Finite Group Databasebeta

← #26185902

Subgroups — #26185902

Order 12032.

700 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252504C211yesmaximal, characteristic, complemented
2#20252409C211yesmaximal, characteristic, complemented
3C4×#13094051C211yesmaximal, characteristic
4#20242252C211yesmaximal, characteristic, complemented
5#20242251C211yesmaximal, characteristic, complemented
6#20242246C211yesmaximal, characteristic, complemented
7#20242243C211yesmaximal, characteristic, complemented
8#20242227C211yesmaximal, characteristic, complemented
9#20242219C211yesmaximal, characteristic, complemented
10#20242214C211yesmaximal, characteristic, complemented
11#20242137C211yesmaximal, characteristic, complemented
12#20241769C211yesmaximal, characteristic, complemented
13D4⋊C8×D47C211yesmaximal, characteristic, complemented
14#20241676C211yesmaximal, characteristic, complemented
15C47×#1582C211yesmaximal, characteristic, complemented
16C2×#13094051C411yescharacteristic
17C4×#10625354C2211yescharacteristic
18C4×D4×D47C2211yescharacteristic
19#15689254C2211yescharacteristic, complemented
20#15689234C411yescharacteristic
21#15689232C411yescharacteristic
22#15689226C411yescharacteristic
23#15689224C411yescharacteristic
24C4×#10625336C2211yescharacteristic
25C4×D376C2211yescharacteristic
26C4×#10625334C2211yescharacteristic
27#15689202C2211yescharacteristic, complemented
28#15689200C2211yescharacteristic, complemented
29#15689196—21no—
30#15689193—21no—
31C8×D188C2211yescharacteristic
32D4×C47⋊C8C2211yescharacteristic
33#15689188C2211yescharacteristic
34C8×C22⋊D47—21no—
35(C22⋊D47)⋊C8—21no—
36#15689178—21no—
37#15689175C2211yescharacteristic
38#15689174C2211yescharacteristic
39#15689162C411yescharacteristic
40#15689145—21no—
41C4⋊C8×D47C2211yescharacteristic
42#15689126C411yescharacteristic
43#15689123C411yescharacteristic
44C2×(C2×D47)⋊C8—21no—
45C22⋊C8×D47—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic, complemented
49#15688875C2211yescharacteristic, complemented
50#15688873C2211yescharacteristic, complemented
51Q8⋊(C47⋊C8)C2211yescharacteristic, complemented
52#15688852C2211yescharacteristic
53#15688852C2211yescharacteristic, complemented
54D4⋊(C47⋊C8)C2211yescharacteristic
55#15688850C2211yescharacteristic
56#15688850C2211yescharacteristic, complemented
57#15688845C2211yescharacteristic, complemented
58#15688845C2211yescharacteristic, complemented
59#15688845C2211yescharacteristic, complemented
60C47×#446C2211yescharacteristic, complemented
61C4×SD16×C47C2211yescharacteristic
62C8×D4×C47C2211yescharacteristic
63C47×#334C2211yescharacteristic, complemented
64C47×Q8⋊C8C2211yescharacteristic, complemented
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic, complemented
67C2×#10625354C2×C411yescharacteristic
68C2×D4×D47C2×C411yescharacteristic
69C22×C4×D47—21no—
70#13094051—24no—
71C2×#10625336C2×C411yescharacteristic
72C2×D376C2×C411yescharacteristic
73C2×#10625334C2×C411yescharacteristic
74C2×#10625324—21no—
75C2×#10625324—21no—
76C2×#10625324C2×C411yescharacteristic
