⋊C4—28no—
296C3×C22⋊C4C2×D412yes—
297C3×C42C2412yes—
298#217—22no—
299#217—28no—
300#217—24no—
301#216—28no—
302#214—216no—
303#214—28no—
304(C2×D3)⋊C4—24no—
305(C2×D3)⋊C4C2×D414yes—
306(C2×D3)⋊C4C2×D414yes—
307C4⋊Dic3C2×D414yes—
308C4⋊Dic3C2414yes—
309Dic3⋊C4—28no—
310Dic3⋊C4—24no—
311C4×Dic3—28no—
312C4×Dic3—22no—
313#208—22no—
314C2×C3⋊C8—22no—
315C2×C3⋊C8C2412yes—
316C2×C3⋊C8C2412yes—
317Dic12C2×D418yes—
318Dic12C2×D418yes—
319ΓL(1,𝔽25)—24no—
320D3×C8—24no—
321D3×C8—24no—
322D3×C8—24no—
3232−1+4—68no—
3242−1+4—124no—
325C2×#41—31no—
326C2×#41—38no—
327C2×#41—34no—
328C2×#41—62no—
329C2×#41—64no—
330C2×#41—64no—
331C2×#41—68no—
332C2×#41—124no—
333C2×#41C22×D312yes—
334C22×Q8—34no—
335C22×Q8—32no—
336C22×Q8—62no—
337C22×D4—62no—
338C22×D4C22×D312yes—
339C23×C4—31no—
340C23×C4—62no—
341#137—616no—
342#137—68no—
343AGL(1,ℤ/8)—44no—
344#135—24no—
345#135—616no—
346#135—616no—
347#135—68no—
348C2×Q16—34no—
349C2×Q16—38no—
350C2×Q16—32no—
351C2×Q16—32no—
352C2×Q16—68no—
353C2×Q16C22×D311yescharacteristic
354C2×SD16—38no—
355C2×SD16—34no—
356C2×SD16—68no—
357C2×SD16C22×D312yes—
358C2×SD16C22×D314yes—
359C2×D8—24no—
360C2×D8C22×D312yes—
361C2×D8C22×D312yes—
362C2×D8C22×D311yescharacteristic
363C2×D8C22×D314yes—
364#131—64no—
365C2×M16—31no—
366C2×M16—32no—
367C2×M16C22×D311yescharacteristic
368C22×C8—31no—
369C22×C8—32no—
370C22×C8—32no—
371C22×C8—32no—
372C22×C8—64no—
373C22×C8C22×D311yescharacteristic
374#127—62no—
375#127—68no—
376#127C22×D314yes—
377#124—68no—
378#124—62no—
379#124C22×D314yes—
380#123—32no—
381#123—62no—
382C22⋊Q8—32no—
383C22⋊Q8—34no—
384C22⋊Q8—68no—
385#121—24no—
386#121—32no—
387#121—34no—
388#121—38no—
389#121—68no—
390#121—68no—
391#121—68no—
392#120—24no—
393#120—32no—
394#120—62no—
395#120—64no—
396C4×Q8—38no—
397C4×Q8—64no—
398C4×D4—34no—
399C4×D4—68no—
400C4×D4—68no—
401C4×D4—64no—
402#117—32no—
403#117—64no—
404#117C22×D311yescharacteristic
405#107—32no—
406#106—32no—
407C4⋊C8—34no—
408#103—34no—
409#103—38no—
410#102—34no—
411#102—38no—
412C22⋊C8—34no—
413C22⋊C8—34no—
414#97C22×D312yes—
415C4×C8C22×D312yes—
416C23×C3—24no—
417C23×C3—22no—
418C23×C3—44no—
419C23×C3—84no—
420C23×C3C22×D411yescharacteristic
421C22×D3—22no—
422C22×D3—42no—
423C22×D3C22×D412yes—
424C22×D3C22×D412yes—
425C3×Q8—22no—
426C3×Q8—22no—
427C3×D4—24no—
428C3×D4—22no—
429C3×D4—22no—
430C3×D4—24no—
431C3×D4—44no—
432C3×D4—44no—
433C3×D4—88no—
434C2×C3×C4—22no—
435C2×C3×C4—44no—
436C2×C3×C4C22×D414yes—
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D412yes—
439C2×C3×C4H(5,𝔽2)12yes—
440C2×C3×C4C2511yescharacteristic
441#67—24no—
442#67—48no—
443#67—416no—
444#67—88no—
445#67—84no—
446C2×Dic3—28no—
447C2×Dic3—22no—
448C2×Dic3—24no—
449C2×Dic3—24no—
450C2×Dic3—48no—
451C2×Dic3—416no—
452C2×Dic3C22×D412yes—
453C2×Dic3C22×D412yes—
454D12—22no—
455D12—44no—
456C4×D3—24no—
457C4×D3—22no—
458C4×D3—22no—
459C4×D3—22no—
460C4×D3—22no—
461C4×D3—48no—
462Dic6—28no—
463Dic6—22no—
464Dic6—24no—
465Dic6—48no—
466Dic6C22×D418yes—
467C3×C8—22no—
468C3×C8C22×D412yes—
469C3×C8C22×D412yes—
470C3⋊C8—22no—
471C3⋊C8—22no—
472C24—22no—
473C24—31no—
474#41—24no—
475#41—24no—
476#41—62no—
477#41—62no—
478#41—616no—
479#41—68no—
480#41—128no—
481#41—128no—
482#41—124no—
483#41—1216no—
484#41—1216no—
485C2×Q8—31no—
486C2×Q8—38no—
487C2×Q8—34no—
