Finite Group Databasebeta

← #26222194

Subgroups — #26222194

Order 12032.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094069C211yesmaximal, characteristic, complemented
2C2×#15689290C211yesmaximal, characteristic
3C4×#13094050C212yesmaximal, complemented
4#20243787C211yesmaximal, characteristic, complemented
5#20243767C212yesmaximal, complemented
6#20243620C212yesmaximal, complemented
7#20243597C211yesmaximal, characteristic, complemented
8#20243341C211yesmaximal, characteristic, complemented
9#20243199C212yesmaximal, complemented
10C4×#13093941C211yesmaximal, characteristic, complemented
11C47×#2328C211yesmaximal, characteristic, complemented
12C2×#13094069C2211yescharacteristic
13C2×#13094050C2212yes—
14#15689604C2212yes—
15#15689602—22no—
16#15689598C2211yescharacteristic
17#15689596—22no—
18#15689595—22no—
19C4×#10625350C2212yes—
20C4×#10625352—22no—
21C2×C4×C22⋊D47—22no—
22C4×D4×D47—22no—
23#15689290C418yescomplemented
24#15689233—22no—
25#15689223—22no—
26C4×#10625335—22no—
27C4×#10625334—22no—
28#15689163C2212yescomplemented
29C2×#13093952C2212yes—
30C2×#13093952C2212yes—
31#15689128C2212yescomplemented
32#15689122C2212yes—
33C2×#13093941C2211yescharacteristic
34C2×#13093941C2211yescharacteristic
35C4×#10625329C2212yescomplemented
36#15688892C2212yescomplemented
37#15688883C2212yescomplemented
38#15688883C2212yescomplemented
39C2×#127×C47C2211yescharacteristic
40C2×C4×D4×C47C2211yescharacteristic
41C2×C4×D4×C47C2211yescharacteristic, complemented
42C47×#390C2211yescharacteristic
43C47×#390C2211yescharacteristic, complemented
44C47×#386—22no—
45C47×#386—22no—
46C47×#382C2211yescharacteristic, complemented
47C47×#376C2211yescharacteristic, complemented
48#13094069—22no—
49#13094069—24no—
50#13094069C2×C414yescomplemented
51C2×#10625350C2312yes—
52C2×#10625352—22no—
53C22×C22⋊D47—22no—
54C2×D4×D47—22no—
55C22×C4×D47—22no—
56#13094050—24no—
57#13094050C2×C418yescomplemented
58C2×#10625335—22no—
59C2×#10625334—22no—
60C2×#10625329C2312yes—
61C2×#10625329C2312yes—
62C4×Dic94D412yescomplemented
63C4×Dic94D412yescomplemented
64#13093989—22no—
65C4×D188D412yescomplemented
66C4×D188D412yescomplemented
67D4×Dic47—22no—
68D4×Dic47—22no—
69#13093986—22no—
70C4×C22⋊D47—24no—
71C4×C22⋊D47—24no—
72C4×C22⋊D47—22no—
73C4×C22⋊D47—24no—
74C4×C22⋊D47—22no—
75(C22⋊D47)⋊C4—24no—
76(C22⋊D47)⋊C4—24no—
77#13093983C2312yes—
78#13093982—22no—
79#13093981—22no—
80#13093980—24no—
81C2×Dic47⋊C4—22no—
82C4⋊C4×D47—22no—
83C2×C22⋊Dic47—22no—
84C2×(C2×D47)⋊C4—22no—
85C22⋊C4×D47—24no—
86C2×C4×Dic47—22no—
87C42×D47—22no—
88#13093957—22no—
89#13093955—22no—
90#13093952—24no—
91#13093952C2×C418yescomplemented
92#13093947—22no—
93#13093945—22no—
94#13093945—22no—
95#13093944—22no—
96#13093941—22no—
97#13093941C2×C414yescomplemented
98#13093935—22no—
99C4×C47⋊C8—22no—
100C22×D4×C47—22no—
101C22×D4×C47C2311yescharacteristic
102C22×D4×C47C2311yescharacteristic
103C23×C4×C47—21no—
104C23×C4×C47D412yes—
105#127×C47—22no—
106#127×C47C2×C414yescomplemented
107C4×D4×C47—22no—
108C4×D4×C47—22no—
109C4×D4×C47—22no—
110C4×D4×C47—22no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—22no—
115C2×C22⋊C4×C47—22no—
116C2×C22⋊C4×C47D412yes—
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—22no—