77C2×#10625324C2×C411yescharacteristic
78C22×C47⋊C8—21no—
79C2×C8×D47—21no—
80C2×C8×D47—21no—
81C2×C8×D47—21no—
82C2×C8×D47—21no—
83Q8×Dic47C2×C411yescharacteristic
84C4×D188—21no—
85C4×D188C2311yescharacteristic
86C4×D188C2311yescharacteristic
87D4×Dic47C2×C411yescharacteristic
88#13093986—21no—
89#13093986C2×C411yescharacteristic
90#13093986C2×C411yescharacteristic
91C4×C22⋊D47—21no—
92(C22⋊D47)⋊C4—21no—
93#13093982—21no—
94#13093982C2×C411yescharacteristic
95C4⋊C4×D47C2×C411yescharacteristic
96C22⋊C4×D47—21no—
97C42×D47—21no—
98C42×D47C2311yescharacteristic
99#13093958C2×C411yescharacteristic
100#13093957C2×C411yescharacteristic
101#13093955C2×C411yescharacteristic
102C4⋊(C47⋊C8)—21no—
103C4⋊(C47⋊C8)C2311yescharacteristic
104C4⋊(C47⋊C8)C2311yescharacteristic
105Dic47⋊C8—21no—
106Dic47⋊C8D411yescharacteristic
107Dic47⋊C8D411yescharacteristic
108#13093948C2×C411yescharacteristic
109#13093946C2×C411yescharacteristic
110#13093946C2×C411yescharacteristic
111#13093945C2×C411yescharacteristic
112#13093944C2×C411yescharacteristic
113#13093944C2×C411yescharacteristic
114C22⋊(C47⋊C8)—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8D411yescharacteristic
121(C2×D47)⋊C8D411yescharacteristic
122#13093935—21no—
123#13093934C2×C411yescharacteristic
124#13093934C2×C411yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C4×C47⋊C8C2311yescharacteristic
127C4×C47⋊C8C2311yescharacteristic
128C8×Dic47—21no—
129C8×Dic47—21no—
130C2×SD16×C47C2×C411yescharacteristic
131C22×C8×C47—21no—
132C4×Q8×C47C2311yescharacteristic
133C4×D4×C47C2311yescharacteristic
134#106×C47C2×C411yescharacteristic
135C4⋊C8×C47—21no—
136C4⋊C8×C47C2311yescharacteristic
137C4⋊C8×C47C2311yescharacteristic
138#103×C47C2×C411yescharacteristic
139#102×C47C2×C411yescharacteristic
140C22⋊C8×C47—21no—
141#97×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C23×D47—21no—
146C2×Q8×C47C22×C411yescharacteristic
147C2×D4×C47C22×C411yescharacteristic
148C22×C4×C47—21no—
149C2×C22⋊D47—21no—
150C22×Dic47—21no—
151#10625354—22no—
152#10625354—22no—
153D4×D47—22no—
154D4×D47M1614yescomplemented
155C2×D188—21no—
156C2×D188C22×C411yescharacteristic
157C2×D188C22×C411yescharacteristic
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—21no—
162C2×C4×D47—21no—
163C2×C4×D47—21no—
164C2×C4×D47—81no—
165C2×C4×D47—81no—
166C2×C4×D47C22×C411yescharacteristic
167C2×C4×D47C2×D411yescharacteristic
168C2×C4×D47#4111yescharacteristic
169SD16×C47—22no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—81no—
174C2×C8×C47C22×C411yescharacteristic
175C2×C8×C47C22×C411yescharacteristic
176C2×C8×C47C2×D411yescharacteristic
177C2×C8×C47C2×D411yescharacteristic
178C4⋊C4×C47—21no—
179C4⋊C4×C47C22×C411yescharacteristic
180C4⋊C4×C47C22×C411yescharacteristic
181C22⋊C4×C47—21no—
182C42×C47—21no—
183C42×C47C2411yescharacteristic
184C22⋊Dic47—21no—
185#10625336—22no—
186#10625334—22no—
187(C2×D47)⋊C4—21no—
188(C2×D47)⋊C4—21no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191(C2×D47)⋊C4—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47C22×C411yescharacteristic
