Finite Group Databasebeta

← #26423091

Subgroups — #26423091

Order 12032.

2326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic, complemented
2C22×#13094001C211yesmaximal, characteristic
3#20251673C212yesmaximal
4#20249859C214yesmaximal, complemented
5#20249855C214yesmaximal, complemented
6#20249704C214yesmaximal, complemented
7#20249700C214yesmaximal, complemented
8#20249207C212yesmaximal, complemented
9#20248543C212yesmaximal
10#20247677C211yesmaximal, characteristic, complemented
11#20247671C211yesmaximal, characteristic, complemented
12#20247662C211yesmaximal, characteristic
13#20247030C212yesmaximal, complemented
14#20245989C211yesmaximal, characteristic, complemented
15C47×#2482C211yesmaximal, characteristic
16C23×Dic94C2211yescharacteristic
17C24×Dic47C2211yescharacteristic
18#15689751—28no—
19C2×#13094024C2212yescomplemented
20C2×#13094023C2212yescomplemented
21C2×#13094023C2214yescomplemented
22C2×#13094016C2212yescomplemented
23C2×#13094015C2212yescomplemented
24C2×#13094001—28no—
25C2×#13094001C2214yes—
26C2×#13094001C2212yes—
27C2×#13094001C2212yes—
28C2×#13094001C2214yes—
29C2×#13094001C2214yes—
30C2×#13093983C2211yescharacteristic
31C22×C4⋊Dic47C2211yescharacteristic
32C22×Dic47⋊C4C2211yescharacteristic
33C22×Dic47⋊C4C2211yescharacteristic
34C22×C22⋊Dic47C2211yescharacteristic
35#15689064C2214yescomplemented
36#15689063C2212yescomplemented
37#15689061C2212yescomplemented
38#15689059C2212yes—
39#15689057C2214yescomplemented
40#15689055C2212yescomplemented
41#15689054C2212yescomplemented
42#15689054C2214yescomplemented
43#15689051C2214yes—
44#15689048C2212yes—
45#15689044C2214yescomplemented
46#15689043C2214yescomplemented
47#15689043C2214yescomplemented
48#15689035C2214yes—
49#15689027C2214yescomplemented
50#15689016C2214yes—
51#15689011C2214yes—
52#15689011C2214yes—
53#15689008C2214yescomplemented
54#15689008C2214yescomplemented
55#15689007C2214yescomplemented
56#15689006C2214yescomplemented
57#15688998C2214yes—
58#15688998C2212yes—
59#15688996C2212yes—
60#15688995C2214yes—
61#15688995C2212yes—
62#15688986C2211yescharacteristic
63#15688984C2212yes—
64#15688984C2212yes—
65#15688981C2214yes—
66#15688981C2212yes—
67#15688979C2212yes—
68C22⋊C4×Dic47C2212yes—
69C2×#117×C47C2211yescharacteristic
70C22×C22⋊C4×C47C2211yescharacteristic
71C47×#399C2212yes—
72C47×#397C2212yes—
73C47×#388C2214yes—
74C47×#385C2212yes—
75C47×#381C2211yescharacteristic
76C47×#379C2212yes—
77C22×Dic94—24no—
78C22×Dic94—44no—
79C22×Dic94C2314yes—
80C22×Dic94C2312yes—
81C22×Dic94C2312yes—
82C23×Dic47—24no—
83C23×Dic47—44no—
84C23×Dic47C2311yescharacteristic
85C23×Dic47C2311yescharacteristic
86C23×Dic47C2314yes—
87C23×Dic47C2311yescharacteristic
88C23×Dic47C2311yescharacteristic
89#13094024—28no—
90#13094023—28no—
91#13094023—48no—
92#13094016—28no—
93#13094015—28no—