123C24×C47—21no—
124C24×C47C2×D412yes—
125C23×D47—22no—
126C2×D4×C47—28no—
127C2×D4×C47—22no—
128C2×D4×C47—22no—
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47—21no—
132C2×D4×C47—82no—
133C2×D4×C47—82no—
134C2×D4×C47C22×C412yes—
135C2×D4×C47C22×C412yescomplemented
136C2×D4×C47#4112yes—
137C22×C4×C47—21no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—42no—
143C22×C4×C47—82no—
144C22×C4×C47—82no—
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C2×C22⋊D47—24no—
150C2×C22⋊D47—28no—
151C2×C22⋊D47—28no—
152C2×C22⋊D47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156#10625352—28no—
157#10625352—28no—
158D4×D47—28no—
159D4×D47—28no—
160#10625350—22no—
161#10625350—22no—
162#10625350C22×C414yes—
163#10625350#4114yes—
164C2×D188C2×D412yes—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47—24no—
168C2×C4×D47—22no—
169C2×C4×D47—84no—
170C2×Dic94C2×D412yes—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C22⋊C4×C47—24no—
174C22⋊C4×C47—41no—
175C22⋊C4×C47—41no—
176C22⋊C4×C47—82no—
177C22⋊C4×C47—82no—
178C42×C47—21no—
179C42×C47—22no—
180C42×C47C22×C412yescomplemented
181C42×C47C2×D412yes—
182C42×C47C2×D412yes—
183C22⋊Dic47—24no—
184C22⋊Dic47—22no—
185#10625335—28no—
186#10625334—28no—
187(C2×D47)⋊C4—24no—
188(C2×D47)⋊C4—22no—
189(C2×D47)⋊C4—22no—
190C4⋊Dic47C2×D412yes—
191C4⋊Dic47C2×D412yes—
192Dic47⋊C4—24no—
193Dic47⋊C4—22no—
194Dic47⋊C4—22no—
195C4×Dic47—24no—
196C4×Dic47—22no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199#10625329—22no—
200#10625329—22no—
201#10625329—22no—
202#10625329C22×C414yescomplemented
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—42no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47C4×D414yes—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—28no—
218C22×D47—82no—
219C22×D47—82no—
220D4×C47—24no—
221D4×C47—24no—
222D4×C47—24no—
223D4×C47—42no—
224D4×C47—88no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—44no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—42no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—82no—
239C2×C4×C47—84no—
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47#12012yes—
242C2×C4×C47#12011yescharacteristic
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C22×D412yes—
245C2×C4×C47C2×#4112yes—
246C2×C4×C47C2×#4111yescharacteristic
247C22⋊D47—216no—
248C22⋊D47—24no—
249C22⋊D47—24no—
250C2×Dic47—28no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—82no—
255C2×Dic47—82no—
256D188—22no—
257D188C4×D414yes—
258C4×D47—24no—
259C4×D47—24no—
260C4×D47—44no—
261C4×D47—82no—
262C4×D47—84no—
263Dic94—22no—
264Dic94C4×D414yes—
265C47⋊C8—24no—
266C47⋊C8—42no—
267#21473—471nomaximal, Sylow
268C22×C47—21no—
269C22×C47—21no—
270C22×C47—24no—
271C22×C47—44no—
272C22×C47—44no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—82no—
277C22×C47—82no—
278C22×C47—81no—
279C22×C47—81no—
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×#12011yescharacteristic
282C22×C47#52511yescharacteristic
283C2×D47—24no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—88no—
287C4×C47—21no—
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47—81no—
294C4×C47C2×C4×D412yes—
295C4×C47#52512yes—
296Dic47—24no—
297Dic47—42no—