194C4⋊Dic47C22×C411yescharacteristic
195Dic47⋊C4—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47C22×C411yescharacteristic
199C4×Dic47C2×D411yescharacteristic
200C4×Dic47#4111yescharacteristic
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—81no—
205C2×C47⋊C8C22×C411yescharacteristic
206C2×C47⋊C8C22×C411yescharacteristic
207C2×C47⋊C8C2×D411yescharacteristic
208C2×C47⋊C8C2×D411yescharacteristic
209D376—22no—
210#10625324—22no—
211#10625324—22no—
212#10625324—81no—
213#10625324—81no—
214C8×D47—42no—
215C8×D47—81no—
216C8×D47—81no—
217C8×D47—81no—
218C23×C47—21no—
219C22×D47—21no—
220C22×D47—24no—
221C22×D47—21no—
222C22×D47—81no—
223C22×D47—81no—
224C22×D47C4×D411yescharacteristic
225Q8×C47—21no—
226Q8×C47—21no—
227D4×C47—21no—
228D4×C47C2×M1612yes—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C22⋊D47—24no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—81no—
241C2×Dic47C4×D411yescharacteristic
242C2×Dic47C4×D411yescharacteristic
243C2×Dic47C4×D411yescharacteristic
244D188—21no—
245D188—21no—
246D188—21no—
247D188—42no—
248D188C2×M1612yes—
249C4×D47—21no—
250C4×D47—21no—
251C4×D47—41no—
252C4×D47—41no—
253C4×D47—41no—
254C4×D47—41no—
255C4×D47—41no—
256C4×D47—42no—
257C4×D47—81no—
258C4×D47—81no—
259C4×D47—81no—
260C4×D47C2×M1612yes—
261C4×D47#13112yes—
262C8×C47—21no—
263C8×C47—21no—
264C8×C47—21no—
265C8×C47—21no—
266C8×C47—41no—
267C8×C47—41no—
268C8×C47—41no—
269C47⋊C8—22no—
270C47⋊C8—21no—
271C47⋊C8—21no—
272C47⋊C8—21no—
273C47⋊C8—21no—
274C47⋊C8—41no—
275C47⋊C8—41no—
276#17873—471nomaximal, Sylow
277C22×C47—22no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47C2×C4×D411yescharacteristic
281C2×D47—21no—
282C2×D47—21no—
283C2×D47—22no—
284C2×D47—41no—
285C2×D47—41no—
286C2×D47—41no—
287C2×D47—41no—
288C2×D47—41no—
289C2×D47—84no—
290C2×D47#43512yes—
291C4×C47—21no—
292C4×C47—41no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47C22×M1611yescharacteristic
296C4×C47C2×#13111yescharacteristic
297C4×C47#57611yescharacteristic
298C4×C47#57711yescharacteristic
299Dic47—21no—
300Dic47—21no—
301Dic47—41no—
302Dic47—41no—
303Dic47#43512yes—
304#2989—471no—
305#2972—471no—
306C4×AGL(1,ℤ/8)—471no—
307#1586—471no—
308#1586—471no—
309#1585—471no—
310#1585—471no—
311#1582—471no—
312#1582D4711yescharacteristic, complemented
313#1581—471no—
314#1581—471no—
315#1565—471no—
316#1476—471no—
317C2×D4⋊C8—471no—
318#1451—471no—
319C2×C47—42no—
320C2×C47C2×#43511yescharacteristic
321C2×C47#294611yescharacteristic
322C2×C47#295611yescharacteristic
323D47—22no—
324D47—42no—
325D47—81no—
326C2×AGL(1,ℤ/8)—471no—
327C4×#41—471no—
328C2×C4×D4—471no—
329#446—471no—
330#446C2×D4711yescharacteristic, complemented
331#442—471no—
332#442—471no—
333#440—471no—
334#440—471no—
335C4×SD16—471no—
336C4×SD16C2×D4711yescharacteristic
337C4×D8—471no—
338C4×D8—471no—
339#436—471no—
340#436—471no—
341#435—941no—
342#435—941no—
343C8×D4—471no—
344C8×D4—471no—
345C8×D4—471no—
346C8×D4—941no—
347C8×D4—941no—
348C8×D4C2×D4711yescharacteristic
349#432—941no—
350#431—471no—
351#431—471no—
352#428—471no—
353#423—941no—
