Finite Group Databasebeta

← #26217169

Subgroups — #26217169

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#20256591C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic
4#20246011C211yesmaximal, characteristic, complemented
5#20243623C214yesmaximal, complemented
6#20243616C214yesmaximal, complemented
7#20243126C211yesmaximal, characteristic, complemented
8#20243098C211yesmaximal, characteristic, complemented
9C2×C47×#421C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11#15689606—21no—
12#15689606—21no—
13#15689606C2212yes—
14#15689606C2212yes—
15#15689601—21no—
16#15689598—22no—
17#15689598—21no—
18#41×Dic47—22no—
19#41×Dic47—22no—
20C2×#13093983—22no—
21C2×#13093983C2211yescharacteristic
22C2×#13093983C2211yescharacteristic
23C22×C4⋊Dic47C2211yescharacteristic
24#15689165—22no—
25#15689165C2214yescomplemented
26C2×#13093958—21no—
27C2×#13093956—21no—
28#15689124—22no—
29#15689124C2214yes—
30C2×#13093948—21no—
31C2×#13093945—21no—
32#15689022—22no—
33#15688998—22no—
34#15688993C2212yescomplemented
35#15688989C2212yescomplemented
36#15688891C2212yescomplemented
37#15688891C2212yescomplemented
38#15688883C2214yescomplemented
39C22×#41×C47C2211yescharacteristic
40C22×M16×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C47×#421C418yescomplemented
43C2×#104×C47C2214yes—
44C23×Dic47—21no—
45C2×#10625329—21no—
46C2×#10625329—22no—
47C2×#10625329—22no—
48C2×#10625329C2312yes—
49C22×C47⋊C8—21no—
50Q8×Dic47—22no—
51Q8×Dic47—22no—
52Q8×Dic47—22no—
53Q8×Dic47—22no—
54D4×Dic47—24no—
55D4×Dic47—24no—
56D4×Dic47—22no—
57D4×Dic47—22no—
58D4×Dic47—22no—
59D4×Dic47—22no—
60D4×Dic47—24no—
61D4×Dic47—24no—
62#13093983—24no—
63#13093983—24no—
64#13093983—21no—
65#13093983—21no—
66#13093983—21no—
67#13093983—21no—
68#13093983—21no—
69#13093983—21no—
70#13093983—22no—
71#13093983—22no—
72#13093983D412yes—
73#13093983C2312yes—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47—22no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47—24no—
82C2×C4⋊Dic47D411yescharacteristic
83C2×C4⋊Dic47D411yescharacteristic
84C2×C4⋊Dic47C2312yes—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C22⋊Dic47—24no—
88C2×C22⋊Dic47—21no—
89C2×C22⋊Dic47D412yes—
90C2×C4×Dic47—24no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47D412yes—
93#13093958—24no—
94#13093956—24no—
95#13093948—24no—
96#13093945—24no—
97C47⋊#95—24no—
98C2×#41×C47—21no—
99C2×#41×C47—21no—
100C2×#41×C47—22no—
101C2×#41×C47—22no—
102C2×#41×C47C2×C414yes—
103C2×#41×C47D412yes—
104C2×#41×C47Q814yes—
105C2×#41×C47C2312yes—
106C22×Q8×C47D411yescharacteristic
107C22×D4×C47—21no—
108C22×D4×C47D411yescharacteristic
109C23×C4×C47—21no—
110C23×C4×C47C2311yescharacteristic
111C2×M16×C47—21no—
112C2×M16×C47—22no—
113C2×M16×C47C2×C414yescomplemented
114C2×M16×C47C2312yes—
115C22×C8×C47—21no—
116#117×C47—22no—
117#117×C47C2×C414yescomplemented
118C2×C4⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120C2×C42×C47C2312yes—
121#104×C47C2×C4116yescomplemented
122C24×C47—21no—
123C24×C47C2×D411yescharacteristic
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—24no—
129#41×C47—22no—
130#41×C47—22no—
131#41×C47C4⋊C418yescomplemented
132#41×C47C22×C414yes—
133#41×C47C2×Q814yes—
134C2×Q8×C47—21no—
135C2×Q8×C47—22no—
136C2×Q8×C47C4⋊C414yescomplemented
137C2×Q8×C47C2×D412yes—
138C2×D4×C47—24no—
139C2×D4×C47—22no—
140C2×D4×C47—24no—
141C2×D4×C47—22no—
142C2×D4×C47—21no—
143C2×D4×C47—22no—
144C2×D4×C47C4⋊C414yescomplemented
145C2×D4×C47C2×D412yes—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—24no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47C22×C412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×Q812yes—
159C22×C4×C47#4112yes—
160C22×C4×C47#4112yes—
161C22×C4×C47C2411yescharacteristic
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—22no—
167C22×Dic47—41no—
168C22×Dic47—41no—
169C22×Dic47—82no—
170C22×Dic47C2×D412yes—
171C22×Dic47C2×D412yes—
172M16×C47—24no—
173M16×C47—22no—
174M16×C47C22×C414yes—
175C2×C8×C47—24no—
176C2×C8×C47—22no—
177C4⋊C4×C47—24no—
178C22⋊C4×C47—24no—
179C42×C47—22no—
180C42×C47C22×C414yes—
181C22⋊Dic47—24no—
