Finite Group Databasebeta

← #26243815

Subgroups — #26243815

Order 12032.

1482 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689701C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689089C211yesmaximal, characteristic, complemented
5#20245976C211yesmaximal, characteristic, complemented
6#20244589C212yesmaximal, complemented
7#20244582C212yesmaximal, complemented
8#20244575C212yesmaximal, complemented
9#20244568C212yesmaximal, complemented
10C2×#15688886C211yesmaximal, characteristic, complemented
11C2×C47×#416C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689701—24no—
15#15689701—24no—
16C2×#13094003—21no—
17C2×#13094000C2212yes—
18C2×#13093998—21no—
19C2×#13093996—22no—
20C2×#13093995C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689166—24no—
24C2×#13093957C2212yes—
25C2×#13093955C2212yes—
26#15689124—24no—
27C2×#13093948C2212yes—
28C2×#13093945C2212yes—
29#15689089—24no—
30C2×(C2×D47)⋊C8C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32#15689029—21no—
33#15689025—21no—
34#15689000—22no—
35#15688980C2212yes—
36#15688980C2212yes—
37#15688886C2212yes—
38#15688886C2212yes—
39C22×#41×C47C2211yescharacteristic
40C23×C8×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C47×#416—24no—
43C2×#103×C47C2211yescharacteristic
44C2×#103×C47C2211yescharacteristic
45C2×#102×C47C2211yescharacteristic
46C2×#102×C47C2211yescharacteristic
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350—22no—
53C22×Dic94D411yescharacteristic
54C22×D188D411yescharacteristic
55C22×C22⋊D47—22no—
56C22×C22⋊D47D412yes—
57C23×Dic47—21no—
58C23×Dic47—21no—
59C22×C4×D47C2312yes—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329D414yescomplemented
63C22×C47⋊C8C2312yes—
64#13094003—28no—
65#13094000—24no—
66#13094000—24no—
67#13093998—28no—
68#13093996—216no—
69#13093995—24no—
70#13093995—24no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47D414yes—
77C2×C4⋊Dic47C2312yes—
78C2×C4⋊Dic47C2312yes—
79C2×Dic47⋊C4—22no—
80C2×C22⋊Dic47—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—22no—
84#13093957—24no—
85#13093955—24no—
86#13093948—24no—
87#13093945—24no—
88(C2×D47)⋊C8—24no—
89(C2×D47)⋊C8—24no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47—22no—
96C2×#41×C47—22no—
97C2×#41×C47D414yes—
98C22×Q8×C47C2311yescharacteristic
99C22×D4×C47—21no—
100C22×D4×C47C2311yescharacteristic
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103C22×C8×C47—22no—
104C22×C8×C47—22no—
105C22×C8×C47—22no—
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108#117×C47—22no—
109#117×C47—22no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114#103×C47—22no—
115#103×C47—22no—
116#102×C47—22no—
117#102×C47—22no—
118C24×C47—21no—
119C24×C47C2×D411yescharacteristic
120C23×D47C2×D412yes—
121#41×C47—28no—
122#41×C47—28no—
123#41×C47—44no—
124#41×C47—44no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C2×D412yes—
131C2×D4×C47—24no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—42no—
138C2×D4×C47—42no—
139C2×D4×C47C2×D412yes—
140C22×C4×C47—21no—
141C22×C4×C47—24no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47#4112yes—
153C22×C4×C47C2411yescharacteristic
154C2×C22⋊D47—24no—
155C2×C22⋊D47—24no—
156C2×C22⋊D47—24no—
157C2×C22⋊D47—28no—
158C2×C22⋊D47—82no—
159C2×C22⋊D47—82no—
160C22×Dic47—22no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—42no—
164C22×Dic47—81no—
165C22×Dic47—81no—
166C22×Dic47—81no—
167C22×Dic47—81no—
168C22×Dic47C2×D412yes—
169#10625350—44no—
170#10625350—44no—
171#10625350—44no—
172#10625350—44no—
173C2×D188—22no—
174C2×D188—22no—
175C2×D188—22no—