354C2×C4⋊C8—471no—
355#418—471no—
356#417—471no—
357C2×C22⋊C8—941no—
358C2×C22⋊C8—941no—
359C4×M16—471no—
360C2×C4×C8—471no—
361#335—471no—
362#335—471no—
363#334—471no—
364#334C2×D4711yescharacteristic, complemented
365Q8⋊C8—471no—
366Q8⋊C8C2×D4711yescharacteristic, complemented
367D4⋊C8—471no—
368D4⋊C8—471no—
369D4⋊C8—471no—
370D4⋊C8—471no—
371D4⋊C8—471no—
372D4⋊C8C2×D4711yescharacteristic
373#322—471no—
374#322—471no—
375#322—471no—
376#322C2×D4711yescharacteristic, complemented
377C47#1787311yescharacteristic, complemented, Sylow
378C2×#41—471no—
379C22×D4—471no—
380C23×C4—941no—
381AGL(1,ℤ/8)—944no—
382C2×SD16—471no—
383C2×SD16C4×D4711yescharacteristic
384C2×D8—471no—
385C2×D8—471no—
386C2×M16—471no—
387C2×M16—471no—
388C2×M16—941no—
389C2×M16—941no—
390C22×C8—21no—
391C22×C8—941no—
392C22×C8—941no—
393C22×C8—941no—
394C22×C8—941no—
395C22×C8—941no—
396C4×Q8—471no—
397C4×Q8C22×D4711yescharacteristic
398C4×D4—471no—
399C4×D4—471no—
400C4×D4—471no—
401C4×D4—471no—
402C4×D4—471no—
403C4×D4—941no—
404C4×D4—941no—
405C4×D4—941no—
406C4×D4—941no—
407C4×D4C22×D4711yescharacteristic
408#117—471no—
409#117—941no—
410C2×C4⋊C4—471no—
411C2×C22⋊C4—941no—
412C2×C42—471no—
413C2×C42—941no—
414#107—471no—
415#107—471no—
416#106—471no—
417#106C4×D4711yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—471no—
420C4⋊C8—471no—
421C4⋊C8—471no—
422C4⋊C8—471no—
423C4⋊C8—941no—
424C4⋊C8—941no—
425C4⋊C8C22×D4711yescharacteristic
426C4⋊C8C22×D4711yescharacteristic
427#103—471no—
428#103C4×D4711yescharacteristic
429#102—471no—
430#102—471no—
431#102—471no—
432#102—471no—
433#102—471no—
434#102C4×D4711yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—471no—
437C22⋊C8—471no—
438C22⋊C8—941no—
439C22⋊C8—941no—
440C22⋊C8—941no—
441C22⋊C8—941no—
442C22⋊C8—941no—
443C22⋊C8—941no—
444#97—21no—
445#97—471no—
446#97—471no—
447#97—941no—
448C4×C8—471no—
449C4×C8—471no—
450C4×C8—471no—
451C4×C8—941no—
452C4×C8—941no—
453C4×C8C22×D4711yescharacteristic
454C4×C8C22×D4711yescharacteristic
455C4×C8C22×D4711yescharacteristic
456C24—941no—
457#41—942no—
458#41—942no—
459C2×Q8C2×C4×D4711yescharacteristic
460C2×D4—474no—
461C2×D4—471no—
462C2×D4—471no—
463C2×D4—941no—
464C2×D4—942no—
465C2×D4—941no—
466C2×D4C2×C4×D4711yescharacteristic
467C22×C4—21no—
468C22×C4—471no—
469C22×C4—471no—
470C22×C4—471no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—3761no—
479C22×C4—3761no—
480SD16—22no—
481SD16—942no—
482D8—942no—
483D8—942no—
484M16—942no—
485M16—942no—
486M16—3761no—
487M16—3761no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—81no—
492C2×C8—471no—
493C2×C8—471no—
494C2×C8—471no—
495C2×C8—471no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—3761no—
500C2×C8—3761no—
501C2×C8—3761no—
502C2×C8—3761no—
503C2×C8—1882no—
504C2×C8C2×C4×D4711yescharacteristic
505C2×C8C2×C4×D4711yescharacteristic
506C2×C8D4×D4711yescharacteristic
507C2×C8D4×D4711yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—471no—
510C4⋊C4—471no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4C2×C4×D4711yescharacteristic