182C22⋊Dic47—24no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—22no—
187C4⋊Dic47—24no—
188C4⋊Dic47—24no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—22no—
196C4⋊Dic47—22no—
197C4⋊Dic47—41no—
198C4⋊Dic47—41no—
199C4⋊Dic47—84no—
200C4⋊Dic47C2×D412yes—
201C4⋊Dic47C2×D412yes—
202C4×Dic47—24no—
203C4×Dic47—24no—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208#10625329—22no—
209#10625329—22no—
210#10625329—42no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—42no—
213C23×C47—22no—
214C23×C47—24no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—81no—
218C23×C47—81no—
219C23×C47C4×D412yes—
220C23×C47#12011yescharacteristic
221C23×C47#12011yescharacteristic
222C23×C47C22⋊Q812yes—
223C23×C47C22×D411yescharacteristic
224Q8×C47—22no—
225Q8×C47—24no—
226Q8×C47C2×C4⋊C414yes—
227D4×C47—24no—
228D4×C47—24no—
229D4×C47—28no—
230D4×C47—22no—
231D4×C47—24no—
232D4×C47C2×C4⋊C414yes—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—28no—
238C2×C4×C47—24no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47C2×C4⋊C412yes—
242C2×C4×C47C2×C4⋊C412yes—
243C2×C4×C47C4×D412yes—
244C2×C4×C47C4×D412yes—
245C2×C4×C47C4×D412yes—
246C2×C4×C47#12011yescharacteristic
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47C22⋊Q812yes—
249C2×C4×C47C22⋊Q812yes—
250C2×C4×C47C22⋊Q812yes—
251C2×C4×C47#12312yes—
252C2×C4×C47#12312yes—
253C2×C4×C47C23×C411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C22×Q811yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—44no—
266C2×Dic47—42no—
267C2×Dic47—81no—
268C2×Dic47—81no—
269C2×Dic47—84no—
270C2×Dic47—84no—
271C2×Dic47—82no—
272C2×Dic47—82no—
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—24no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—84no—
288C22×C47#38512yes—
289C22×C47#38512yes—
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×#12011yescharacteristic
292C22×C47C2×C22⋊Q811yescharacteristic
293C4×C47—24no—
294C4×C47—44no—
295C4×C47#38512yes—
296C4×C47#38512yes—
297C4×C47C22×C4⋊C411yescharacteristic
298C4×C47C2×C4×D411yescharacteristic
299C4×C47C2×C22⋊Q811yescharacteristic
300C4×C47C2×#12311yescharacteristic
301Dic47—42no—
302Dic47—42no—
303Dic47—42no—
304Dic47—42no—
305Dic47—82no—
306#3428—471no—
307C2×#428—471no—
308C2×#421D4711yescharacteristic
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×C47—44no—
320C2×C47C2×#38511yescharacteristic
321C2×C47#302111yescharacteristic
322C2×C47#302211yescharacteristic
323C22×#41C2×D4711yescharacteristic
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⋊C4—471no—
340#428—474no—
341#428—942no—
342C2×#107—941no—
343C2×#106—941no—
344#421Dic4718yescomplemented
345C2×#104C2×D4714yes—
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—21no—
359C2×#41—21no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41Dic9414yes—
363C2×#41D18812yes—
364C2×#41C2×Dic4714yes—
365C2×#41C22×D4712yes—
366C22×Q8D18811yescharacteristic
367C22×D4—21no—
368C22×D4D18811yescharacteristic
369C23×C4—21no—
370C23×C4—941no—
371C23×C4C22×D4711yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—472no—
375C2×M16—941no—
376C2×M16—942no—
377C2×M16—942no—
378C2×M16C2×Dic4714yescomplemented
379C2×M16C22×D4712yes—
380C22×C8—21no—
381C22×C8—941no—
382C4×Q8—942no—
383C4×Q8—942no—
384C4×Q8—942no—
385C4×Q8—942no—
386C4×D4—944no—
387C4×D4—944no—
388C4×D4—944no—
389C4×D4—944no—
390C4×D4—942no—
391C4×D4—942no—
392C4×D4—942no—
393C4×D4—942no—
394#117—22no—
395#117—472no—
396#117—472no—
397#117—942no—
398#117—942no—
399#117—944no—
400#117—944no—
401#117—941no—
402#117—941no—
403#117—941no—
404#117—941no—
405#117—941no—
406#117—941no—
407#117C2×Dic4714yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—472no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—941no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—472no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—941no—
426C2×C42—472no—
427C2×C42—944no—
428C2×C42—941no—
429C2×C42C22×D4712yes—
430#107—944no—
431#106—944no—
432#104C2×Dic47116yescomplemented
433#103—944no—
434#102—944no—
435#95—944no—
436C24—21no—
437C24D4×D4711yescharacteristic
438#41—24no—
439#41—24no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—22no—
