Finite Group Databasebeta

← #26248699

Subgroups — #26248699

Order 12032.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689273C211yesmaximal, characteristic, complemented
2C2×#15689127C211yesmaximal, characteristic, complemented
3#20246744C211yesmaximal, characteristic
4#20244531C212yesmaximal, complemented
5#20244389C212yesmaximal, complemented
6#20243822C211yesmaximal, characteristic, complemented
7#20243743C212yesmaximal, complemented
8#20243705C212yesmaximal, complemented
9#20243686C211yesmaximal, characteristic, complemented
10#20243483C211yesmaximal, characteristic, complemented
11C2×C47×#552C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C22×#10625336C2212yes—
14C22×#10625329C2211yescharacteristic
15C2×#13093998C2212yes—
16C2×#13093992—21no—
17#15689273—24no—
18#15689127—24no—
19C2×#13093948C2212yescomplemented
20C2×#13093944C2212yes—
21C2×#13093944C2212yes—
22C2×#13093944C2212yes—
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2212yescomplemented
25#15689040—21no—
26#15689032—21no—
27#15689025C2212yes—
28#15688983C2211yescharacteristic
29#15688978—21no—
30#15688885C2212yescomplemented
31#15688885C2212yescomplemented
32#15688885C2212yescomplemented
33#15688878C2212yescomplemented
34C47×#552—24no—
35C47×#552—24no—
36C2×#126×C47—21no—
37C2×#125×C47—21no—
38C2×#125×C47C2211yescharacteristic, complemented
39C2×#123×C47—21no—
40C2×C22⋊Q8×C47C2211yescharacteristic
41C2×C22⋊Q8×C47C2211yescharacteristic, complemented
42C2×C4×Q8×C47C2211yescharacteristic, complemented
43C2×C4×D4×C47C2211yescharacteristic
44C2×#117×C47—21no—
45C2×#117×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic, complemented
47C24×D47—21no—
48C22×D188—22no—
49C22×D188—24no—
50C22×D188—22no—
51C22×D188—21no—
52C22×D188—42no—
53C22×D188C2312yes—
54C22×C22⋊D47—22no—
55C2×#10625336—24no—
56C2×#10625336—24no—
57C2×#10625336—42no—
58C2×#10625329—22no—
59C2×#10625329—42no—
60C22×C47⋊C8C2312yes—
61C22×C47⋊C8C2312yes—
62#13093998—24no—
63#13093998—42no—
64#13093992—44no—
65C2×C4⋊Dic47C2312yes—
66C2×C22⋊Dic47—21no—
67C2×(C2×D47)⋊C4—22no—
68C2×(C2×D47)⋊C4—22no—
69C2×(C2×D47)⋊C4—22no—
70#13093948—42no—
71#13093944—24no—
72#13093944—24no—
73#13093944—24no—
74C22⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)—42no—
76C47⋊#95—21no—
77C22×Q8×C47C2311yescharacteristic
78C22×D4×C47D411yescharacteristic
79C23×C4×C47—21no—
80C23×C4×C47C2311yescharacteristic
81C23×C4×C47C2311yescharacteristic
82#126×C47—28no—
83#125×C47—28no—
84#125×C47—22no—
85#125×C47—22no—
86#123×C47—28no—
87C22⋊Q8×C47—22no—
88C22⋊Q8×C47—22no—
89C22⋊Q8×C47—22no—
90C22⋊Q8×C47D414yescomplemented
91C4×Q8×C47—22no—
92C4×Q8×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95#117×C47—28no—
96#117×C47—22no—
97#117×C47D414yescomplemented
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47D411yescharacteristic
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118C24×C47C2×D411yescharacteristic
119C24×C47C2×D411yescharacteristic
120C23×D47—22no—
121C23×D47—44no—
122C23×D47—42no—
123C23×D47—42no—
124C23×D47—42no—
125C23×D47—82no—
126C2×Q8×C47—21no—
127C2×Q8×C47—22no—
128C2×Q8×C47—22no—
129C2×Q8×C47C2×D412yes—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47—41no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2411yescharacteristic
153C2×C22⋊D47—44no—
154C2×C22⋊D47—44no—
155C2×C22⋊D47—82no—
156C22×Dic47—22no—
157C2×D188—28no—
158C2×D188—22no—
159C2×D188—22no—
160C2×D188—22no—
161C2×D188—22no—
162C2×D188—22no—
163C2×D188—48no—
164C2×D188—44no—
165C2×D188—44no—
166C2×D188—42no—
167C2×D188#4114yes—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47C2×D412yes—
179C4⋊C4×C47C2×D412yes—
180C4⋊C4×C47C2×D412yes—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—24no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47—22no—
186C42×C47—24no—
187C42×C47—24no—
188C42×C47—21no—
189C42×C47—21no—
190C22⋊Dic47—42no—
191#10625336—48no—