514C4⋊C4C2×C4×D4711yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—941no—
517C22⋊C4—941no—
518C22⋊C4—941no—
519C22⋊C4—941no—
520C22⋊C4—941no—
521C22⋊C4—941no—
522C42—21no—
523C42—471no—
524C42—471no—
525C42—471no—
526C42—941no—
527C42—941no—
528C42C23×D4711yescharacteristic
529C23—21no—
530C23—471no—
531C23—944no—
532C23—941no—
533C23—941no—
534C23—3761no—
535C23—3761no—
536Q8—21no—
537Q8—21no—
538D4—21no—
539D4—472no—
540D4—944no—
541D4—941no—
542D4—941no—
543D4—941no—
544D4—1882no—
545D4M16×D4712yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—471no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—471no—
555C2×C4—471no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—3761no—
561C2×C4—3761no—
562C2×C4—3761no—
563C2×C4—3761no—
564C2×C4—1881no—
565C2×C4—1882no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4#41×D4711yescharacteristic
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—942no—
581C8—941no—
582C8—941no—
583C8—941no—
584C8—941no—
585C8—1881no—
586C8—1881no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—472no—
591C22—941no—
592C22—941no—
593C22—942no—
594C22—3764no—
595C22—1881no—
596C22—1881no—
597C22—1881no—
598C22—1881no—
599C22—1881no—
600C22C4×D4×D4711yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—472no—
606C4—941no—
607C4—941no—
608C4—1881no—
609C4—1881no—
610C4C2×M16×D4711yescharacteristic
611C4#131×D4711yescharacteristic
612C4#1568992511yescharacteristic
613C4#1568992911yescharacteristic
614C2—42no—
615C2—942no—
616C2—3761no—
617C2—1882no—
618C2#435×D4711yescharacteristic
619C2#2025221511yescharacteristic
620C2#2025229611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1582C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094051C4
cosocleC42×C47C24
hypercenter#1582D47
hypocenterC47#17873
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #20252504C2semidirect
2. #20252409C2semidirect
4. #20242252C2semidirect
5. #20242251C2semidirect
6. #20242246C2semidirect
7. #20242243C2semidirect
8. #20242227C2semidirect
9. #20242219C2semidirect
10. #20242214C2semidirect
11. #20242137C2semidirect
12. #20241769C2semidirect
13. D4⋊C8×D47C2semidirect
14. #20241676C2semidirect
15. C47×#1582C2semidirect
19. #15689254C22semidirect
27. #15689202C22semidirect
28. #15689200C22semidirect
48. #15688876C22semidirect
49. #15688875C22semidirect
50. #15688873C22semidirect
51. Q8⋊(C47⋊C8)C22semidirect
53. #15688852C22semidirect
56. #15688850C22semidirect
57. #15688845C22semidirect
58. #15688845C22semidirect
59. #15688845C22semidirect
60. C47×#446C22semidirect
63. C47×#334C22semidirect
64. C47×Q8⋊C8C22semidirect
66. C47×#322C22semidirect
154. D4×D47M16semidirect
312. #1582D47semidirect
330. #446C2×D47semidirect
364. #334C2×D47semidirect
366. Q8⋊C8C2×D47semidirect
376. #322C2×D47semidirect
377. C47#17873semidirect

Maximal subgroups

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

subgroupindex
#202525042
#202524092
C4×#130940512
#202422522
#202422512
#202422462
#202422432
#202422272
#202422192
#202422142
#202421372
#202417692
D4⋊C8×D472
#202416762
C47×#15822
#1787347