Finite Group Databasebeta

← #10858727

Subgroups — #10858727

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83930C211yesmaximal, characteristic
2#82561C211yesmaximal, characteristic, complemented
3#82560C211yesmaximal, characteristic, complemented
4#78918C211yesmaximal, characteristic, complemented
5C3×#1883C211yesmaximal, characteristic
6#72091C211yesmaximal, characteristic
7#70376C211yesmaximal, characteristic, complemented
8#70363C211yesmaximal, characteristic, complemented
9#70317C211yesmaximal, characteristic, complemented
10#69656C211yesmaximal, characteristic, complemented
11#69642C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68981C211yesmaximal, characteristic, complemented
14#68738C211yesmaximal, characteristic, complemented
15#68735C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5798C2211yescharacteristic
22#5519C2211yescharacteristic
23C2×C3×#104C2211yescharacteristic
24C3×#418C2211yescharacteristic
25C3×#390C2211yescharacteristic
26#5233C2211yescharacteristic
27#5232C2211yescharacteristic
28C2×#833C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5164—21no—
32Dic3×SD16—21no—
33#5152—21no—
34Dic3×D8—21no—
35C2×#822C2211yescharacteristic
36#5122C2211yescharacteristic
37#5121C2211yescharacteristic
38#5121C2211yescharacteristic
39#5119C2211yescharacteristic
40#5022—21no—
41#5020—21no—
42#5009—21no—
43#5008—21no—
44#4999C2211yescharacteristic
45#4998—21no—
46#4998C2211yescharacteristic
47#4995—21no—
48#4995C2211yescharacteristic
49#4964—21no—
50#4964C2211yescharacteristic
51#4964C2211yescharacteristic
52C2×#808—21no—
53C2×#808C2211yescharacteristic
54C2×#808C2211yescharacteristic
55#4935C2211yescharacteristic
56#4766—21no—
57#4763—21no—
58#4761—21no—
59#4759—21no—
60C3×#343C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4524C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2912—31no—
72#2094—31no—
73#2090—31no—
74#2063—31no—
75#2060—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883D311yescharacteristic
80#1866—31no—
81#1859—31no—
82#1856—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C3×AGL(1,ℤ/8)—22no—
86C3×AGL(1,ℤ/8)Q814yescomplemented
87C2×C3×SD16—21no—
88C2×C3×D8—21no—
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×D4—21no—
93C3×C4×D4—41no—
94C3×#117—21no—
95C3×#117C2311yescharacteristic
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—41no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—41no—
101#948—22no—
102#948—22no—
103Q8×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3—21no—
108D4×Dic3—21no—
109D4×Dic3—41no—
110#926—21no—
111#926C2311yescharacteristic
112#926C2311yescharacteristic
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—41no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#890—21no—
124#888—21no—
125#878—41no—
126#870—21no—
127#869—21no—
128C3×#104—22no—
129C3×#102—21no—
130C3×#102—21no—
131#835—21no—
132#835D411yescharacteristic
133#835D411yescharacteristic
134#833—22no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#832—21no—
140#832D411yescharacteristic
141#832D411yescharacteristic
142#822—22no—
143#818—21no—
144#818—21no—
145#817—21no—
146#817—21no—
147#815—21no—
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150C8×Dic3—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808—21no—
158#808D411yescharacteristic
159#808D411yescharacteristic
160#807—21no—
161#807—21no—
162#807—21no—
163#807D411yescharacteristic
164#807D411yescharacteristic
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167C2×AGL(1,ℤ/8)C2×D311yescharacteristic
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520C2×D311yescharacteristic
172#518—31no—
173#479—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#476—61no—
179#475—61no—
180#474—61no—
181#442—61no—
182#440—61no—
183C4×SD16—61no—
184C4×D8—61no—
185C2×#108—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#106—31no—
196C2×#106—31no—
197C2×#106—61no—
198#424—31no—
199#424—31no—
200C2×#104C2×D311yescharacteristic
201#418C2×D311yescharacteristic
202#417—31no—
203#416—31no—
204C2×#102—31no—
205C2×#102—31no—
206C2×#100—31no—
207#405—31no—
208#390C2×D311yescharacteristic
209#384—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339C2×D311yescharacteristic
217#336—31no—
218C24×C3—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41C2×Q812yes—
222C3×#41#4112yes—
223C2×C3×Q8C2×D411yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4C2×D411yescharacteristic
231C2×C3×D4C2×Q812yes—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—41no—
236C22×C3×C4C2411yescharacteristic
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—41no—
240C22×Dic3—41no—
241C22×Dic3—41no—
242C2×Dic6—41no—
243C3×SD16—24no—
244C3×D8—24no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16C2×Q812yes—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—82no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—41no—
263#217—21no—
264#217—21no—
265#217—41no—
266#217—41no—
267#217—41no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—41no—
273C4⋊Dic3—41no—
274C4⋊Dic3—82no—
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278C4⋊Dic3C2×D411yescharacteristic
279Dic3⋊C4—41no—
280Dic3⋊C4—41no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—41no—
284#208—24no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208#4112yes—
289C2×C3⋊C8—24no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8C2×D411yescharacteristic
297C2×C3⋊C8C2×D411yescharacteristic