94#13094001—28no—
95#13094001—28no—
96#13094001—216no—
97#13094001—216no—
98#13094001—48no—
99#13093983—24no—
100C2×C4⋊Dic47D418yes—
101C2×C4⋊Dic47C2314yes—
102C2×C4⋊Dic47C2312yes—
103C2×C4⋊Dic47C2312yes—
104C2×C4⋊Dic47C2311yescharacteristic
105C2×C4⋊Dic47C2311yescharacteristic
106C2×C4⋊Dic47C2312yes—
107C2×C4⋊Dic47C2312yes—
108C2×Dic47⋊C4—24no—
109C2×Dic47⋊C4—24no—
110C2×Dic47⋊C4C2312yes—
111C2×Dic47⋊C4C2314yes—
112C2×Dic47⋊C4C2312yes—
113C2×Dic47⋊C4C2312yes—
114C2×Dic47⋊C4C2312yes—
115C2×Dic47⋊C4C2312yes—
116C2×Dic47⋊C4C2312yes—
117C2×Dic47⋊C4C2314yes—
118C2×Dic47⋊C4C2314yes—
119C2×Dic47⋊C4C2314yes—
120C2×C22⋊Dic47—24no—
121C2×C22⋊Dic47C2312yes—
122C2×C22⋊Dic47C2314yes—
123C2×C22⋊Dic47C2312yes—
124C2×C22⋊Dic47C2314yes—
125C2×C22⋊Dic47C2314yes—
126C2×C4×Dic47C2312yes—
127C2×C4×Dic47C2314yes—
128C47⋊#95C2314yes—
129C47⋊#95C2314yes—
130C47⋊#95C2314yes—
131C47⋊#95C2314yes—
132C47⋊#95C2314yes—
133C47⋊#95C2314yes—
134C47⋊#95C2314yes—
135C47⋊#95C2314yes—
136C47⋊#95C2312yes—
137C47⋊#95C2312yes—
138C47⋊#95C2314yes—
139C25×C47C2311yescharacteristic
140C23×C4×C47C2311yescharacteristic
141C23×C4×C47C2311yescharacteristic
142#117×C47—24no—
143C2×C4⋊C4×C47C2311yescharacteristic
144C2×C4⋊C4×C47C2311yescharacteristic
145C2×C4⋊C4×C47C2314yes—
146C2×C22⋊C4×C47—24no—
147C2×C22⋊C4×C47C2312yes—
148C2×C22⋊C4×C47C2314yes—
149C2×C22⋊C4×C47C2312yes—
150C2×C22⋊C4×C47C2312yes—
151C2×C22⋊C4×C47C2312yes—
152C2×C22⋊C4×C47C2314yes—
153C2×C42×C47C2312yes—
154#95×C47C2312yes—
155#95×C47C2312yes—
156#95×C47C2314yes—
157C24×C47—22no—
158C24×C47—24no—
159C24×C47—24no—
160C24×C47—42no—
161C24×C47C2412yes—
162C24×C47C2411yescharacteristic
163C22×C4×C47—24no—
164C22×C4×C47—22no—
165C22×C4×C47—42no—
166C22×C4×C47C2×D414yes—
167C22×C4×C47C2412yes—
168C22×C4×C47C2412yes—
169C22×C4×C47C2414yes—
170C22×C4×C47C2412yes—
171C22×C4×C47C2411yescharacteristic
172C22×C4×C47C2411yescharacteristic
173C22×Dic47—22no—
174C22×Dic47—24no—
175C22×Dic47—22no—
176C22×Dic47—24no—
177C22×Dic47—28no—
178C22×Dic47—28no—
179C22×Dic47—42no—
180C22×Dic47—44no—
181C22×Dic47—42no—
182C22×Dic47—416no—
183C22×Dic47C2×D414yes—
184C22×Dic47C2×D414yes—
185C22×Dic47C2414yes—
186C22×Dic47C2412yes—
187C22×Dic47C2412yes—
188C22×Dic47C2412yes—
189C22×Dic47C2412yes—
190C22×Dic47C2412yes—
191C22×Dic47C2412yes—
192C2×Dic94—28no—
193C2×Dic94—28no—
194C2×Dic94—24no—
195C2×Dic94—24no—
196C2×Dic94—44no—
197C2×Dic94—416no—
198C2×Dic94—44no—
199C2×Dic94—44no—
200C4⋊C4×C47—22no—
201C4⋊C4×C47—22no—
202C4⋊C4×C47—44no—
203C22⋊C4×C47—24no—
204C22⋊C4×C47—28no—
205C22⋊C4×C47—24no—
206C22⋊C4×C47—24no—
207C22⋊C4×C47—24no—
208C22⋊C4×C47—44no—
209C42×C47—24no—
210C22⋊Dic47—24no—
211C22⋊Dic47—28no—
212C22⋊Dic47—24no—
213C22⋊Dic47—28no—
214C22⋊Dic47—44no—
215C4⋊Dic47—22no—
216C4⋊Dic47—22no—
217C4⋊Dic47—24no—
218C4⋊Dic47—24no—
219C4⋊Dic47—28no—
