Finite Group Databasebeta

← #26217163

Subgroups — #26217163

Order 12032.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256589C211yesmaximal, characteristic
2C2×#15689139C211yesmaximal, characteristic
3C2×#15689122C211yesmaximal, characteristic
4#20246012C211yesmaximal, characteristic, complemented
5#20243618C214yesmaximal, complemented
6#20243611C214yesmaximal, complemented
7#20243126C211yesmaximal, characteristic, complemented
8#20243097C211yesmaximal, characteristic, complemented
9C2×C47×#428C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689604—21no—
13#15689604—21no—
14#15689603C2212yes—
15#15689603C2212yes—
16#15689600—21no—
17#15689597—21no—
18#15689596—22no—
19C4×#10625350—22no—
20#15689591—22no—
21C2×#13093983C2211yescharacteristic
22C2×#13093983C2211yescharacteristic
23#117×D47—22no—
24#15689139C418yescomplemented
25C2×#13093950C2214yes—
26#15689122—22no—
27#15689122C2214yes—
28C2×#13093947—21no—
29C2×#13093944—21no—
30#15689019—22no—
31#15688996—22no—
32#15688992C2212yescomplemented
33#15688988C2212yescomplemented
34#15688891C2212yescomplemented
35#15688891C2212yescomplemented
36#15688882C2214yescomplemented
37C22×M16×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C47×#428—22no—
41C47×#428C2214yescomplemented
42C2×#107×C47—21no—
43C2×#106×C47—21no—
44C2×#10625350—21no—
45C2×#10625350—21no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350C2×C414yes—
49C2×#10625350D412yes—
50C2×#10625350Q814yes—
51C2×#10625350C2312yes—
52C22×Dic94D411yescharacteristic
53C22×D188D411yescharacteristic
54C22×C22⋊D47—21no—
55C22×C4×D47—21no—
56C2×#10625329—21no—
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329C2312yes—
60C22×C47⋊C8—21no—
61C4×Dic94—22no—
62C4×Dic94—22no—
63#13093989—22no—
64#13093989—22no—
65C4×D188—22no—
66C4×D188—22no—
67#13093986—22no—
68#13093986—22no—
69C4×C22⋊D47—24no—
70C4×C22⋊D47—24no—
71(C22⋊D47)⋊C4—24no—
72(C22⋊D47)⋊C4—24no—
73#13093983—21no—
74#13093983—21no—
75#13093983—21no—
76#13093983—21no—
77#13093983—22no—
78#13093983C2×C414yescomplemented
79#13093982—22no—
80#13093982—22no—
81#13093981—24no—
82#13093980—24no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×Dic47⋊C4—24no—
87C4⋊C4×D47—22no—
88C4⋊C4×D47—22no—
89C2×C22⋊Dic47—21no—
90C2×C22⋊Dic47D412yes—
91C22⋊C4×D47—24no—
92C2×C4×Dic47D412yes—
93C2×C4×Dic47C2312yes—
94C42×D47—24no—
95#13093950C2×C4116yescomplemented
96#13093947—24no—
97#13093944—24no—
98C47⋊#95—24no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—21no—
102C2×M16×C47—22no—
103C2×M16×C47C2×C414yescomplemented
104C2×M16×C47C2312yes—
105C22×C8×C47—21no—
106#117×C47—21no—
107#117×C47—21no—
108#117×C47D412yes—
109#117×C47C2312yes—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47C2312yes—
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120#107×C47—24no—
121#106×C47—24no—
122C24×C47C2×D411yescharacteristic
123C23×D47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—41no—
128C22×C4×C47—41no—
129C22×C4×C47—82no—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×Q812yes—
137C22×C4×C47#4112yes—
138C22×C4×C47#4112yes—
139C22×C4×C47C2411yescharacteristic
140C2×C22⋊D47—24no—
141C2×C22⋊D47—22no—
142C2×C22⋊D47—24no—
143C2×C22⋊D47—22no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47C2×D412yes—