444#41—22no—
445#41C4⋊Dic4718yescomplemented
446#41C2×Dic9414yes—
447#41C22×Dic4714yes—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8C4⋊Dic4714yescomplemented
451C2×Q8C2×D18812yes—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4C4⋊Dic4714yescomplemented
459C2×D4C2×D18812yes—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—942no—
473C22×C4—3762no—
474C22×C4—1881no—
475C22×C4—1881no—
476C22×C4C2×Dic9412yes—
477C22×C4C2×D18811yescharacteristic
478C22×C4C2×D18811yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4#1062535212yes—
483C22×C4#1062535212yes—
484C22×C4C22×Dic4712yes—
485C22×C4C23×D4711yescharacteristic
486M16—24no—
487M16—22no—
488M16—942no—
489M16—942no—
490M16—1882no—
491M16C22×Dic4714yes—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—942no—
495C2×C8—1882no—
496C4⋊C4—24no—
497C4⋊C4—472no—
498C4⋊C4—472no—
499C4⋊C4—944no—
500C4⋊C4—944no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—942no—
508C4⋊C4—942no—
509C4⋊C4—3764no—
510C4⋊C4—1881no—
511C4⋊C4—1881no—
512C22⋊C4—24no—
513C22⋊C4—944no—
514C22⋊C4—944no—
515C22⋊C4—942no—
516C22⋊C4—942no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C42—22no—
520C42—944no—
521C42—944no—
522C42—942no—
523C42—942no—
524C42—942no—
525C42—942no—
526C42C22×Dic4714yes—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23D4×Dic4712yes—
534C23#1309399411yescharacteristic
535C23#1309399411yescharacteristic
536C23#1309400112yes—
537C23C2×D4×D4711yescharacteristic
538Q8—24no—
539Q8—22no—
540Q8C2×C4⋊Dic4714yes—
541D4—28no—
542D4—24no—
543D4—24no—
544D4—24no—
545D4—22no—
546D4C2×C4⋊Dic4714yes—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—3762no—
560C2×C4—3762no—
561C2×C4—3761no—
562C2×C4—3761no—
563C2×C4—3764no—
564C2×C4—3764no—
565C2×C4—1884no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4C2×C4⋊Dic4712yes—
570C2×C4C2×C4⋊Dic4712yes—
571C2×C4D4×Dic4712yes—
572C2×C4D4×Dic4712yes—
573C2×C4D4×Dic4712yes—
574C2×C4#1309399411yescharacteristic
575C2×C4#1309399411yescharacteristic
576C2×C4#1309400112yes—
577C2×C4#1309400112yes—
578C2×C4#1309400112yes—
579C2×C4#1309400812yes—
580C2×C4#1309400812yes—
581C2×C4C23×Dic4711yescharacteristic
582C2×C4C2×D4×D4711yescharacteristic
583C2×C4C22×D18811yescharacteristic
584C2×C4C22×Dic9411yescharacteristic
585C2×C4C2×#1062535211yescharacteristic
586C2×C4C2×#1062535211yescharacteristic
587C8—24no—
588C8—1882no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1568899812yes—
602C22#1568899812yes—
603C22C2×D4×Dic4711yescharacteristic
604C22C2×#1309399411yescharacteristic
605C22C2×#1309400111yescharacteristic
606C4—24no—
607C4—44no—
608C4—3762no—
609C4—1882no—
610C4—1882no—
611C4—1882no—
612C4—1882no—
613C4#1568899812yes—
614C4#1568899812yes—
615C4C22×C4⋊Dic4711yescharacteristic
616C4C2×D4×Dic4711yescharacteristic
617C4C2×#1309400111yescharacteristic
618C4C2×#1309400811yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—44no—
624C2C2×#1568899811yescharacteristic
625C2#2025276111yescharacteristic
626C2#2025276811yescharacteristic
subgroupG/H
centerC22C2×#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#421C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#421D47
hypocenterC47#20865
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
4. #20246011C2semidirect
5. #20243623C2semidirect
6. #20243616C2semidirect
7. #20243126C2semidirect
8. #20243098C2semidirect
25. #15689165C22semidirect
34. #15688993C22semidirect
35. #15688989C22semidirect
36. #15688891C22semidirect
37. #15688891C22semidirect
38. #15688883C22semidirect
42. C47×#421C4semidirect
113. C2×M16×C47C2×C4semidirect
117. #117×C47C2×C4semidirect
121. #104×C47C2×C4semidirect
131. #41×C47C4⋊C4semidirect
136. C2×Q8×C47C4⋊C4semidirect
144. C2×D4×C47C4⋊C4semidirect
344. #421Dic47semidirect
357. C47#20865semidirect
378. C2×M16C2×Dic47semidirect
407. #117C2×Dic47semidirect
432. #104C2×Dic47semidirect
445. #41C4⋊Dic47semidirect
450. C2×Q8C4⋊Dic47semidirect
458. C2×D4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
#202565912
C2×#156891652
C2×#156891242
#202460112
#202436232
#202436162
#202431262
#202430982
C2×C47×#4212
#2086547