220C4⋊Dic47C2×D418yes—
221C4⋊Dic47C2×D418yes—
222Dic47⋊C4—24no—
223Dic47⋊C4—24no—
224Dic47⋊C4—28no—
225Dic47⋊C4—24no—
226Dic47⋊C4—24no—
227Dic47⋊C4—24no—
228Dic47⋊C4—24no—
229Dic47⋊C4—28no—
230Dic47⋊C4—44no—
231Dic47⋊C4—44no—
232C4×Dic47—24no—
233C4×Dic47—44no—
234C23×C47—22no—
235C23×C47—22no—
236C23×C47—28no—
237C23×C47—22no—
238C23×C47—24no—
239C23×C47—22no—
240C23×C47—24no—
241C23×C47—416no—
242C23×C47—44no—
243C23×C47—44no—
244C23×C47C22×D412yes—
245C23×C472−1+414yes—
246C23×C472−1+414yes—
247C23×C47C2511yescharacteristic
248C2×C4×C47—28no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—22no—
254C2×C4×C47—24no—
255C2×C4×C47—24no—
256C2×C4×C47—24no—
257C2×C4×C47—24no—
258C2×C4×C47—24no—
259C2×C4×C47—24no—
260C2×C4×C47—44no—
261C2×C4×C47C22×D412yes—
262C2×C4×C47C22×D412yes—
263C2×Dic47—28no—
264C2×Dic47—22no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—24no—
275C2×Dic47—24no—
276C2×Dic47—24no—
277C2×Dic47—44no—
278C2×Dic47—44no—
279C2×Dic47—416no—
280C2×Dic47—44no—
281C2×Dic47C22×D414yes—
282C2×Dic47C22×D414yes—
283C2×Dic47H(5,𝔽2)14yes—
284Dic94—416no—
285Dic94—84no—
286Dic94—84no—
287#36708—471nomaximal, Sylow
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—28no—
291C22×C47—24no—
292C22×C47—28no—
293C22×C47—416no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—44no—
297C22×C47#53614yes—
298C22×C47C23×D411yescharacteristic
299C22×C47C2×H(5,𝔽2)12yes—
300C22×C47C2×2−1+412yes—
301C22×C47C2×2−1+412yes—
302C4×C47—22no—
303C4×C47—22no—
304C4×C47—42no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47—44no—
308Dic47—24no—
309Dic47—24no—
310Dic47—24no—
311Dic47—42no—
312Dic47—42no—
313Dic47—42no—
314Dic47—44no—
315C2×#544—471no—
316C22×C22⋊Q8—471no—
317#2848—472no—
318#2690—474no—
319#2690—474no—
320#2677—474no—
321#2677—474no—
322#2627—472no—
323#2561—472no—
324#2484—471no—
325#2484—471no—
326#2482—471no—
327#2482D4711yescharacteristic
328#2434—472no—
329#2313—471no—
330C2×C47—24no—
331C2×C47—44no—
332C2×C47C2×#53611yescharacteristic
333C2×C47#348411yescharacteristic
334C2×C47#348611yescharacteristic
335C2×C47#353814yes—
336C23×Q8—471no—
337C24×C4—471no—
338#544—948no—
339C2×C4⋊Q8—472no—
340C2×C4⋊Q8—474no—
341C2×C4⋊Q8—472no—
342C2×#125—472no—
343C2×#125—472no—
344C2×C22⋊Q8—472no—
345C2×C22⋊Q8—472no—
346C2×C22⋊Q8—474no—
347C2×C22⋊Q8—474no—
348C2×C22⋊Q8—474no—
349C2×C22⋊Q8—948no—
350C2×#117—471no—
351C2×#117C2×D4711yescharacteristic
352C22×C4⋊C4—471no—
353C22×C4⋊C4—471no—
354C22×C4⋊C4—471no—
355C22×C22⋊C4—471no—
356C22×C22⋊C4C2×D4711yescharacteristic
357#400—472no—
358#400—474no—
359#400—472no—
360#399—472no—
361#399C2×D4712yes—
362#398—474no—
363#398—472no—
364#398—474no—
365#398—472no—
366#397—474no—
367#397—472no—
368#397C2×D4712yes—
369#395—474no—
370#395—474no—
371#395—474no—
372#393—474no—
373#391—474no—
374#388—474no—
375#388—474no—
