Finite Group Databasebeta

← #11670577

Subgroups — #11670577

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×#904C211yesmaximal, characteristic, complemented
3#86023C212yesmaximal, complemented
4C3×#3050C211yesmaximal, characteristic, complemented
5#83906C211yesmaximal, characteristic, complemented
6#83734C211yesmaximal, characteristic, complemented
7#82928C212yesmaximal, complemented
8#80779C214yesmaximal, complemented
9#80735C214yesmaximal, complemented
10#80688C218yesmaximal, complemented
11#79183C211yesmaximal, characteristic, complemented
12#79169C211yesmaximal, characteristic, complemented
13#78864C212yesmaximal, complemented
14#78836C212yesmaximal, complemented
15#61054—31nomaximal, Sylow
16C2×#1010C2212yes—
17C22×#235C2212yes—
18C2×C3×#135C2211yescharacteristic
19C23×C3×C8C2211yescharacteristic
20C3×#525C2212yes—
21#5830C2212yes—
22C2×#951C2212yes—
23C2×#949C2212yes—
24#5791C2211yescharacteristic
25#5771—28no—
26C2×D3×Q16C2212yescomplemented
27C2×#915C2214yescomplemented
28C2×#912C2212yescomplemented
29C22×Dic12C2211yescharacteristic
30C2×#904—28no—
31C2×#904C2214yes—
32C2×#904C2214yes—
33C22×D24C2211yescharacteristic
34C22×#204C2212yes—
35C2×#901C2211yescharacteristic
36#5656—24no—
37#5633C2218yescomplemented
38#5599C2218yescomplemented
39#5592—24no—
40#5553—24no—
41#5552—24no—
42#5525C2212yes—
43#5516C2212yes—
44C3×#467C2212yescomplemented
45C3×#466C2212yescomplemented
46C3×#465C2214yescomplemented
47C3×#427C2211yescharacteristic
48C3×#416C2212yes—
49#5207C2211yescharacteristic
50#5191C2212yescomplemented
51#5176C2214yescomplemented
52#5151C2212yescomplemented
53#5111C2214yescomplemented
54#5110C2211yescharacteristic
55C8×#67C2214yescomplemented
56#5102C2211yescharacteristic
57#5101C2211yescharacteristic
58#5047C2218yescomplemented
59#5037C2218yescomplemented
60#5004C2212yes—
61#5001C2212yes—
62#4883C2214yescomplemented
63#4873C2214yescomplemented
64#4865C2214yescomplemented
65#4853C2214yescomplemented
66#4806C2218yescomplemented
67#4798C2218yescomplemented
68#4781C2218yescomplemented
69#4762C2218yescomplemented
70C2×#578—31no—
71C22×#135—31no—
72#3487—32no—
73#3287—34no—
74#3285—34no—
75#3271—38no—
76#3155—31no—
77#3058—31no—
78#3050D311yescharacteristic, complemented
79#3043—32no—
80#3010—32no—
81#3005—32no—
82#2950—31no—
83#2929—31no—
84C2×C3×#41C2312yes—
85C23×C3×C4C2311yescharacteristic
86C22×#67—22no—
87#1010—28no—
88#1010—44no—
89C2×D3×Q8C2314yes—
90C2×#237—24no—
91C2×#237C2314yes—
92C2×#235—24no—
93C2×#235—22no—
94C2×#235—28no—
95C2×#235—44no—
96C2×#235C2311yescharacteristic
97C2×#235C2318yes—
98C22×D12C2312yes—
99C22×C4×D3—22no—
100C22×Dic6—22no—
101C22×Dic6C2312yes—
102C3×#135—24no—
103C2×C3×Q16C2311yescharacteristic
104C2×C3×SD16C2312yes—
105C2×C3×D8C2311yescharacteristic
106C22×C3×C8—24no—
107C22×C3×C8C2311yescharacteristic
108C22×C3×C8C2312yes—
109C22×C3×C8C2312yes—
110C22×C3×C8C2311yescharacteristic
111C3×#123—22no—
112C3×C22⋊Q8C2314yes—
113C3×#121C2314yes—
114C3×#120—22no—
115C3×C4×D4—24no—
116C3×#117C2312yes—
117#953D412yes—
118#951—28no—
119#949—44no—
120#947C2314yes—
121C2×#216C2314yes—
122C2×#215C2314yes—
123#940—28no—
124#940C2314yes—
125#939—28no—
126#937—24no—
127C2×#214C2314yes—
128C2×#213C2314yes—
129#930—24no—
130#930D412yes—
131#929D412yes—
132C4×#67C2314yes—
133#926C2311yescharacteristic
134#924D412yes—
