Finite Group Databasebeta

← #26218980

Subgroups — #26218980

Order 12032.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689124C211yesmaximal, characteristic
2C2×M16×Dic47C211yesmaximal, characteristic, complemented
3#20245879C211yesmaximal, characteristic
4C2×#15689019C211yesmaximal, characteristic, complemented
5#20244082C212yesmaximal, complemented
6#20244075C212yesmaximal, complemented
7#20244068C212yesmaximal, complemented
8#20244061C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10#20243107C211yesmaximal, characteristic, complemented
11C2×C47×#417C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×Dic47⋊C4—21no—
16C22×C4×Dic47C2211yescharacteristic
17#15689124C418yescomplemented
18C2×#13093948C2212yes—
19C2×#13093945C2212yes—
20M16×Dic47—24no—
21C2×#13093934C2212yes—
22C2×C8×Dic47C2212yes—
23#15689030C2211yescharacteristic
24#15689030C2211yescharacteristic
25#15689026C2211yescharacteristic
26#15689026C2211yescharacteristic
27#15689019—21no—
28#15689019—21no—
29#15689019C2212yes—
30#15689019C2212yes—
31#15689003—21no—
32#15689003—21no—
33C4×C22⋊Dic47—21no—
34C4×C22⋊Dic47—21no—
35C2×C47⋊#95—21no—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40C22×#41×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#417—24no—
44C2×#103×C47C2212yes—
45C2×#102×C47C2212yes—
46C23×Dic47—21no—
47C23×Dic47D411yescharacteristic
48C23×Dic47D411yescharacteristic
49C2×#10625329—22no—
50C2×#10625329—22no—
51C2×#10625329C2×C414yes—
52C22×C47⋊C8C2312yes—
53#13093983—22no—
54#13093983—22no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47C2×C414yes—
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2311yescharacteristic
60C2×C4⋊Dic47C2312yes—
61C2×Dic47⋊C4—22no—
62C2×Dic47⋊C4—22no—
63C2×Dic47⋊C4—44no—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47—21no—
66C2×C22⋊Dic47—21no—
67C2×C22⋊Dic47—21no—
68C2×C22⋊Dic47—21no—
69C2×C4×Dic47—22no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47D411yescharacteristic
73C2×C4×Dic47D411yescharacteristic
74C2×C4×Dic47C2311yescharacteristic
75C2×C4×Dic47C2311yescharacteristic
76#13093948C2×C418yescomplemented
77#13093945C2×C418yescomplemented
78#13093934—24no—
79C8×Dic47—24no—
80C47⋊#95—22no—
81C47⋊#95—22no—
82C47⋊#95—21no—
83C47⋊#95—21no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47C2×C414yes—
92C22×Q8×C47C2311yescharacteristic
93C22×D4×C47—21no—
94C22×D4×C47C2311yescharacteristic
95C23×C4×C47—21no—
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98C2×M16×C47—22no—
99C2×M16×C47—22no—
100C2×M16×C47D414yes—
101C22×C8×C47C2312yes—
102C2×C4⋊C4×C47—22no—
103C2×C4⋊C4×C47—22no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47C2312yes—
106#103×C47—24no—
107#102×C47—24no—
108C24×C47—21no—
109C24×C47C2×D411yescharacteristic
110#41×C47—28no—
111#41×C47—28no—
112#41×C47—44no—
113#41×C47—44no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47C22×C412yes—
120C2×D4×C47—24no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—42no—
127C2×D4×C47—42no—
128C2×D4×C47C22×C412yes—
129C22×C4×C47—21no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47—42no—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D412yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47#4112yes—
143C22×C4×C47#4112yes—
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—41no—
154C22×Dic47—41no—
155C22×Dic47—42no—
156C22×Dic47—42no—
157C22×Dic47C2×D411yescharacteristic
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161M16×C47—28no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47C2×D414yes—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—42no—
168C22⋊Dic47—42no—
169C22⋊Dic47—42no—
170C22⋊Dic47—42no—
171C22⋊Dic47—42no—
172C22⋊Dic47—42no—
173C4⋊Dic47—24no—
174C4⋊Dic47—22no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47C22×C414yes—
180Dic47⋊C4—44no—
181Dic47⋊C4—44no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—21no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—42no—
189#10625329—28no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8C22×C414yes—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—22no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47C4×D412yes—
205C23×C47#12112yes—
206C23×C47#12412yes—
207C23×C47C22×D411yescharacteristic
208Q8×C47—22no—
209Q8×C47—22no—
210Q8×C47—41no—
211Q8×C47—41no—
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—48no—
215D4×C47—41no—
216D4×C47—41no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—48no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—84no—
225C2×C4×C47#11714yes—
226C2×C4×C47C4×D412yes—
227C2×C4×C47C4×D414yes—
228C2×C4×C47#12112yes—
229C2×C4×C47#12314yes—
230C2×C4×C47#12412yes—
231C2×C4×C47#12614yes—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—24no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251C2×Dic47—48no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47AGL(1,ℤ/8)12yes—
261C2×Dic47#13712yes—
262C8×C47—24no—
263C47⋊C8—24no—
264#21093—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47—81no—
278C22×C47#38814yes—
279C22×C47C2×C4×D411yescharacteristic
280C22×C47C2×#12111yescharacteristic
281C22×C47C2×#12411yescharacteristic
282C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C47C2×#13711yescharacteristic
284C22×C47#57511yescharacteristic
285C22×C47#57511yescharacteristic
286C4×C47—42no—
287C4×C47—82no—
288C4×C47#38814yes—
289C4×C47C2×#11711yescharacteristic
290C4×C47C2×C4×D411yescharacteristic