376#388—474no—
377#388C2×D4714yes—
378#387—474no—
379#387—474no—
380#387—474no—
381#387—474no—
382#385—472no—
383#385—474no—
384#385—472no—
385#385—474no—
386#385—472no—
387#385C2×D4712yes—
388#381—471no—
389#381—472no—
390#381—472no—
391#381C2×D4711yescharacteristic
392#380—472no—
393#380—474no—
394#379—472no—
395#379C2×D4712yes—
396C4×C22⋊C4—472no—
397C47#3670811yescharacteristic, complemented, Sylow
398C25C22×D4711yescharacteristic
399C22×Q8—474no—
400C22×Q8—472no—
401C22×Q8—472no—
402C22×Q8—944no—
403C22×Q8—1884no—
404C23×C4—471no—
405C23×C4—471no—
406C23×C4—471no—
407C23×C4—471no—
408C23×C4—474no—
409C23×C4—944no—
410C23×C4—1884no—
411C23×C4C22×D4711yescharacteristic
412C23×C4C22×D4711yescharacteristic
413C4⋊Q8—948no—
414C4⋊Q8—948no—
415C4⋊Q8—1888no—
416#125—948no—
417#125—948no—
418C22⋊Q8—9416no—
419C22⋊Q8—9416no—
420C22⋊Q8—948no—
421C22⋊Q8—948no—
422C22⋊Q8—1888no—
423#117—24no—
424#117—944no—
425C2×C4⋊C4—478no—
426C2×C4⋊C4—472no—
427C2×C4⋊C4—472no—
428C2×C4⋊C4—472no—
429C2×C4⋊C4—472no—
430C2×C4⋊C4—474no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—472no—
434C2×C4⋊C4—472no—
435C2×C4⋊C4—472no—
436C2×C4⋊C4—472no—
437C2×C4⋊C4—472no—
438C2×C4⋊C4—472no—
439C2×C4⋊C4—474no—
440C2×C4⋊C4—474no—
441C2×C4⋊C4—474no—
442C2×C4⋊C4—474no—
443C2×C4⋊C4—944no—
444C2×C4⋊C4—944no—
445C2×C4⋊C4C22×D4711yescharacteristic
446C2×C4⋊C4C22×D4711yescharacteristic
447C2×C4⋊C4C22×D4714yes—
448C2×C22⋊C4—24no—
449C2×C22⋊C4—474no—
450C2×C22⋊C4—472no—
451C2×C22⋊C4—472no—
452C2×C22⋊C4—474no—
453C2×C22⋊C4—474no—
454C2×C22⋊C4—944no—
455C2×C22⋊C4C22×D4712yes—
456C2×C22⋊C4C22×D4712yes—
457C2×C22⋊C4C22×D4712yes—
458C2×C22⋊C4C22×D4712yes—
459C2×C22⋊C4C22×D4714yes—
460C2×C22⋊C4C22×D4714yes—
461C2×C42—472no—
462C2×C42—474no—
463C2×C42C22×D4712yes—
464#95—474no—
465#95—474no—
466#95—474no—
467#95—474no—
468#95—472no—
469#95—472no—
470#95—474no—
471#95—474no—
472#95—474no—
473#95—474no—
474#95—474no—
475#95C22×D4712yes—
476#95C22×D4712yes—
477#95C22×D4714yes—
478C24—22no—
479C24—24no—
480C24—24no—
481C24—42no—
482C24C23×D4712yes—
483C24C23×D4711yescharacteristic
484C2×Q8—944no—
485C2×Q8—944no—
486C2×Q8—948no—
487C2×Q8—948no—
488C2×Q8—1884no—
489C2×Q8—1884no—
490C2×Q8—1884no—
491C2×Q8—18816no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—472no—
496C22×C4—472no—
497C22×C4—472no—
498C22×C4—474no—
499C22×C4—474no—
500C22×C4—474no—
501C22×C4—472no—
502C22×C4—472no—
503C22×C4—472no—
504C22×C4—944no—
505C22×C4—948no—
506C22×C4—948no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—944no—
510C22×C4—1882no—
511C22×C4—1884no—
512C22×C4—1882no—
513C22×C4—18816no—
514C22×C4D4×D4714yes—
515C22×C4C23×D4714yes—
516C22×C4C23×D4712yes—
517C22×C4C23×D4712yes—
518C22×C4C23×D4712yes—
519C22×C4C23×D4711yescharacteristic
520C22×C4C23×D4711yescharacteristic
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—478no—