298Dic47—42no—
299C4×H(5,𝔽2)—471no—
300C2×#453—471no—
301C4×AGL(1,ℤ/8)—472no—
302#2328D4711yescharacteristic, complemented
303#1887—471no—
304#1883—472no—
305#1860—472no—
306#1856—471no—
307#1792—471no—
308#1759—472no—
309C4×#100—471no—
310C2×C47—21no—
311C2×C47—21no—
312C2×C47—44no—
313C2×C47—82no—
314C2×C47C4×#12011yescharacteristic
315C2×C47#301911yescharacteristic
316C2×C47#302111yescharacteristic
317D47—44no—
318C2×H(5,𝔽2)—471no—
319C2×AGL(1,ℤ/8)—472no—
320C2×#127C2×D4711yescharacteristic
321#520—472no—
322#520—942no—
323#518—471no—
324#518—942no—
325#518—942no—
326C4×#41—472no—
327C4×#41—942no—
328C2×C4×D4—942no—
329C2×C4×D4—942no—
330C2×C4×D4C2×D4711yescharacteristic
331C2×C4×D4C2×D4711yescharacteristic, complemented
332#453—478no—
333#442—942no—
334#440—942no—
335C4×SD16—942no—
336C4×D8—942no—
337#428—472no—
338C2×#104—472no—
339C2×#104—472no—
340#418—472no—
341#417—472no—
342C2×#100—471no—
343C2×#100—471no—
344C4×M16—472no—
345#390C2×D4711yescharacteristic
346#390C2×D4711yescharacteristic, complemented
347#386—22no—
348#386—22no—
349#382C2×D4711yescharacteristic, complemented
350#376C2×D4711yescharacteristic, complemented
351#343—472no—
352#339—472no—
353#339—472no—
354C47#2147311yescharacteristic, complemented, Sylow
355H(5,𝔽2)—474no—
356H(5,𝔽2)—944no—
357H(5,𝔽2)—942no—
358C2×#41—472no—
359C2×#41—942no—
360C22×D4—22no—
361C22×D4—942no—
362C22×D4—942no—
363C22×D4C22×D4711yescharacteristic
364C22×D4C22×D4711yescharacteristic
365C23×C4—21no—
366C23×C4—942no—
367C23×C4C22⋊D4712yes—
368AGL(1,ℤ/8)—478no—
369AGL(1,ℤ/8)—944no—
370C2×SD16—942no—
371C2×D8—942no—
372C2×M16—472no—
373C2×M16—472no—
374#127—22no—
375#127C4×D4714yescomplemented
376C4×Q8—472no—
377C4×Q8—472no—
378C4×Q8—942no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—472no—
384C4×D4—472no—
385C4×D4—944no—
386C4×D4—944no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4—942no—
390C4×D4—944no—
391C4×D4—944no—
392C4×D4—942no—
393C4×D4—944no—
394C4×D4—942no—
395#117—472no—
396#117—944no—
397#117—942no—
398#117—942no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—942no—
409C2×C22⋊C4C22⋊D4712yes—
410C2×C42—942no—
411C2×C42—942no—
412C2×C42C22×D4711yescharacteristic
413C2×C42C22×D4711yescharacteristic
414C2×C42C22×D4711yescharacteristic
415#107—942no—
416#106—942no—
417#104—478no—
418#104—944no—
419#103—942no—
420#102—942no—
421#102—942no—
422#102—942no—
423#100—474no—
424#100—942no—
425#97—942no—
426C4×C8—942no—
427#95—21no—
428#95—21no—
429#95—22no—
430C24—21no—
431C24—942no—
432C24C2×C22⋊D4712yes—
433#41—474no—
434#41—474no—
435#41—948no—
436#41—948no—
437#41—942no—
438#41—942no—
439C2×Q8—472no—
440C2×D4—28no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—472no—
449C2×D4—948no—
450C2×D4—948no—
451C2×D4—944no—
452C2×D4—948no—
453C2×D4—948no—
454C2×D4—942no—
455C2×D4C2×C4×D4712yes—
456C2×D4C2×C4×D4712yescomplemented
457C2×D4#1062535012yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—944no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—3764no—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478SD16—948no—
479D8—948no—
480M16—474no—
481M16—942no—
482M16—942no—
483M16—942no—
484C2×C8—942no—
485C2×C8—942no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—472no—