147C22×Dic47C2×D412yes—
148#10625350—24no—
149#10625350—24no—
150#10625350—24no—
151#10625350—24no—
152#10625350—24no—
153#10625350—22no—
154#10625350—22no—
155#10625350C4⋊C418yescomplemented
156#10625350C22×C414yes—
157#10625350C2×Q814yes—
158C2×D188—21no—
159C2×D188—22no—
160C2×D188C4⋊C414yescomplemented
161C2×D188C2×D412yes—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—24no—
165C2×C4×D47—22no—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×Dic94—21no—
169C2×Dic94—22no—
170C2×Dic94C4⋊C414yescomplemented
171C2×Dic94C2×D412yes—
172M16×C47—24no—
173M16×C47—22no—
174M16×C47C22×C414yes—
175C2×C8×C47—24no—
176C2×C8×C47—22no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—41no—
182C4⋊C4×C47—41no—
183C4⋊C4×C47C2×D412yes—
184C4⋊C4×C47C2×D412yes—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C42×C47—22no—
188C42×C47—22no—
189C22⋊Dic47—24no—
190C22⋊Dic47—22no—
191C22⋊Dic47—22no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—24no—
194C4⋊Dic47—24no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199Dic47⋊C4—24no—
200Dic47⋊C4—24no—
201Dic47⋊C4—84no—
202C4×Dic47—24no—
203C4×Dic47—24no—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207C4×Dic47C22×C414yes—
208#10625329—22no—
209#10625329—22no—
210#10625329—42no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—42no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C4×D412yes—
216C23×C47#12011yescharacteristic
217C23×C47#12011yescharacteristic
218C23×C47C22⋊Q812yes—
219C23×C47C22×D411yescharacteristic
220C22×D47—22no—
221C22×D47—24no—
222C22×D47—81no—
223C22×D47—81no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—44no—
227C2×C4×C47—42no—
228C2×C4×C47—81no—
229C2×C4×C47—81no—
230C2×C4×C47—82no—
231C2×C4×C47—82no—
232C2×C4×C47C2×C4⋊C412yes—
233C2×C4×C47C2×C4⋊C412yes—
234C2×C4×C47C4×D412yes—
235C2×C4×C47C4×D412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47#12011yescharacteristic
238C2×C4×C47#12011yescharacteristic
239C2×C4×C47C22⋊Q812yes—
240C2×C4×C47C22⋊Q812yes—
241C2×C4×C47C22⋊Q812yes—
242C2×C4×C47#12312yes—
243C2×C4×C47#12312yes—
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×Q811yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C22⋊D47—24no—
251C22⋊D47—24no—
252C22⋊D47—28no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262D188—22no—
263D188—24no—
264D188C2×C4⋊C414yes—
265C4×D47—24no—
266C4×D47—24no—
267C4×D47—28no—
268C4×D47—84no—
269C4×D47—84no—
270Dic94—22no—
271Dic94—24no—
272Dic94C2×C4⋊C414yes—
273C8×C47—24no—
274C47⋊C8—42no—
275#20865—471nomaximal, Sylow
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—21no—
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—21no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47#38512yes—
285C22×C47#38512yes—
286C22×C47C2×C4×D411yescharacteristic
287C22×C47C2×#12011yescharacteristic
288C22×C47C2×C22⋊Q811yescharacteristic
289C2×D47—24no—
290C2×D47—42no—
291C2×D47—42no—
292C2×D47—84no—
293C4×C47—42no—
294C4×C47—42no—
295C4×C47—82no—
296C4×C47#38512yes—
297C4×C47#38512yes—
298C4×C47C22×C4⋊C411yescharacteristic
299C4×C47C2×C4×D411yescharacteristic
300C4×C47C2×C22⋊Q811yescharacteristic
301C4×C47C2×#12311yescharacteristic
302Dic47—24no—
303Dic47—44no—