192(C2×D47)⋊C4—44no—
193(C2×D47)⋊C4—44no—
194(C2×D47)⋊C4—82no—
195C4⋊Dic47—22no—
196C4⋊Dic47—22no—
197#10625329—44no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C23×C47—21no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47#12012yes—
213C23×C47AGL(1,ℤ/8)12yes—
214C23×C47C22×D411yescharacteristic
215C22×D47—24no—
216C22×D47—48no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47—88no—
224C22×D47—88no—
225C22×D47—84no—
226C22×D47—84no—
227Q8×C47—24no—
228Q8×C47—41no—
229Q8×C47—41no—
230D4×C47—44no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—21no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—48no—
247C2×C4×C47—42no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—42no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—42no—
256C2×C4×C47#12012yes—
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C2×#4112yes—
260C2×C4×C47H(5,𝔽2)12yes—
261C22⋊D47—88no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265D188—24no—
266D188—24no—
267D188—48no—
268D188—42no—
269D188—42no—
270C47⋊C8—42no—
271C47⋊C8—42no—
272#23982—471nomaximal, Sylow
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—24no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47C2×#12011yescharacteristic
284C22×C47#54612yes—
285C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C47#57511yescharacteristic
287C22×C47#57712yes—
288C2×D47—44no—
289C2×D47—44no—
290C2×D47—44no—
291C2×D47—816no—
292C2×D47—82no—
293C2×D47—82no—
294C2×D47—82no—
295C2×D47—82no—
296C4×C47—22no—
297C4×C47—41no—
298C4×C47—42no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—42no—
302C4×C47—41no—
303C4×C47—42no—
304C4×C47#52511yescharacteristic
305C4×C47#53511yescharacteristic
306C4×C47#54612yes—
307Dic47—82no—
308C2×#552D4711yescharacteristic, complemented
309C2×#450—471no—
310C2×#418—471no—
311#2408—471no—
312#2025—472no—
313#2000—472no—
314#1894—471no—
315#1880—472no—
316#1874—472no—
317#1870—471no—
318#1825—471no—
319C2×C47—42no—
320C2×C47—42no—
321C2×C47#240411yescharacteristic
322C2×C47#300611yescharacteristic
323C2×C47#301011yescharacteristic
324C2×C47#331012yes—
325C2×C47#332312yes—
326D47—84no—
327C23×D4—471no—
328C22×SD16—472no—
329C22×M16—471no—
330#552—24no—
331#552—24no—
332C2×#126—21no—
333C2×#125—21no—
334C2×#125C2×D4711yescharacteristic, complemented
335C2×#123—21no—
336C2×C22⋊Q8C2×D4711yescharacteristic
337C2×C22⋊Q8C2×D4711yescharacteristic, complemented
338C2×#121—472no—
339C2×#120—941no—
340C2×C4×Q8C2×D4711yescharacteristic, complemented
341C2×C4×D4C2×D4711yescharacteristic
342C2×#117—21no—
343C2×#117C2×D4711yescharacteristic
344C22×C4⋊C4C2×D4711yescharacteristic, complemented
345#450—944no—
346#418—944no—
347C2×#103—472no—
348C2×#102—472no—
349C2×#102—472no—
350C2×#102—472no—
351C2×C22⋊C8—471no—
352C2×C22⋊C8—472no—
353#394—941no—
354#392—941no—
355#390—472no—
356#380—471no—
357#379—941no—
358#340—472no—
359#340—472no—
360#340—472no—
361#336—472no—
362C47#2398211yescharacteristic, complemented, Sylow
363C25—941no—
364C22×Q8C22×D4711yescharacteristic
365C22×D4—472no—
366C22×D4—944no—
367C22×D4—942no—
368C22×D4—942no—
369C22×D4—942no—
370C22×D4—941no—
371C22×D4—1882no—
372C22×D4C22⋊D4711yescharacteristic
373C23×C4—21no—
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C2×SD16—944no—
377C2×SD16—944no—
378C2×SD16—1882no—
379C2×M16—942no—
380C2×M16—1882no—
381C22×C8—472no—
382C22×C8—472no—
383#126—28no—
384#125—28no—
385#125—22no—
386#125—22no—
387#123—28no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8C22⋊D4714yescomplemented
392#121—944no—
393#121—1882no—
394#120—1884no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—28no—
400#117—22no—
401#117C22⋊D4714yescomplemented
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—472no—
411C2×C4⋊C4C22⋊D4711yescharacteristic
412C2×C4⋊C4C22×D4711yescharacteristic
413C2×C4⋊C4C22×D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4C22⋊D4711yescharacteristic