489C4⋊C4—472no—
490C4⋊C4—944no—
491C4⋊C4—942no—
492C4⋊C4—942no—
493C22⋊C4—24no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C22⋊C4—942no—
501C22⋊C4—942no—
502C22⋊C4—942no—
503C42—21no—
504C42—22no—
505C42—944no—
506C42—942no—
507C42—942no—
508C42—942no—
509C42C2×C4×D4712yescomplemented
510C42D4×D4712yes—
511C42D4×D4712yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—948no—
522C23—942no—
523C23—3762no—
524C23—3762no—
525C23C4×C22⋊D4714yes—
526C23C22×C22⋊D4711yescharacteristic
527Q8—474no—
528Q8—942no—
529D4—24no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—88no—
534D4—474no—
535D4—9416no—
536D4—944no—
537D4—944no—
538D4—942no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—948no—
555C2×C4—942no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—3762no—
561C2×C4—3764no—
562C2×C4—3762no—
563C2×C4—3762no—
564C2×C4—1884no—
565C2×C4#1309399211yescharacteristic
566C2×C4#1309399212yes—
567C2×C4#1309399211yescharacteristic
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4C2×D4×D4712yes—
570C2×C4#41×D4712yes—
571C2×C4C2×#1062535011yescharacteristic
572C8—944no—
573C8—1882no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—81no—
585C22—81no—
586C22—944no—
587C22—3768no—
588C22—1882no—
589C22—1882no—
590C22C2×C4×C22⋊D4711yescharacteristic
591C22C2×#1309399211yescharacteristic
592C22#1568963011yescharacteristic
593C4—21no—
594C4—21no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—944no—
601C4—1882no—
602C4—1882no—
603C4C4×D4×D4712yes—
604C4#1568963212yes—
605C2—21no—
606C2—21no—
607C2—44no—
608C2—82no—
609C2—1884no—
610C2C4×#1309399211yescharacteristic
611C2#2025274411yescharacteristic
612C2#2025275911yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2328C2
socleC22×C47C2×#120
prime radicalC2×#15689290C2
cosocleC22×C4×C47C24
hypercenter#2328D47
hypocenterC47#21473
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. C4×#13094069C2semidirect
3. C4×#13094050C2semidirect
4. #20243787C2semidirect
5. #20243767C2semidirect
6. #20243620C2semidirect
7. #20243597C2semidirect
8. #20243341C2semidirect
9. #20243199C2semidirect
10. C4×#13093941C2semidirect
11. C47×#2328C2semidirect
23. #15689290C4semidirect
28. #15689163C22semidirect
31. #15689128C22semidirect
35. C4×#10625329C22semidirect
36. #15688892C22semidirect
37. #15688883C22semidirect
38. #15688883C22semidirect
41. C2×C4×D4×C47C22semidirect
43. C47×#390C22semidirect
46. C47×#382C22semidirect
47. C47×#376C22semidirect
50. #13094069C2×C4semidirect
57. #13094050C2×C4semidirect
62. C4×Dic94D4semidirect
63. C4×Dic94D4semidirect
65. C4×D188D4semidirect
66. C4×D188D4semidirect
91. #13093952C2×C4semidirect
97. #13093941C2×C4semidirect
106. #127×C47C2×C4semidirect
135. C2×D4×C47C22×C4semidirect
180. C42×C47C22×C4semidirect
202. #10625329C22×C4semidirect
302. #2328D47semidirect
331. C2×C4×D4C2×D47semidirect
346. #390C2×D47semidirect
349. #382C2×D47semidirect
350. #376C2×D47semidirect
354. C47#21473semidirect
375. #127C4×D47semidirect
456. C2×D4C2×C4×D47semidirect
509. C42C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C4×#130940692
C2×#156892902
C4×#130940502
#202437872
#202437672
#202436202
#202435972
#202433412
#202431992
C4×#130939412
C47×#23282
#2147347