304Dic47—42no—
305Dic47—42no—
306#3428—471no—
307C2×#428D4711yescharacteristic, complemented
308C2×#421—471no—
309C2×#417—471no—
310#2317—471no—
311#1860—474no—
312#1859—474no—
313#1742—471no—
314#1733—471no—
315C2×C47—21no—
316C2×C47—22no—
317C2×C47—21no—
318C2×C47—22no—
319C2×C47C2×#38511yescharacteristic
320C2×C47#302111yescharacteristic
321C2×C47#302211yescharacteristic
322D47—44no—
323C22×#41—471no—
324C22×M16—471no—
325C22×M16C2×D4711yescharacteristic
326#520—472no—
327#520—472no—
328#520—941no—
329#520—941no—
330#519—941no—
331#518—942no—
332#518—941no—
333C4×#41—942no—
334C4×#41—942no—
335C2×#117—471no—
336C2×#117—471no—
337C2×#117—942no—
338C2×#117C2×D4711yescharacteristic
339C22×C4⋊C4C2×D4711yescharacteristic
340#428—22no—
341#428C2×D4714yescomplemented
342C2×#107—21no—
343C2×#106—21no—
344#421—478no—
345C2×#104—474no—
346#417—474no—
347#417—942no—
348C2×#103—941no—
349C2×#102—941no—
350#389—942no—
351#385—942no—
352#384—472no—
353#382—472no—
354#343—472no—
355#343—472no—
356#339—474no—
357C47#2086511yescharacteristic, complemented, Sylow
358C2×#41—474no—
359C2×#41—472no—
360C2×#41—474no—
361C2×#41—472no—
362C2×#41—942no—
363C2×#41—941no—
364C2×#41—942no—
365C2×#41—941no—
366C22×Q8—471no—
367C22×D4—471no—
368C22×D4—941no—
369C23×C4—21no—
370C23×C4—941no—
371C23×C4C22×D4711yescharacteristic
372C2×M16—22no—
373C2×M16—21no—
374C2×M16—472no—
375C2×M16—941no—
376C2×M16—942no—
377C2×M16—942no—
378C2×M16C4×D4714yescomplemented
379C2×M16C22×D4712yes—
380C22×C8—21no—
381C22×C8—941no—
382C4×Q8—942no—
383C4×Q8—942no—
384C4×Q8—942no—
385C4×Q8—942no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—944no—
389C4×D4—944no—
390C4×D4—942no—
391C4×D4—942no—
392C4×D4—944no—
393C4×D4—944no—
394#117—21no—
395#117—21no—
396#117—474no—
397#117—944no—
398#117—944no—
399#117—941no—
400#117—941no—
401#117—942no—
402#117—941no—
403#117—941no—
404#117—942no—
405#117—942no—
406#117C22⋊D4712yes—
407#117C22×D4712yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—944no—
419C2×C4⋊C4C22⋊D4711yescharacteristic
420C2×C4⋊C4C22⋊D4711yescharacteristic
421C2×C4⋊C4C22×D4712yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—472no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—472no—
428C2×C42—472no—
429C2×C42—944no—
430#107—24no—
431#106—24no—
432#104—4716no—
433#103—944no—
434#102—944no—
435#95—944no—
436C24—941no—
437C24D4×D4711yescharacteristic
438#41—478no—
439#41—474no—
440#41—474no—
441#41—944no—
442#41—944no—
443#41—944no—
444#41—944no—
445#41—944no—
446#41—942no—
447#41—942no—
448C2×Q8—474no—
449C2×Q8—472no—
450C2×Q8—941no—
451C2×Q8—942no—
452C2×D4—474no—
453C2×D4—472no—
454C2×D4—944no—
455C2×D4—942no—
456C2×D4—944no—
457C2×D4—942no—
458C2×D4—941no—
459C2×D4—942no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—82no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—941no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4—944no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4C2×C4×D4712yes—
477C22×C4#1062535012yes—
478C22×C4D4×D4711yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4Q8×D4712yes—
482C22×C4#1062535412yes—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486M16—24no—
487M16—22no—
488M16—942no—
489M16—942no—
490M16—1882no—