488C2×Q8—64no—
489C2×Q8—62no—
490C2×Q8—64no—
491C2×Q8—68no—
492C2×Q8—128no—
493C2×Q8—124no—
494C2×Q8C23×D312yes—
495C2×D4—22no—
496C2×D4—24no—
497C2×D4—22no—
498C2×D4—28no—
499C2×D4—24no—
500C2×D4—32no—
501C2×D4—31no—
502C2×D4—34no—
503C2×D4—68no—
504C2×D4—64no—
505C2×D4—68no—
506C2×D4—64no—
507C2×D4—128no—
508C2×D4C23×D314yes—
509C2×D4C23×D312yes—
510C22×C4—22no—
511C22×C4—31no—
512C22×C4—32no—
513C22×C4—32no—
514C22×C4—32no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—68no—
518C22×C4—62no—
519C22×C4—62no—
520C22×C4—62no—
521C22×C4—128no—
522C22×C4C23×D311yescharacteristic
523Q16—22no—
524Q16—38no—
525Q16—38no—
526Q16—68no—
527SD16—24no—
528SD16—44no—
529SD16—616no—
530SD16—68no—
531D8—24no—
532D8—24no—
533D8—22no—
534D8—44no—
535M16—22no—
536M16—62no—
537M16—64no—
538C2×C8—32no—
539C2×C8—32no—
540C2×C8—62no—
541C2×C8—64no—
542C2×C8—64no—
543C2×C8—64no—
544C2×C8D3×D414yes—
545C2×C8C23×D312yes—
546C2×C8C23×D311yescharacteristic
547C2×C8C23×D311yescharacteristic
548C4⋊C4—34no—
549C4⋊C4—34no—
550C4⋊C4—64no—
551C4⋊C4—68no—
552C4⋊C4D3×D412yes—
553C22⋊C4—28no—
554C22⋊C4—34no—
555C22⋊C4—34no—
556C22⋊C4—64no—
557C22⋊C4—64no—
558C22⋊C4—68no—
559C22⋊C4—62no—
560C22⋊C4D3×D412yes—
561C42—62no—
562C42—68no—
563C42C23×D312yes—
564C22×C3—22no—
565C22×C3—44no—
566C22×C3—48no—
567C22×C3—42no—
568C22×C3—44no—
569C22×C3—88no—
570C22×C3D4212yes—
571C22×C3C23×D411yescharacteristic
572C2×D3—22no—
573C2×D3—24no—
574C2×D3—22no—
575C2×D3—22no—
576C2×D3—22no—
577C2×D3—44no—
578C3×C4—24no—
579C3×C4—42no—
580C3×C4D4212yes—
581C3×C4C23×D411yescharacteristic
582C3×C4C2×H(5,𝔽2)11yescharacteristic
583Dic3—28no—
584Dic3—22no—
585Dic3—22no—
586Dic3—44no—
587C23—24no—
588C23—22no—
589C23—32no—
590C23—32no—
591C23—44no—
592C23—62no—
593C23—84no—
594C23—122no—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—38no—
599Q8—64no—
600Q8—68no—
601Q8—62no—
602Q8—128no—
603D4—24no—
604D4—22no—
605D4—22no—
606D4—24no—
607D4—44no—
608D4—44no—
609D4—64no—
610D4—62no—
611D4—88no—
612D4—124no—
613D4—1216no—
614D4—128no—
615D4—244no—
616D4—248no—
617C2×C4—22no—
618C2×C4—32no—
619C2×C4—32no—
620C2×C4—44no—
621C2×C4—64no—
622C2×C4—62no—
623C2×C4—68no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—128no—
631C2×C4—1216no—
632C2×C4—128no—
633C2×C4C2×D3×D414yes—
634C2×C4C2×D3×D412yes—
635C2×C4C2×D3×D411yescharacteristic
636C2×C4#100412yes—
637C2×C4C24×D311yescharacteristic
638C8—22no—
639C8—62no—
640C8—62no—
641C8C2×D3×D412yes—
642C8C2×D3×D412yes—
643C2×C3—22no—
644C2×C3—82no—
645C2×C3—84no—
646C2×C3C2×D4211yescharacteristic
647C2×C3#358412yes—
648D3—42no—
649D3—42no—
650C22—22no—
651C22—44no—
652C22—48no—
653C22—42no—
654C22—44no—
655C22—62no—
656C22—64no—
657C22—62no—
658C22—62no—
659C22—62no—
660C22—88no—
661C22—124no—
662C22#571712yes—
663C22C22×D3×D411yescharacteristic
664C4—24no—
665C4—42no—
666C4—62no—
667C4—62no—
668C4—68no—
669C4—124no—
670C4#571712yes—
671C4C22×D3×D411yescharacteristic
672C4C2×#100411yescharacteristic
673C3#6105411yescharacteristic, complemented, Sylow
674C2—22no—
675C2—84no—
676C2—82no—
677C2—122no—
678C2—122no—
679C2C2×#571711yescharacteristic
680C2#8720312yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3155C2
socleC22×C3C23×D4
hypercenter#3155D3
hypocenterC3#61054
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #3155