423C2×C22⋊C4C22⋊D4711yescharacteristic
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D4711yescharacteristic
427#103—1882no—
428#102—944no—
429#102—944no—
430#102—944no—
431C22⋊C8—942no—
432C22⋊C8—1882no—
433#95—941no—
434C24—942no—
435C24—3762no—
436C24—1884no—
437C24—1882no—
438C24—1882no—
439C24—1882no—
440C24C2×C22⋊D4711yescharacteristic
441C24C2×C22⋊D4711yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8C2×C22⋊D4712yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—474no—
450C2×D4—948no—
451C2×D4—942no—
452C2×D4—942no—
453C2×D4—942no—
454C2×D4—942no—
455C2×D4—942no—
456C2×D4—3762no—
457C2×D4—1888no—
458C2×D4—1884no—
459C2×D4—1884no—
460C2×D4—1884no—
461C2×D4—1884no—
462C2×D4—1882no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—942no—
477C22×C4C2×C22⋊D4711yescharacteristic
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4712yes—
482C22×C4C2×C22⋊D4712yes—
483C22×C4C23×D4711yescharacteristic
484SD16—1888no—
485M16—1884no—
486C2×C8—942no—
487C2×C8—942no—
488C2×C8—942no—
489C2×C8—942no—
490C2×C8—942no—
491C2×C8—942no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C4⋊C4C2×C22⋊D4712yes—
505C4⋊C4C2×C22⋊D4712yes—
506C4⋊C4C2×C22⋊D4712yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—3762no—
513C22⋊C4—1884no—
514C22⋊C4—1884no—
515C22⋊C4—1882no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—944no—
529C23—3764no—
530C23—3764no—
531C23—3768no—
532C23—3768no—
533C23—1888no—
534C23—1882no—
535C23—1882no—
536C23—1882no—
537C23—1882no—
538C23—1882no—
539C23—1882no—
540C23#1309399312yes—
541C23#1309404912yes—
542C23C22×C22⋊D4711yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546D4—44no—
547D4—944no—
548D4—944no—
549D4—3768no—
550D4—1888no—
551D4—1882no—
552D4—1882no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4#1309399312yes—
582C2×C4C22×C22⋊D4711yescharacteristic
583C2×C4C22×C22⋊D4711yescharacteristic
584C2×C4#41×D4712yes—
585C2×C4#1309407012yes—
586C8—1882no—
587C8—1882no—
588C22—22no—
589C22—24no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—37616no—
599C22—3762no—
600C22—3762no—
601C22—3762no—
602C22—3762no—
603C22—1884no—
604C22—1884no—
605C22—1884no—
606C22C2×#1309399311yescharacteristic
607C22#1568976112yes—
608C22C2×#1309404911yescharacteristic
609C22#1568992111yescharacteristic
610C22#1568993112yes—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—3762no—
620C4#1568963011yescharacteristic
621C4#1568968711yescharacteristic
622C4#1568976112yes—
623C2—42no—
624C2—42no—
625C2—3764no—
626C2#2024670311yescharacteristic
627C2#2025264011yescharacteristic
628C2#2025267411yescharacteristic
629C2#2025540312yes—
630C2#2025559812yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#552C2
socleC23×C47C22×D4
prime radical#20246744C2
hypercenterC2×#552D47
hypocenterC47#23982
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#15689273C2semidirect
2. C2×#15689127C2semidirect
4. #20244531C2semidirect
5. #20244389C2semidirect
6. #20243822C2semidirect
7. #20243743C2semidirect
8. #20243705C2semidirect
9. #20243686C2semidirect
10. #20243483C2semidirect
11. C2×C47×#552C2semidirect
19. C2×#13093948C22semidirect
24. C2×C22⋊(C47⋊C8)C22semidirect
30. #15688885C22semidirect
31. #15688885C22semidirect
32. #15688885C22semidirect
33. #15688878C22semidirect
38. C2×#125×C47C22semidirect
41. C2×C22⋊Q8×C47C22semidirect
42. C2×C4×Q8×C47C22semidirect
46. C22×C4⋊C4×C47C22semidirect
90. C22⋊Q8×C47D4semidirect
97. #117×C47D4semidirect
308. C2×#552D47semidirect
334. C2×#125C2×D47semidirect
337. C2×C22⋊Q8C2×D47semidirect
340. C2×C4×Q8C2×D47semidirect
344. C22×C4⋊C4C2×D47semidirect
362. C47#23982semidirect
391. C22⋊Q8C22⋊D47semidirect
401. #117C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892732
C2×#156891272
#202467442
#202445312
#202443892
#202438222
#202437432
#202437052
#202436862
#202434832
C2×C47×#5522
#2398247