491M16C2×C4×D4714yes—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—942no—
495C2×C8—1882no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—944no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—3764no—
510C4⋊C4C2×C22⋊D4712yes—
511C4⋊C4C2×C22⋊D4712yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—944no—
515C22⋊C4—944no—
516C22⋊C4—944no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C42—22no—
520C42—22no—
521C42—474no—
522C42—944no—
523C42—944no—
524C42—942no—
525C42—942no—
526C42—942no—
527C23—41no—
528C23—41no—
529C23—944no—
530C23—942no—
531C23—3761no—
532C23—3761no—
533C23#1309398612yes—
534C23#1309399211yescharacteristic
535C23#1309399211yescharacteristic
536C23#1309400212yes—
537C23C2×D4×D4711yescharacteristic
538Q8—474no—
539Q8—944no—
540Q8—942no—
541D4—474no—
542D4—948no—
543D4—944no—
544D4—944no—
545D4—944no—
546D4—942no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—944no—
556C2×C4—942no—
557C2×C4—948no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—3764no—
564C2×C4—3764no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4C4⋊C4×D4712yes—
570C2×C4C4⋊C4×D4712yes—
571C2×C4C4×C22⋊D4712yes—
572C2×C4C4×C22⋊D4712yes—
573C2×C4#1309398612yes—
574C2×C4#1309399211yescharacteristic
575C2×C4#1309399211yescharacteristic
576C2×C4#1309400012yes—
577C2×C4#1309400012yes—
578C2×C4#1309400212yes—
579C2×C4#1309400912yes—
580C2×C4#1309400912yes—
581C2×C4C22×C4×D4711yescharacteristic
582C2×C4C2×D4×D4711yescharacteristic
583C2×C4C22×C22⋊D4711yescharacteristic
584C2×C4C2×Q8×D4711yescharacteristic
585C2×C4C2×#1062535011yescharacteristic
586C2×C4C2×#1062535411yescharacteristic
587C8—24no—
588C8—1882no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—944no—
598C22—3764no—
599C22—1882no—
600C22—1882no—
601C22#1568899412yes—
602C22#1568899412yes—
603C22C2×#1309398611yescharacteristic
604C22C2×#1309399211yescharacteristic
605C22C2×#1309400211yescharacteristic
606C4—42no—
607C4—42no—
608C4—82no—
609C4—944no—
610C4—1884no—
611C4—1882no—
612C4—1882no—
613C4#1568899412yes—
614C4#1568899412yes—
615C4C2×C4⋊C4×D4711yescharacteristic
616C4C2×C4×C22⋊D4711yescharacteristic
617C4C2×#1309400011yescharacteristic
618C4C2×#1309400911yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—1884no—
624C2C2×#1568899411yescharacteristic
625C2#2025275711yescharacteristic
626C2#2025276411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#20865
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #20246012C2semidirect
5. #20243618C2semidirect
6. #20243611C2semidirect
7. #20243126C2semidirect
8. #20243097C2semidirect
9. C2×C47×#428C2semidirect
24. #15689139C4semidirect
32. #15688992C22semidirect
33. #15688988C22semidirect
34. #15688891C22semidirect
35. #15688891C22semidirect
36. #15688882C22semidirect
41. C47×#428C22semidirect
78. #13093983C2×C4semidirect
95. #13093950C2×C4semidirect
103. C2×M16×C47C2×C4semidirect
155. #10625350C4⋊C4semidirect
160. C2×D188C4⋊C4semidirect
170. C2×Dic94C4⋊C4semidirect
307. C2×#428D47semidirect
341. #428C2×D47semidirect
357. C47#20865semidirect
378. C2×M16C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565892
C2×#156891392
C2×#156891222
#202460122
#202436182
#202436112
#202431262
#202430972
C2×C47×#4282
#2086547