Finite Group Databasebeta

← #26220536

Subgroups — #26220536

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689214C211yesmaximal, characteristic
2#20246035C211yesmaximal, characteristic, complemented
3#20246025C211yesmaximal, characteristic
4#20244243C211yesmaximal, characteristic
5#20244241C211yesmaximal, characteristic
6#20244115C211yesmaximal, characteristic, complemented
7#20244113C211yesmaximal, characteristic, complemented
8#20244087C211yesmaximal, characteristic
9#20244078C211yesmaximal, characteristic, complemented
10#20244076C211yesmaximal, characteristic, complemented
11#20243276C211yesmaximal, characteristic, complemented
12#20243217C211yesmaximal, characteristic, complemented
13#20243215C211yesmaximal, characteristic
14#20243076C211yesmaximal, characteristic, complemented
15C47×#1936C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×#13093989C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689214C418yescomplemented
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093949C2211yescharacteristic
24C2×#13093949C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15689029C2211yescharacteristic
33#15689028C2211yescharacteristic
34#15689026C2211yescharacteristic
35#15689022C2211yescharacteristic
36#15689021C2211yescharacteristic
37#15689019C2211yescharacteristic
38#15689017—21no—
39#15689011—21no—
40#15689010—21no—
41#15689004—21no—
42#15688990—21no—
43#15688988—21no—
44#15688984—21no—
45#15688981—21no—
46C4⋊C4×Dic47C2211yescharacteristic
47C4⋊C4×Dic47C2211yescharacteristic
48#15688969C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#102×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C47×#390C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C23×Dic47—21no—
62C2×#10625335—22no—
63C2×#10625335—22no—
64C2×#10625335C2×C414yes—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093989—22no—
68#13093989C2×C414yes—
69D4×Dic47—22no—
70D4×Dic47C2×C414yes—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2311yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86#13093955—22no—
87#13093955—22no—
88#13093955C2×C414yes—
89#13093949—22no—
90#13093949C2×C414yes—
91#13093947—22no—
92#13093945—22no—
93#13093934—22no—
94#13093934—22no—
95C4×C47⋊C8—22no—
96C8×Dic47—22no—
97C8×Dic47C2×C414yes—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C47⋊#95D411yescharacteristic
110C47⋊#95D411yescharacteristic
111C22×D4×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yes—
123#97×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2411yescharacteristic
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—24no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—81no—
144C22×Dic47—81no—
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94C22×C412yes—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47C22×C412yes—
160C2×C8×C47#4112yes—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—81no—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—81no—
169C22⋊C4×C47—81no—
170C42×C47—21no—
171C42×C47—21no—
172C22⋊Dic47—24no—
173C22⋊Dic47—81no—
174C22⋊Dic47—81no—
175#10625335—28no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—81no—
182C4⋊Dic47C22×C412yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—81no—
185Dic47⋊C4—81no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—24no—
189C4×Dic47—81no—
190C4×Dic47—81no—
191C4×Dic47C22×C412yes—
192C4×Dic47#4112yes—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8C22×C412yes—
196C2×C47⋊C8#4112yes—
197C2×C47⋊C8#4112yes—
198C2×C47⋊C8#4112yes—
199C23×C47—22no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C22×D411yescharacteristic
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—41no—
210D4×C47—41no—
211C2×C4×C47—22no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—84no—
255C2×Dic47C4×D412yes—
256C2×Dic47C4×D412yes—
257Dic94—22no—
258Dic94—22no—
259Dic94—41no—
260Dic94—41no—
261C8×C47—22no—
262C8×C47—22no—
263C47⋊C8—22no—
264C47⋊C8#11714yes—
265#21280—471nomaximal, Sylow
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—81no—
270C22×C47—81no—
271C22×C47—81no—
272C22×C47—81no—
273C22×C47C2×C4×D411yescharacteristic
274C22×C47#53211yescharacteristic
275C22×C47#53211yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57711yescharacteristic
279C22×C47#57711yescharacteristic
280C4×C47—41no—
281C4×C47—41no—
282C4×C47—42no—
283C4×C47—81no—
284C4×C47—81no—
285C4×C47C2×#11711yescharacteristic
286C4×C47C4×#4111yescharacteristic
287C4×C47#52911yescharacteristic
288C4×C47#52911yescharacteristic
289Dic47—22no—
290Dic47—22no—
291Dic47—41no—
292Dic47—41no—
293Dic47—42no—
294Dic47—42no—
295Dic47—81no—
296Dic47—81no—
297C2×C4×SD16—471no—
298#2320—471no—
299#2319—471no—
300#1968—471no—
301#1968—471no—
302#1941—471no—
303#1941—471no—
304#1936—471no—
305#1936D4711yescharacteristic
306#1935—471no—
307#1935—471no—
308C4×#106—471no—