298C2×C3⋊C8C2×D411yescharacteristic
299C2×C3⋊C8C2×D411yescharacteristic
300C2×#41C22×D311yescharacteristic
301C22×D4C22×D311yescharacteristic
302AGL(1,ℤ/8)—22no—
303AGL(1,ℤ/8)Dic614yescomplemented
304C2×SD16—21no—
305C2×D8—21no—
306#131—62no—
307#131—62no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—61no—
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—31no—
315C22×C8—61no—
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318#125—61no—
319#125—61no—
320C22⋊Q8—121no—
321C4×Q8—21no—
322C4×Q8—61no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—61no—
329C4×D4—61no—
330C4×D4—121no—
331#117—21no—
332#117—31no—
333#117—31no—
334#117—61no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—41no—
343C2×C22⋊C4—121no—
344C2×C42C22×D311yescharacteristic
345#108—62no—
346#107—31no—
347#107—31no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#106—31no—
354#106—31no—
355#106—61no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—22no—
369#103—31no—
370#103—31no—
371#103—61no—
372#102—21no—
373#102—21no—
374#102—31no—
375#102—31no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102—61no—
380#102—61no—
381#100—62no—
382#97—61no—
383C4×C8—61no—
384C23×C3—21no—
385C23×C3—22no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—81no—
389C23×C3—81no—
390C23×C3H(5,𝔽2)11yescharacteristic
391C3×Q8—21no—
392C3×Q8C22⋊Q812yes—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—21no—
397C3×D4—22no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4C22⋊Q812yes—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—81no—
411C2×C3×C4—161no—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×Q811yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—81no—
425C2×Dic3—81no—
426C2×Dic3—161no—
427C2×Dic3—161no—
428C2×Dic3—161no—
429Dic6—161no—
430C3×C8—22no—
431C3×C8—41no—
432C3⋊C8—21no—
433C3⋊C8—22no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C3⋊C8—41no—
438C24—21no—
439#41—21no—
440#41—21no—
441#41C2×Dic612yes—
442#41#23712yes—
443C2×Q8—121no—
444C2×Q8D3×D411yescharacteristic
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C2×Dic612yes—
452C2×D4D3×D411yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—24no—
464D8—24no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—32no—
469M16—64no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16C2×Dic612yes—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—64no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—41no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—242no—
503C4⋊C4D3×D411yescharacteristic
504C4⋊C4D3×D411yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—61no—
508C22⋊C4—61no—
509C22⋊C4—82no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—121no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—61no—
518C42—61no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—82no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—161no—
530C22×C3#53911yescharacteristic
531C22×C3D4211yescharacteristic
532C22×C3#55411yescharacteristic
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—81no—
538C3×C4C2×C22⋊Q811yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#54711yescharacteristic
541C3×C4D4×Q811yescharacteristic
542Dic3—41no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—81no—
553C23—81no—
554C23#100411yescharacteristic
555Q8—21no—
556Q8—481no—
557Q8#87812yes—
558D4—22no—
559D4—22no—
560D4—21no—
561D4—22no—
562D4—22no—
563D4—41no—
564D4—41no—
565D4#87812yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—81no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—161no—
583C2×C4—241no—
584C2×C4—241no—
585C2×C4—481no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C22×Dic611yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×#23711yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4#101011yescharacteristic
594C8—22no—
595C8—41no—
596C8—61no—
597C8—61no—
598C8—61no—
599C8—61no—
600C8—62no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—42no—
605C2×C3—82no—
606C2×C3#244911yescharacteristic
607C2×C3#328811yescharacteristic
608C2×C3#329111yescharacteristic
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—81no—
617C22—81no—
618C22—161no—
619C22#553911yescharacteristic
620C22#559011yescharacteristic
621C22#560211yescharacteristic
622C4—22no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—121no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4C2×#87811yescharacteristic
634C4D4×Dic611yescharacteristic
635C4#560611yescharacteristic
636C4#565511yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—42no—
641C2—82no—
642C2#7470911yescharacteristic
643C2#8028111yescharacteristic
644C2#8028711yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1883C2
socleC22×C3D42
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenter#1883D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1883

Complements

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

NH ≅ G/Ntype
2. #82561C2semidirect
3. #82560C2semidirect
4. #78918C2semidirect
7. #70376C2semidirect
8. #70363C2semidirect
9. #70317C2semidirect
10. #69656C2semidirect
11. #69642C2semidirect
12. #68984C2semidirect
13. #68981C2semidirect
14. #68738C2semidirect
15. #68735C2semidirect
86. C3×AGL(1,ℤ/8)Q8semidirect
303. AGL(1,ℤ/8)Dic6semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
#839302
#825612
#825602
#789182
C3×#18832
#720912
#703762
#703632
#703172
#696562
#696422
#689842
#689812
#687382
#687352
#264073