Finite Group Databasebeta

← #26217114

Subgroups — #26217114

Order 12032.

1394 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256589C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689165C211yesmaximal, characteristic, complemented
4C2×#15689105C211yesmaximal, characteristic
5#20245976C211yesmaximal, characteristic
6#20243604C214yesmaximal, complemented
7#20243599C214yesmaximal, complemented
8#20243098C211yesmaximal, characteristic, complemented
9C47×#1733C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689604—21no—
13#15689604—21no—
14#15689603—21no—
15#15689603—21no—
16#15689600C2212yes—
17#15689597C2212yes—
18#15689596—22no—
19C4×#10625350—22no—
20#15689591—22no—
21C2×#13093983C2211yescharacteristic
22#117×D47—22no—
23C22×C4⋊Dic47C2211yescharacteristic
24#15689166—22no—
25#15689166C2214yes—
26#15689165—22no—
27#15689165C2214yes—
28C2×#13093958—21no—
29C2×#13093957—21no—
30C2×#13093956—21no—
31C2×#13093955—21no—
32#15689105C418yescomplemented
33C2×#13093942C2212yes—
34C2×#13093940C2212yes—
35#15689029C2212yes—
36#15689025C2212yes—
37#15689000—22no—
38#15688980—22no—
39#15688883C2214yes—
40C22×M16×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C2×#117×C47C2211yescharacteristic
43C47×#339C2214yes—
44C2×#10625350—21no—
45C2×#10625350—21no—
46C2×#10625350—21no—
47C2×#10625350—21no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350C2×C414yes—
51C2×#10625350Q814yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47D412yes—
55C23×Dic47—21no—
56C22×C4×D47D412yes—
57C2×#10625329—21no—
58C2×#10625329—22no—
59C2×#10625329C2×C414yescomplemented
60C2×#10625329C2312yes—
61C22×C47⋊C8—21no—
62C4×Dic94—22no—
63C4×Dic94—22no—
64#13093989—22no—
65#13093989—22no—
66C4×D188—22no—
67C4×D188—22no—
68#13093986—22no—
69#13093986—22no—
70C4×C22⋊D47—24no—
71C4×C22⋊D47—24no—
72(C22⋊D47)⋊C4—24no—
73(C22⋊D47)⋊C4—24no—
74#13093983—21no—
75#13093983—21no—
76#13093983D412yes—
77#13093983C2312yes—
78#13093982—22no—
79#13093982—22no—
80#13093981—24no—
81#13093980—24no—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47D411yescharacteristic
88C2×C4⋊Dic47D411yescharacteristic
89C2×C4⋊Dic47C2312yes—
90C4⋊C4×D47—22no—
91C4⋊C4×D47—22no—
92C2×C22⋊Dic47—21no—
93C2×(C2×D47)⋊C4—24no—
94C22⋊C4×D47—24no—
95C2×C4×Dic47—21no—
96C42×D47—24no—
97#13093958—24no—
98#13093957—24no—
99#13093956—24no—
100#13093955—24no—
101#13093942C2×C418yescomplemented
102#13093940C2×C418yescomplemented
103C47⋊#95—24no—
104C23×C4×C47C2311yescharacteristic
105C2×M16×C47—21no—
106C2×M16×C47—22no—
107C2×M16×C47C2×C414yescomplemented
108C2×M16×C47C2312yes—
109C22×C8×C47—21no—
110#117×C47—21no—
111#117×C47—21no—
112#117×C47—22no—
113#117×C47—22no—
114#117×C47D412yes—
115#117×C47C2312yes—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47D411yescharacteristic
118C2×C4⋊C4×C47D411yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C42×C47—21no—
122C2×C42×C47C2312yes—
123C24×C47C2×D411yescharacteristic
124C23×D47C2×D412yes—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×Q812yes—
135C22×C4×C47#4112yes—
136C22×C4×C47#4112yes—
137C22×C4×C47C2411yescharacteristic
138C2×C22⋊D47—22no—
139C2×C22⋊D47—22no—
140C2×C22⋊D47—24no—
141C2×C22⋊D47C4⋊C418yes—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—41no—
146C22×Dic47—41no—
147C22×Dic47—82no—
148C22×Dic47C2×D412yes—
149#10625350—24no—