Complements

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

NH ≅ G/Ntype
1. C2×#5771C2semidirect
2. C22×#912C2semidirect
3. #86315C2semidirect
4. C3×#3155C2semidirect
5. #83948C2semidirect
6. #83945C2semidirect
7. #83914C2semidirect
8. #83836C2semidirect
9. #83391C2semidirect
10. #81859C2semidirect
11. #81834C2semidirect
12. #81790C2semidirect
13. #78573C2semidirect
14. #78553C2semidirect
27. C2×D3×Q16C22semidirect
28. C2×#915C22semidirect
32. C2×#912C22semidirect
36. #5718C22semidirect
37. #5677C22semidirect
44. C3×#496C22semidirect
45. C3×#495C22semidirect
46. C3×#493C22semidirect
50. #5188C22semidirect
51. #5173C22semidirect
52. #5171C22semidirect
53. #5170C22semidirect
54. #5159C22semidirect
55. #5158C22semidirect
56. #5158C22semidirect
57. #5157C22semidirect
58. #5157C22semidirect
63. #5075C22semidirect
64. #5054C22semidirect
66. #4716C22semidirect
67. #4706C22semidirect
68. C4×Dic12C22semidirect
69. C8×D12C22semidirect
76. #3155D3semidirect
145. C2×Dic12D4semidirect
205. #496C2×D3semidirect
206. #495C2×D3semidirect
207. #493C2×D3semidirect
673. C3#61054semidirect

Maximal subgroups

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

subgroupindex
C2×#57712
C22×#9122
#863152
C3×#31552
#839482
#839452
#839142
#838362
#833912
#818592
#818342
#817902
#785732
#785532
#610543