176C2×C4×D47—22no—
177C2×C4×D47—22no—
178C2×C4×D47—22no—
179C2×C4×D47—22no—
180C2×C4×D47—22no—
181C2×C4×D47—22no—
182C2×Dic94—22no—
183C2×Dic94—22no—
184C2×Dic94—22no—
185C2×C8×C47—21no—
186C2×C8×C47—21no—
187C2×C8×C47—21no—
188C2×C8×C47—21no—
189C2×C8×C47—21no—
190C2×C8×C47—21no—
191C2×C8×C47—44no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47—21no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47—21no—
196C22⋊C4×C47—42no—
197C42×C47—42no—
198C22⋊Dic47—28no—
199(C2×D47)⋊C4—28no—
200(C2×D47)⋊C4—82no—
201(C2×D47)⋊C4—82no—
202C4⋊Dic47—24no—
203C4⋊Dic47—22no—
204C4⋊Dic47—22no—
205C4⋊Dic47—22no—
206C4⋊Dic47—42no—
207C4⋊Dic47C2×D414yes—
208Dic47⋊C4—28no—
209#10625329—28no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8C2×D414yes—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47—22no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47#12012yes—
225C23×C47#12112yes—
226C23×C47#12312yes—
227C23×C47C22×D411yescharacteristic
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—22no—
234C22×D47—22no—
235Q8×C47—22no—
236Q8×C47—22no—
237Q8×C47—41no—
238Q8×C47—41no—
239D4×C47—22no—
240D4×C47—22no—
241D4×C47—48no—
242D4×C47—41no—
243D4×C47—41no—
244C2×C4×C47—22no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—48no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47#12012yes—
255C2×C4×C47#12114yes—
256C2×C4×C47#12114yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12312yes—
259C2×C4×C47#12314yes—
260C2×C4×C47#12414yes—
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C22⋊D47—48no—
268C22⋊D47—88no—
269C2×Dic47—22no—
270C2×Dic47—24no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—84no—
279C2×Dic47—84no—
280D188—44no—
281C4×D47—42no—
282C4×D47—42no—
283C4×D47—42no—
284C4×D47—42no—
285Dic94—44no—
286C8×C47—41no—
287C8×C47—41no—
288C47⋊C8—24no—
289#23485—471nomaximal, Sylow
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—22no—
295C22×C47—22no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—81no—
300C22×C47—81no—
301C22×C47—81no—
302C22×C47—81no—
303C22×C47#39414yes—
304C22×C47C2×#12011yescharacteristic
305C22×C47C2×#12111yescharacteristic
306C22×C47C2×#12311yescharacteristic
307C22×C47#57611yescharacteristic
308C22×C47#57711yescharacteristic
309C22×C47#57711yescharacteristic
310C22×C47#57811yescharacteristic
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—42no—
314C2×D47—42no—
315C2×D47—42no—
316C2×D47—42no—
317C2×D47—42no—
318C4×C47—42no—
319C4×C47—81no—
320C4×C47—81no—
321C4×C47#39414yes—
322C4×C47C2×#12111yescharacteristic
323C4×C47C2×#12111yescharacteristic
324C4×C47C2×#12311yescharacteristic
325C4×C47C2×#12411yescharacteristic
326Dic47—42no—
327Dic47—44no—
328Dic47—82no—
329C2×#537—471no—
330C2×#428—471no—
331C2×#417—471no—
332C2×#416D4711yescharacteristic, complemented
333C2×#408—471no—
334#2311—471no—
335#2036—472no—
336#2035—472no—
337#2034—472no—
338#2033—472no—
339C2×#340—471no—
340C2×C47—24no—
341C2×C47—82no—
342C2×C47C2×#39411yescharacteristic
343C2×C47#301211yescharacteristic
344C2×C47#301311yescharacteristic
345C2×C47#306211yescharacteristic
346C2×C47#306311yescharacteristic
347C2×C47#309811yescharacteristic
348C2×C47#309911yescharacteristic
349D47—82no—
350C22×#41—471no—
351C22×#41C2×D4711yescharacteristic
352C22×M16—471no—
353C23×C8C2×D4711yescharacteristic
354#537—944no—
355#537—944no—
356C2×C22⋊Q8—472no—
357C2×C22⋊Q8—941no—
358C2×#121—472no—
359C2×#121—941no—
360C2×#121—942no—
361C2×#117C2×D4711yescharacteristic
362C22×C4⋊C4—471no—
363C22×C4⋊C4—471no—
364#428—944no—
365C2×#107—472no—
366C2×#106—472no—
367#417—944no—