150#10625350—24no—
151#10625350—24no—
152#10625350—24no—
153#10625350—24no—
154#10625350—24no—
155#10625350—24no—
156#10625350—24no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188C22×C412yes—
162C2×D188C2×Q812yes—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—24no—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×C4×D47C4⋊C418yes—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94C22×C412yes—
174C2×Dic94C2×Q812yes—
175M16×C47—24no—
176M16×C47—22no—
177M16×C47C22×C414yes—
178C2×C8×C47—24no—
179C2×C8×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—42no—
183C4⋊C4×C47C2×D412yes—
184C4⋊C4×C47C2×D412yes—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—42no—
188C42×C47—22no—
189C42×C47—22no—
190C42×C47—22no—
191C42×C47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194(C2×D47)⋊C4—24no—
195(C2×D47)⋊C4—24no—
196(C2×D47)⋊C4—84no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—22no—
200C4⋊Dic47—22no—
201C4⋊Dic47—41no—
202C4⋊Dic47—41no—
203C4⋊Dic47C2×D412yes—
204C4⋊Dic47C2×D412yes—
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207C4×Dic47—24no—
208C4×Dic47—24no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211#10625329—24no—
212#10625329—22no—
213#10625329C22×C414yes—
214C2×C47⋊C8—24no—
215C2×C47⋊C8—22no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47C2×C4⋊C412yes—
219C23×C47#12011yescharacteristic
220C23×C47#12011yescharacteristic
221C23×C47#12312yes—
222C23×C47C22×D411yescharacteristic
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—42no—
226C22×D47—42no—
227C22×D47C2×C4⋊C414yes—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—81no—
234C2×C4×C47—81no—
235C2×C4×C47C2×C4⋊C412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47C4×D412yes—
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47#12011yescharacteristic
242C2×C4×C47C22⋊Q812yes—
243C2×C4×C47C22⋊Q812yes—
244C2×C4×C47C22⋊Q812yes—
245C2×C4×C47C22⋊Q812yes—
246C2×C4×C47#12312yes—
247C2×C4×C47C23×C411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C22×Q811yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C22⋊D47—28no—
254C22⋊D47—28no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—44no—
260C2×Dic47—42no—
261C2×Dic47—81no—
262C2×Dic47—81no—
263C2×Dic47—82no—
264C2×Dic47—82no—
265C2×Dic47C2×C4⋊C414yes—
266D188—22no—
267D188—22no—
268D188—22no—
269D188—22no—
270C4×D47—28no—
271C4×D47—28no—
272C4×D47—84no—
273C4×D47—84no—
274Dic94—22no—
275Dic94—22no—
276Dic94—22no—
277Dic94—22no—
278C8×C47—24no—
279C47⋊C8—24no—
280#20858—471nomaximal, Sylow
281C22×C47—21no—
282C22×C47—21no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—21no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47#38512yes—
290C22×C47#38512yes—
291C22×C47C22×C4⋊C411yescharacteristic
292C22×C47C2×#12011yescharacteristic
293C22×C47C2×#12311yescharacteristic
294C2×D47—24no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—44no—
298C2×D47—82no—
299C4×C47—42no—
300C4×C47—42no—
301C4×C47—82no—
302C4×C47#38512yes—
303C4×C47#38512yes—
304C4×C47C2×C4×D411yescharacteristic
305C4×C47C2×C4×D411yescharacteristic
306C4×C47C2×C22⋊Q811yescharacteristic
307C4×C47C2×C22⋊Q811yescharacteristic
308Dic47—24no—
309Dic47—42no—
310Dic47—42no—
311Dic47—82no—
312#3428—471no—
313C2×#428—471no—
314C2×#428—471no—
315C2×#413—471no—
316#2311—471no—
317#1857—474no—