135C2×#208C2311yescharacteristic
136C2×#208C2311yescharacteristic
137D3×Q16—28no—
138#915—216no—
139#912—28no—
140C2×Dic12D418yescomplemented
141C2×Dic12C2312yes—
142C2×Dic12C2312yes—
143#904—216no—
144#904—216no—
145C2×D24—24no—
146C2×D24C2312yes—
147C2×D24C2312yes—
148C2×#204—28no—
149C2×#204C2318yes—
150#901—24no—
151C2×ΓL(1,𝔽25)C2312yes—
152C2×D3×C8C2312yes—
153#896—28no—
154#895C2318yes—
155#893—28no—
156#887C2318yes—
157#885—28no—
158#884—28no—
159#884—28no—
160#882—24no—
161#67⋊C4—28no—
162#877—24no—
163#875—22no—
164#874—22no—
165C4×Dic6—24no—
166C3×#107C2312yes—
167C3×#106C2312yes—
168C3×#103C2314yes—
169C3×#102C2314yes—
170#830C2314yes—
171(C2×D3)⋊C8C2314yes—
172#815C2312yes—
173Dic3⋊C8C2314yes—
174C8×Dic3C2312yes—
175#810C2318yes—
176#809C2318yes—
177C2×2−1+4—32no—
178C22×#41—32no—
179#578—68no—
180C2×#137—32no—
181C2×#137—34no—
182C2×AGL(1,ℤ/8)—32no—
183C2×#135—32no—
184C2×#135—34no—
185C2×#135—34no—
186C2×#135—68no—
187C2×#135C2×D311yescharacteristic
188C22×Q16—32no—
189C22×Q16—31no—
190C22×SD16—32no—
191C22×D8—31no—
192C2×#131—31no—
193C23×C8C2×D311yescharacteristic
194#550—38no—
195#550—64no—
196#548—38no—
197#548—64no—
198#535—32no—
199#535—32no—
200#535—64no—
201#525—31no—
202#525—64no—
203#525C2×D312yes—
204#519—32no—
205#496—34no—
206#495—32no—
207#493—32no—
208#469—34no—
209#467C2×D312yescomplemented
210#466—34no—
211#466C2×D312yescomplemented
212#465C2×D314yescomplemented
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#427C2×D311yescharacteristic
224#424—31no—
225#419—32no—
226#418—32no—
227#416C2×D312yes—
228#408—31no—
229#407—31no—
230#405—31no—
231C24×C3C2×D411yescharacteristic
232C23×D3—22no—
233C3×#41—24no—
234C3×#41—24no—
235C2×C3×Q8C2412yes—
236C2×C3×D4—22no—
237C2×C3×D4—24no—
238C2×C3×D4C2412yes—
239C22×C3×C4—22no—
240C22×C3×C4—22no—
241C22×C3×C4—22no—
242C22×C3×C4C2×D412yes—
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C2×#67—28no—
247C2×#67—24no—
248C2×#67—28no—
249C2×#67—48no—
250C2×#67C2×D412yes—
251C2×#67C2×D414yes—
252C22×Dic3—22no—
253C22×Dic3—24no—
254D3×Q8—28no—
255D3×Q8—44no—
256#237—28no—
257#237—44no—
258#237—416no—
259#235—22no—
260#235—22no—
261#235—216no—
262#235—48no—
263#235—48no—
264#235—416no—
265C2×D12—24no—
266C2×D12—24no—
267C2×D12—22no—
268C2×D12—28no—
269C2×D12C2×D411yescharacteristic
270C2×D12C2414yes—
271C2×C4×D3—24no—
272C2×C4×D3—28no—
273C2×C4×D3—48no—
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×D8—22no—
285C2×C3×C8—22no—
286C2×C3×C8—24no—
287C2×C3×C8—24no—
288C2×C3×C8C2×D414yes—
289C2×C3×C8C2412yes—
290C2×C3×C8C2411yescharacteristic
291C2×C3×C8C2411yescharacteristic
292C3×C4⋊C4—24no—
293C3×C4⋊C4C2414yes—
294C3×C22⋊C4—22no—
295C3×C22⋊C4—24no—
296C3×C22⋊C4—24no—
297C3×C42—22no—
298#217C2×D412yes—
299#217C2×D414yes—
300#216—28no—
301#215—44no—
302#214—28no—
303#213—44no—
304(C2×D3)⋊C4—28no—
305(C2×D3)⋊C4—28no—
306(C2×D3)⋊C4C2×D414yes—
307C4⋊Dic3C2×D412yes—
308Dic3⋊C4—28no—
309Dic3⋊C4C2×D414yes—
310C4×Dic3—28no—
311C4×Dic3C2412yes—
312#208—22no—
313#208—22no—
314C2×C3⋊C8C2412yes—
315C2×C3⋊C8C2412yes—
316Dic12C2×D418yes—
317Dic12C2×D418yes—
318D24—24no—
319D24—24no—
320#204—216no—
321ΓL(1,𝔽25)—24no—
322D3×C8—24no—
3232−1+4—68no—
3242−1+4—124no—
325C2×#41—31no—
326C2×#41—34no—
327C2×#41—38no—
328C2×#41—62no—
329C2×#41—68no—
330C2×#41—64no—
331C2×#41—64no—
332C2×#41—124no—
333C2×#41C22×D312yes—