525C4⋊C4—478no—
526C4⋊C4—948no—
527C4⋊C4—944no—
528C4⋊C4—944no—
529C4⋊C4—944no—
530C4⋊C4—944no—
531C4⋊C4—948no—
532C4⋊C4—944no—
533C4⋊C4—944no—
534C4⋊C4—948no—
535C4⋊C4—944no—
536C4⋊C4—944no—
537C4⋊C4—942no—
538C4⋊C4—942no—
539C4⋊C4—1884no—
540C4⋊C4—1884no—
541C22⋊C4—24no—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—948no—
548C22⋊C4—944no—
549C22⋊C4—948no—
550C22⋊C4—944no—
551C22⋊C4—1884no—
552C42—24no—
553C42—944no—
554C42—1884no—
555C23—22no—
556C23—22no—
557C23—28no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—24no—
562C23—416no—
563C23—44no—
564C23—44no—
565C23C2×D4×D4712yes—
566C23#1309407114yes—
567C23#1309407114yes—
568C23C24×D4711yescharacteristic
569Q8—3764no—
570Q8—3764no—
571Q8—18816no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—474no—
586C2×C4—474no—
587C2×C4—474no—
588C2×C4—948no—
589C2×C4—942no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—1884no—
603C2×C4—1884no—
604C2×C4—18816no—
605C2×C4—1884no—
606C2×C4C2×D4×D4712yes—
607C2×C4C2×D4×D4712yes—
608C22—22no—
609C22—28no—
610C22—22no—
611C22—24no—
612C22—28no—
613C22—416no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22#1568969514yes—
618C22C22×D4×D4711yescharacteristic
619C22H(5,𝔽2)×D4712yes—
620C22C2×#1309407112yes—
621C22C2×#1309407112yes—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—944no—
629C4—944no—
630C4—944no—
631C4—1882no—
632C4—1884no—
633C4—1882no—
634C4—1882no—
635C2—24no—
636C2—44no—
637C2C2×#1568969511yescharacteristic
638C2#2025702211yescharacteristic
639C2#2025703511yescharacteristic
640C2#2025762814yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2482C2
prime radicalC25×C47C23
hypercenter#2482D47
hypocenterC47#36708
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. C2×#15689751C2semidirect
4. #20249859C2semidirect
5. #20249855C2semidirect
6. #20249704C2semidirect
7. #20249700C2semidirect
8. #20249207C2semidirect
10. #20247677C2semidirect
11. #20247671C2semidirect
13. #20247030C2semidirect
14. #20245989C2semidirect
19. C2×#13094024C22semidirect
20. C2×#13094023C22semidirect
21. C2×#13094023C22semidirect
22. C2×#13094016C22semidirect
23. C2×#13094015C22semidirect
35. #15689064C22semidirect
36. #15689063C22semidirect
37. #15689061C22semidirect
39. #15689057C22semidirect
40. #15689055C22semidirect
41. #15689054C22semidirect
42. #15689054C22semidirect
45. #15689044C22semidirect
46. #15689043C22semidirect
47. #15689043C22semidirect
49. #15689027C22semidirect
53. #15689008C22semidirect
54. #15689008C22semidirect
55. #15689007C22semidirect
56. #15689006C22semidirect
397. C47#36708semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
C22×#130940012
#202516732
#202498592
#202498552
#202497042
#202497002
#202492072
#202485432
#202476772
#202476712
#202476622
#202470302
#202459892
C47×#24822
#3670847