318#1856—474no—
319#1733—471no—
320#1733D4711yescharacteristic, complemented
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—22no—
324C2×C47—22no—
325C2×C47C2×#38511yescharacteristic
326C2×C47#302111yescharacteristic
327C2×C47#302211yescharacteristic
328D47—44no—
329C22×#41—471no—
330C22×M16—471no—
331C22×M16C2×D4711yescharacteristic
332#520—941no—
333#520—941no—
334#520—941no—
335#520—941no—
336#519—472no—
337#518—472no—
338#518—942no—
339C4×#41—942no—
340C4×#41—942no—
341C2×#117—471no—
342C2×#117—942no—
343C2×#117C2×D4711yescharacteristic
344C2×#117C2×D4711yescharacteristic
345C22×C4⋊C4—471no—
346#428—474no—
347#428—474no—
348#428—942no—
349#428—942no—
350C2×#107—941no—
351C2×#107—941no—
352C2×#106—941no—
353C2×#106—941no—
354#413—478no—
355C2×#101—472no—
356C2×#100—472no—
357#391—472no—
358#390—472no—
359#386—942no—
360#380—942no—
361#339—474no—
362#339C2×D4714yes—
363C47#2085811yescharacteristic, complemented, Sylow
364C2×#41—474no—
365C2×#41—474no—
366C2×#41—941no—
367C2×#41—941no—
368C2×#41—942no—
369C2×#41—941no—
370C2×#41—941no—
371C2×#41—942no—
372C22×Q8—471no—
373C22×D4—472no—
374C22×D4—471no—
375C23×C4—472no—
376C23×C4—941no—
377C23×C4C22×D4711yescharacteristic
378C2×M16—21no—
379C2×M16—22no—
380C2×M16—474no—
381C2×M16—472no—
382C2×M16—941no—
383C2×M16—942no—
384C2×M16C4×D4714yescomplemented
385C2×M16C22×D4712yes—
386C22×C8—21no—
387C22×C8—941no—
388C4×Q8—942no—
389C4×Q8—942no—
390C4×Q8—942no—
391C4×Q8—942no—
392C4×D4—942no—
393C4×D4—942no—
394C4×D4—944no—
395C4×D4—944no—
396C4×D4—944no—
397C4×D4—944no—
398C4×D4—942no—
399C4×D4—942no—
400#117—21no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117—472no—
405#117—472no—
406#117—944no—
407#117—944no—
408#117—942no—
409#117—942no—
410#117—941no—
411#117—941no—
412#117D18812yes—
413#117C22×D4712yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—472no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4—942no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4D18811yescharacteristic
426C2×C4⋊C4D18811yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—941no—
431C2×C22⋊C4—944no—
432C2×C42—21no—
433C2×C42—944no—
434C2×C42—941no—
435C2×C42C22×D4712yes—
436#107—944no—
437#107—944no—
438#106—944no—
439#106—944no—
440#101—478no—
441#100—478no—
442#95—944no—
443C24—472no—
444C24D4×D4711yescharacteristic
445#41—944no—
446#41—944no—
447#41—944no—
448#41—944no—
449#41—944no—
450#41—944no—
451#41—944no—
452#41—944no—
453C2×Q8—472no—
454C2×Q8—472no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×Q8—941no—
458C2×Q8—941no—
459C2×D4—478no—
460C2×D4—472no—
461C2×D4—472no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—944no—
465C2×D4—941no—
466C2×D4—941no—
467C2×D4—941no—
468C2×D4—941no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—472no—
473C22×C4—478no—
474C22×C4—944no—
475C22×C4—944no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—944no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—942no—
482C22×C4—3762no—
483C22×C4—1881no—
484C22×C4—1881no—
485C22×C4C2×C4×D4712yes—
486C22×C4C2×D18811yescharacteristic
487C22×C4C2×D18811yescharacteristic
488C22×C4#1062535012yes—
489C22×C4D4×D4711yescharacteristic
490C22×C4D4×D4711yescharacteristic
491C22×C4D4×D4711yescharacteristic
492C22×C4#1062535212yes—
493C22×C4Q8×D4712yes—
494C22×C4C23×D4711yescharacteristic
495M16—24no—