334C22×Q8—34no—
335C22×Q8—32no—
336C22×Q8—62no—
337C22×D4—32no—
338C22×D4—62no—
339C23×C4—62no—
340C23×C4C22×D311yescharacteristic
341#137—616no—
342#137—68no—
343AGL(1,ℤ/8)—124no—
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—34no—
355C2×SD16—34no—
356C2×SD16—38no—
357C2×SD16—68no—
358C2×SD16C22×D312yes—
359C2×D8—34no—
360C2×D8—32no—
361C2×D8—32no—
362C2×D8—64no—
363C2×D8C22×D311yescharacteristic
364#131—64no—
365C2×M16—31no—
366C2×M16—31no—
367C2×M16—32no—
368C22×C8—24no—
369C22×C8—32no—
370C22×C8C22×D311yescharacteristic
371C22×C8C22×D312yes—
372C22×C8C22×D312yes—
373C22×C8C22×D311yescharacteristic
374#127—34no—
375#127—62no—
376#127—68no—
377#124—34no—
378#124—62no—
379#124—68no—
380#123—22no—
381#123—32no—
382C22⋊Q8—32no—
383C22⋊Q8—68no—
384C22⋊Q8C22×D314yes—
385#121—32no—
386#121—38no—
387#121—68no—
388#121—68no—
389#121—68no—
390#121—64no—
391#121C22×D314yes—
392#120—22no—
393#120—32no—
394#120—64no—
395#120—64no—
396C4×Q8—38no—
397C4×Q8—64no—
398C4×D4—24no—
399C4×D4—34no—
400C4×D4—68no—
401C4×D4—68no—
402#117—31no—
403#117—64no—
404#117C22×D312yes—
405#107C22×D312yes—
406#106C22×D312yes—
407C4⋊C8—34no—
408#103—38no—
409#103C22×D314yes—
410#102—38no—
411#102C22×D314yes—
412C22⋊C8—34no—
413C22⋊C8—34no—
414#97—32no—
415C4×C8—32no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—42no—
419C23×C3C22×D412yes—
420C23×C3C22×D411yescharacteristic
421C22×D3—24no—
422C22×D3—44no—
423C22×D3—84no—
424C22×D3C22×D412yes—
425C3×Q8—22no—
426C3×Q8—22no—
427C3×D4—22no—
428C3×D4—22no—
429C3×D4—44no—
430C3×D4—84no—
431C2×C3×C4—22no—
432C2×C3×C4—24no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—24no—
436C2×C3×C4—22no—
437C2×C3×C4—44no—
438C2×C3×C4C22×D412yes—
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D412yes—
441C2×C3×C4H(5,𝔽2)12yes—
442C2×C3×C4C2511yescharacteristic
443#67—24no—
444#67—44no—
445#67—48no—
446#67—416no—
447#67—88no—
448C2×Dic3—28no—
449C2×Dic3—24no—
450C2×Dic3—48no—
451C2×Dic3C22×D414yes—
452C2×Dic3C22×D412yes—
453D12—22no—
454D12—24no—
455D12—24no—
456D12—44no—
457D12—88no—
458C4×D3—22no—
459C4×D3—22no—
460C4×D3—416no—
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—62no—
473C24D3×D411yescharacteristic
474#41—24no—
475#41—24no—
476#41—616no—
477#41—62no—
478#41—62no—
479#41—68no—
480#41—128no—
481#41—1216no—
482#41—128no—
483#41—124no—
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—32no—
498C2×D4—34no—
499C2×D4—31no—
500C2×D4—34no—
501C2×D4—68no—
502C2×D4—68no—
503C2×D4—64no—
504C2×D4—68no—
505C2×D4—64no—
506C2×D4—62no—
507C2×D4—64no—
508C2×D4—128no—
509C2×D4C23×D312yes—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—32no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—62no—
517C22×C4—68no—
518C22×C4—128no—
519C22×C4D3×D412yes—
520C22×C4D3×D411yescharacteristic
521C22×C4C23×D312yes—
522C22×C4C23×D311yescharacteristic
523Q16—22no—
524Q16—38no—
525Q16—38no—
526Q16—68no—
527SD16—24no—
528SD16—616no—
529SD16—68no—
530SD16—124no—
531D8—22no—
532D8—64no—
533D8—64no—
534D8—124no—
535M16—62no—
536M16—62no—
537M16—64no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—24no—
541C2×C8—32no—
542C2×C8—32no—
543C2×C8—64no—
544C2×C8D3×D414yes—
545C2×C8C23×D312yes—
546C2×C8C23×D311yescharacteristic
547C2×C8C23×D311yescharacteristic
548C4⋊C4—24no—
549C4⋊C4—32no—
550C4⋊C4—34no—
551C4⋊C4—68no—
552C4⋊C4C23×D314yes—