309#1763—471no—
310#1762—471no—
311#1726—471no—
312C2×C47—82no—
313C2×C47#229311yescharacteristic
314C2×C47#294911yescharacteristic
315C2×C47#295211yescharacteristic
316C2×C47#317311yescharacteristic
317C2×C47#317311yescharacteristic
318C2×C47#317411yescharacteristic
319C2×C47#317411yescharacteristic
320C22×SD16—471no—
321C2×C4×Q8—471no—
322C2×C4×D4—471no—
323C4×SD16—478no—
324C2×#106—471no—
325C2×#106—471no—
326C2×#106—471no—
327C2×#103—471no—
328C2×#103—471no—
329C2×#103—471no—
330C2×#102—471no—
331C2×#102C2×D4711yescharacteristic
332C2×#102C2×D4711yescharacteristic
333C2×#97—471no—
334C2×#97—471no—
335C2×#97C2×D4711yescharacteristic
336C2×C4×C8—471no—
337C2×C4×C8—471no—
338C2×C4×C8—471no—
339#391—471no—
340#391—471no—
341#390—471no—
342#390C2×D4711yescharacteristic
343#389—471no—
344#389—471no—
345#389—471no—
346#389C2×D4711yescharacteristic
347#388—941no—
348#388—941no—
349#387—941no—
350#387—941no—
351#383—941no—
352#382—941no—
353#381—941no—
354#380—941no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357#376—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340C2×D4711yescharacteristic
363#340C2×D4711yescharacteristic
364C47#2128011yescharacteristic, complemented, Sylow
365C22×Q8—471no—
366C22×D4C22×D4711yescharacteristic
367C23×C4—941no—
368C2×SD16—474no—
369C2×SD16—942no—
370C2×SD16—942no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375C4×Q8—474no—
376C4×Q8—942no—
377C4×D4—474no—
378C4×D4—942no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42C22×D4711yescharacteristic
401#106—474no—
402#106—942no—
403#106—942no—
404#103—474no—
405#103—942no—
406#103—942no—
407#102—22no—
408#102—942no—
409#102C4×D4714yes—
410#97—22no—
411#97—942no—
412#97—942no—
413C4×C8—474no—
414C4×C8—942no—
415C4×C8—942no—
416#95—21no—
417#95—471no—
418#95—471no—
419#95—471no—
420#95—471no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430C24—21no—
431C2×Q8—472no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×Q8—941no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D4712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—3761no—
456C22×C4—3761no—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460SD16—948no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—472no—
466C2×C8—472no—
467C2×C8—472no—
468C2×C8—472no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8C2×C4×D4712yes—
472C2×C8#1062535212yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—81no—
479C4⋊C4—472no—
480C4⋊C4—944no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—3761no—
487C4⋊C4—3761no—
488C4⋊C4—3761no—
489C4⋊C4C2×C4×D4712yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—944no—
493C22⋊C4—3761no—
494C22⋊C4—3761no—
495C42—21no—
496C42—21no—
497C42—472no—
498C42—472no—
499C42—944no—
500C42—941no—
501C42—941no—
502C42—3761no—
503C42—3761no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D4711yescharacteristic
512Q8—942no—
513Q8—942no—
514Q8—1881no—
515Q8—1881no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—3764no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4C22×C4×D4711yescharacteristic
561C2×C4#41×D4711yescharacteristic
562C2×C4#41×D4711yescharacteristic
563C2×C4#41×D4711yescharacteristic
564C2×C4#41×D4711yescharacteristic
565C2×C4C2×#1062535211yescharacteristic
566C8—22no—
567C8—22no—
568C8—474no—
569C8—942no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D4711yescharacteristic
578C22#1568966811yescharacteristic
579C22#1568967211yescharacteristic
580C22#1568993211yescharacteristic
581C22#1568993311yescharacteristic
582C22#1568993311yescharacteristic
583C22#1568993311yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—942no—
590C4—942no—
591C4—3761no—
592C4—3761no—
593C4—1881no—
594C4—1881no—
595C4—1882no—
596C4—1882no—
597C4#117×D4711yescharacteristic
598C4C4×#1062535211yescharacteristic
599C4#1568965011yescharacteristic
600C4#1568965611yescharacteristic
601C2—82no—
602C2#2024584811yescharacteristic
603C2#2025224411yescharacteristic
604C2#2025227011yescharacteristic
605C2#2025400011yescharacteristic
606C2#2025400411yescharacteristic
607C2#2025401111yescharacteristic
608C2#2025401511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1936C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1936D47
hypocenterC47#21280
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #20246035C2semidirect
6. #20244115C2semidirect
7. #20244113C2semidirect
9. #20244078C2semidirect
10. #20244076C2semidirect
11. #20243276C2semidirect
12. #20243217C2semidirect
14. #20243076C2semidirect
19. #15689214C4semidirect
364. C47#21280semidirect

Maximal subgroups

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

subgroupindex
C2×#156892142
#202460352
#202460252
#202442432
#202442412
#202441152
#202441132
#202440872
#202440782
#202440762
#202432762
#202432172
#202432152
#202430762
C47×#19362
#2128047