496M16—22no—
497M16—474no—
498M16—944no—
499M16—942no—
500M16C2×C4×D4714yes—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—944no—
504C2×C8—942no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—472no—
509C4⋊C4—472no—
510C4⋊C4—944no—
511C4⋊C4—944no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C4⋊C4—1881no—
517C4⋊C4—1881no—
518C4⋊C4C2×D18812yes—
519C4⋊C4C2×D18812yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—42no—
523C22⋊C4—944no—
524C22⋊C4—944no—
525C22⋊C4—942no—
526C22⋊C4—942no—
527C22⋊C4—3764no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—944no—
533C42—944no—
534C42—942no—
535C42—942no—
536C23—41no—
537C23—41no—
538C23—474no—
539C23—942no—
540C23—942no—
541C23—1882no—
542C23—1882no—
543C23C4⋊C4×D4712yes—
544C23#1309399411yescharacteristic
545C23#1309399411yescharacteristic
546C23#1309400612yes—
547C23C2×D4×D4711yescharacteristic
548Q8—942no—
549Q8—942no—
550Q8—942no—
551Q8—942no—
552D4—948no—
553D4—948no—
554D4—942no—
555D4—942no—
556D4—942no—
557D4—942no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—474no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—948no—
569C2×C4—948no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—3764no—
573C2×C4—3764no—
574C2×C4—3762no—
575C2×C4—3762no—
576C2×C4—3761no—
577C2×C4—3761no—
578C2×C4—1884no—
579C2×C4—1882no—
580C2×C4C4⋊C4×D4712yes—
581C2×C4(C22⋊D47)⋊C412yes—
582C2×C4(C22⋊D47)⋊C412yes—
583C2×C4C4×D18812yes—
584C2×C4C4×D18812yes—
585C2×C4#1309399411yescharacteristic
586C2×C4#1309399411yescharacteristic
587C2×C4#1309400212yes—
588C2×C4#1309400212yes—
589C2×C4#1309400412yes—
590C2×C4#1309400412yes—
591C2×C4#1309400612yes—
592C2×C4C22×C4×D4711yescharacteristic
593C2×C4C2×D4×D4711yescharacteristic
594C2×C4C22×D18811yescharacteristic
595C2×C4C2×Q8×D4711yescharacteristic
596C2×C4C2×#1062535211yescharacteristic
597C2×C4C2×#1062535011yescharacteristic
598C8—24no—
599C8—944no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—944no—
609C22—3762no—
610C22—1884no—
611C22—1882no—
612C22—1882no—
613C22#1568899712yes—
614C22#1568899712yes—
615C22C2×C4⋊C4×D4711yescharacteristic
616C22C2×#1309399411yescharacteristic
617C22C2×#1309400611yescharacteristic
618C4—42no—
619C4—42no—
620C4—82no—
621C4—944no—
622C4—3762no—
623C4—1882no—
624C4—1882no—
625C4#1568899712yes—
626C4#1568899712yes—
627C4C2×(C22⋊D47)⋊C411yescharacteristic
628C4C2×C4×D18811yescharacteristic
629C4C2×#1309400211yescharacteristic
630C4C2×#1309400411yescharacteristic
631C2—21no—
632C2—21no—
633C2—22no—
634C2—22no—
635C2—1884no—
636C2C2×#1568899711yescharacteristic
637C2#2025276111yescharacteristic
638C2#2025276811yescharacteristic
subgroupG/H
centerC22C2×#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1733C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#1733D47
hypocenterC47#20858
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689165C2semidirect
6. #20243604C2semidirect
7. #20243599C2semidirect
8. #20243098C2semidirect
9. C47×#1733C2semidirect
32. #15689105C4semidirect
59. C2×#10625329C2×C4semidirect
101. #13093942C2×C4semidirect
102. #13093940C2×C4semidirect
107. C2×M16×C47C2×C4semidirect
320. #1733D47semidirect
363. C47#20858semidirect
384. 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×#156891662
C2×#156891652
C2×#156891052
#202459762
#202436042
#202435992
#202430982
C47×#17332
#2085847