291C4×C47C2×#12311yescharacteristic
292C4×C47C2×#12611yescharacteristic
293Dic47—22no—
294Dic47—22no—
295Dic47—44no—
296Dic47—41no—
297Dic47—41no—
298Dic47—81no—
299Dic47—81no—
300C2×#417—471no—
301C2×#417D4711yescharacteristic
302C2×C4×M16—471no—
303#2297—471no—
304C2×#389—471no—
305#1936—472no—
306#1935—472no—
307#1934—472no—
308#1933—472no—
309#1736—471no—
310#1736—471no—
311C2×C47—24no—
312C2×C47—82no—
313C2×C47C2×#38811yescharacteristic
314C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C47C4×#13711yescharacteristic
316C2×C47#305511yescharacteristic
317C2×C47#305611yescharacteristic
318C2×C47#314111yescharacteristic
319C2×C47#314211yescharacteristic
320C22×#41C2×D4711yescharacteristic
321C22×M16—471no—
322C22×M16C2×D4711yescharacteristic
323C2×#117—471no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—941no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C42—471no—
328#417—24no—
329#417—478no—
330C2×#103—472no—
331C2×#103C2×D4712yes—
332C2×#102—472no—
333C2×#102C2×D4712yes—
334C4×M16—944no—
335C2×#97—472no—
336C2×C4×C8—472no—
337#391—471no—
338#391—471no—
339#390—471no—
340#390—471no—
341#389—472no—
342#389—472no—
343#389—941no—
344#389—941no—
345#386—941no—
346#386—941no—
347C4×C22⋊C4—941no—
348C4×C22⋊C4—941no—
349C2×#95—941no—
350#340—472no—
351#340—472no—
352#340—472no—
353#340—472no—
354C47#2109311yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C4×D4714yes—
361C22×Q8C22×D4711yescharacteristic
362C22×D4—21no—
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—471no—
367C23×C4—471no—
368C23×C4—941no—
369C23×C4C22×D4711yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—474no—
373C2×M16—942no—
374C2×M16—942no—
375C2×M16C22⋊D4714yes—
376C22×C8—472no—
377C22×C8C22×D4712yes—
378#117—942no—
379#117—942no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—472no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—1884no—
392C2×C4⋊C4C22×D4712yes—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—942no—
403C2×C42—942no—
404C2×C42—941no—
405#103—24no—
406#103—478no—
407#102—24no—
408#102—478no—
409#97—944no—
410C4×C8—944no—
411#95—942no—
412#95—942no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417C24—21no—
418C24D4×D4711yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×C4×D4712yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C4×D4712yes—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—1881no—
459C22×C4—1881no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4C2×C4×D4712yes—
463C22×C4#1062535012yes—
464C22×C4D4×D4711yescharacteristic
465C22×C4#1062535212yes—
466C22×C4#1062535212yes—
467C22×C4#1062535212yes—
468C22×C4C2×C22⋊D4712yes—
469C22×C4C23×D4711yescharacteristic
470M16—28no—
471M16—948no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—474no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8C2×C22⋊D4714yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—474no—
482C4⋊C4—944no—
483C4⋊C4—942no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—1884no—
489C4⋊C4—1884no—
490C22⋊C4—1882no—
491C22⋊C4—1882no—
492C22⋊C4—1882no—
493C22⋊C4—1882no—
494C22⋊C4—1882no—
495C42—942no—
496C42—942no—
497C42—941no—
498C42—941no—
499C42—941no—
500C42—941no—
501C42—1882no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23(C22⋊D47)⋊C412yes—
514C23#1309399512yes—
515C23#1309401012yes—
516C23C2×D4×D4711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—944no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1888no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4#1309398014yes—
559C2×C4(C22⋊D47)⋊C412yes—
560C2×C4C4×C22⋊D4714yes—
561C2×C4#1309399512yes—
562C2×C4#1309400514yes—
563C2×C4#1309401012yes—
564C2×C4#1309401714yes—
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C22×C22⋊D4711yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4C2×#1062535211yescharacteristic
569C2×C4C2×#1062535211yescharacteristic
570C2×C4C2×#1062535011yescharacteristic
571C8—24no—
572C8—944no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1568901114yes—
587C22C2×(C22⋊D47)⋊C411yescharacteristic
588C22C2×#1309399511yescharacteristic
589C22C2×#1309401011yescharacteristic
590C22AGL(1,ℤ/8)×D4711yescharacteristic
591C22#137×D4711yescharacteristic
592C22#1568991911yescharacteristic
593C22#1568992211yescharacteristic
594C4—42no—
595C4—82no—
596C4—942no—
597C4—942no—
598C4—3761no—
599C4—3761no—
600C4—1884no—
601C4—1881no—
602C4—1881no—
603C4#1568901114yes—
604C4C2×#1309398011yescharacteristic
605C4C2×C4×C22⋊D4711yescharacteristic
606C4C2×#1309400511yescharacteristic
607C4C2×#1309401711yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1568901111yescharacteristic
611C2#2025220111yescharacteristic
612C2#2025221211yescharacteristic
613C2#2025300111yescharacteristic
614C2#2025301211yescharacteristic
615C2#2025377911yescharacteristic
616C2#2025378611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#21093
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic47C2semidirect
4. C2×#15689019C2semidirect
5. #20244082C2semidirect
6. #20244075C2semidirect
7. #20244068C2semidirect
8. #20244061C2semidirect
9. #20243108C2semidirect
10. #20243107C2semidirect
17. #15689124C4semidirect
76. #13093948C2×C4semidirect
77. #13093945C2×C4semidirect
354. C47#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#156891242
C2×M16×Dic472
#202458792
C2×#156890192
#202440822
#202440752
#202440682
#202440612
#202431082
#202431072
C2×C47×#4172
#2109347