553C22⋊C4—22no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—32no—
557C22⋊C4—34no—
558C22⋊C4—34no—
559C22⋊C4—68no—
560C22⋊C4—68no—
561C42—22no—
562C42—32no—
563C42—68no—
564C22×C3—22no—
565C22×C3—22no—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—44no—
569C22×C3—44no—
570C22×C3—42no—
571C22×C3D4212yes—
572C22×C3C23×D411yescharacteristic
573C2×D3—24no—
574C2×D3—22no—
575C2×D3—48no—
576C2×D3—44no—
577C2×D3—88no—
578C3×C4—22no—
579C3×C4—44no—
580C3×C4—42no—
581C3×C4D4212yes—
582C3×C4C23×D411yescharacteristic
583C3×C4C2×H(5,𝔽2)11yescharacteristic
584Dic3—24no—
585Dic3—28no—
586Dic3—22no—
587C23—22no—
588C23—22no—
589C23—32no—
590C23—42no—
591C23—64no—
592C23—124no—
593C23—244no—
594C23C2×D3×D412yes—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—38no—
599Q8—64no—
600Q8—68no—
601Q8—62no—
602Q8—128no—
603D4—22no—
604D4—22no—
605D4—44no—
606D4—64no—
607D4—64no—
608D4—64no—
609D4—62no—
610D4—84no—
611D4—124no—
612D4—1216no—
613D4—128no—
614D4—124no—
615D4—248no—
616D4—248no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—34no—
624C2×C4—32no—
625C2×C4—44no—
626C2×C4—64no—
627C2×C4—68no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—1216no—
631C2×C4—128no—
632C2×C4—128no—
633C2×C4C2×D3×D412yes—
634C2×C4C2×D3×D412yes—
635C2×C4C2×D3×D411yescharacteristic
636C2×C4#100912yes—
637C2×C4C24×D311yescharacteristic
638C8—22no—
639C8—62no—
640C8—62no—
641C8C2×D3×D412yes—
642C8C2×D3×D412yes—
643C2×C3—22no—
644C2×C3—42no—
645C2×C3—82no—
646C2×C3C2×D4211yescharacteristic
647C2×C3#358412yes—
648D3—42no—
649D3—84no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—64no—
658C22—62no—
659C22—128no—
660C22—124no—
661C22—248no—
662C22#560912yes—
663C22C22×D3×D411yescharacteristic
664C4—22no—
665C4—44no—
666C4—42no—
667C4—64no—
668C4—62no—
669C4—68no—
670C4#560912yes—
671C4C22×D3×D411yescharacteristic
672C4C2×#100911yescharacteristic
673C3#6105411yescharacteristic, complemented, Sylow
674C2—22no—
675C2—42no—
676C2—82no—
677C2—122no—
678C2—244no—
679C2C2×#560911yescharacteristic
680C2#8718712yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3050C2
socleC22×C3C23×D4
hypercenter#3050D3
hypocenterC3#61054
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#5771C2semidirect
2. C22×#904C2semidirect
3. #86023C2semidirect
4. C3×#3050C2semidirect
5. #83906C2semidirect
6. #83734C2semidirect
7. #82928C2semidirect
8. #80779C2semidirect
9. #80735C2semidirect
10. #80688C2semidirect
11. #79183C2semidirect
12. #79169C2semidirect
13. #78864C2semidirect
14. #78836C2semidirect
26. C2×D3×Q16C22semidirect
27. C2×#915C22semidirect
28. C2×#912C22semidirect
37. #5633C22semidirect
38. #5599C22semidirect
44. C3×#467C22semidirect
45. C3×#466C22semidirect
46. C3×#465C22semidirect
50. #5191C22semidirect
51. #5176C22semidirect
52. #5151C22semidirect
53. #5111C22semidirect
55. C8×#67C22semidirect
58. #5047C22semidirect
59. #5037C22semidirect
62. #4883C22semidirect
63. #4873C22semidirect
64. #4865C22semidirect
65. #4853C22semidirect
66. #4806C22semidirect
67. #4798C22semidirect
68. #4781C22semidirect
69. #4762C22semidirect
78. #3050D3semidirect
140. C2×Dic12D4semidirect
209. #467C2×D3semidirect
211. #466C2×D3semidirect
212. #465C2×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×#9042
#860232
C3×#30502
#839062
#837342
#829282
#807792
#807352
#806882
#791832
#791692
#788642
#788362
#610543