368#416—24no—
369C2×#103—472no—
370C2×#103C2×D4711yescharacteristic
371C2×#103C2×D4711yescharacteristic
372C2×#102—472no—
373C2×#102C2×D4711yescharacteristic
374C2×#102C2×D4711yescharacteristic
375#408—944no—
376C2×C22⋊C8—472no—
377C2×C22⋊C8—472no—
378#391—941no—
379#390—941no—
380#386—942no—
381#380—472no—
382#380—472no—
383#340—472no—
384#340—472no—
385C47#2348511yescharacteristic, complemented, Sylow
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—942no—
392C2×#41—942no—
393C2×#41—942no—
394C2×#41—942no—
395C2×#41—942no—
396C2×#41—942no—
397C2×#41C22⋊D4714yes—
398C22×Q8—471no—
399C22×Q8C22×D4711yescharacteristic
400C22×D4—21no—
401C22×D4—472no—
402C22×D4—471no—
403C22×D4—942no—
404C22×D4C22×D4711yescharacteristic
405C23×C4—21no—
406C23×C4—472no—
407C23×C4—941no—
408C23×C4—941no—
409C23×C4C22×D4711yescharacteristic
410C2×M16—474no—
411C2×M16—942no—
412C2×M16—942no—
413C22×C8—22no—
414C22×C8—22no—
415C22×C8—22no—
416C22×C8—472no—
417C22×C8C22×D4711yescharacteristic
418C22×C8C22×D4711yescharacteristic
419C22⋊Q8—944no—
420C22⋊Q8—948no—
421C22⋊Q8—944no—
422#121—944no—
423#121—948no—
424#121—9416no—
425#121—944no—
426#117—22no—
427#117—22no—
428C2×C4⋊C4—474no—
429C2×C4⋊C4—472no—
430C2×C4⋊C4—472no—
431C2×C4⋊C4—942no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—942no—
434C2×C4⋊C4—942no—
435C2×C4⋊C4—942no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C4⋊C4C22×D4711yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—942no—
441C2×C22⋊C4—942no—
442C2×C22⋊C4—942no—
443C2×C22⋊C4—942no—
444C2×C42—21no—
445#107—944no—
446#106—944no—
447#103—22no—
448#103—22no—
449#103—944no—
450#102—22no—
451#102—22no—
452#102—944no—
453C22⋊C8—944no—
454C22⋊C8—944no—
455#95—942no—
456#95—942no—
457C24—21no—
458C24—472no—
459C24D4×D4711yescharacteristic
460#41—28no—
461#41—28no—
462#41—44no—
463#41—44no—
464#41—1884no—
465#41—1884no—
466#41—1884no—
467#41—1884no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—942no—
474C2×Q8—942no—
475C2×Q8—942no—
476C2×Q8C2×C22⋊D4712yes—
477C2×D4—24no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—948no—
486C2×D4—944no—
487C2×D4—944no—
488C2×D4—944no—
489C2×D4—942no—
490C2×D4—942no—
491C2×D4—942no—
492C2×D4—3762no—
493C2×D4—3762no—
494C2×D4C2×C22⋊D4712yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—472no—
502C22×C4—944no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—3761no—
512C22×C4—3761no—
513C22×C4—3761no—
514C22×C4—3761no—
515C22×C4—1882no—
516C22×C4#1062535012yes—
517C22×C4#1062535012yes—
518C22×C4D4×D4711yescharacteristic
519C22×C4D4×D4712yes—
520C22×C4D4×D4712yes—
521C22×C4#1062535212yes—
522C22×C4C2×C22⋊D4712yes—
523C22×C4C23×D4711yescharacteristic
524M16—948no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—44no—
532C2×C8—474no—
533C2×C8—942no—
534C2×C8—942no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—474no—
540C4⋊C4—948no—
541C4⋊C4—944no—
542C4⋊C4—942no—
543C4⋊C4—942no—
544C4⋊C4—942no—
545C4⋊C4—1882no—
546C22⋊C4—42no—
547C22⋊C4—948no—
548C22⋊C4—948no—
549C22⋊C4—3762no—
550C22⋊C4—3762no—
551C42—42no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—942no—
564C23—942no—
565C23—942no—
566C23—942no—
567C23—942no—
568C23—942no—
569C23—942no—
570C23#1309399212yes—
571C23#1309399712yes—
572C23#1309400612yes—
573C23C2×D4×D4711yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—1884no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—41no—
583D4—41no—
584D4—3768no—
585D4—1888no—
586D4—1884no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—944no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—3764no—
602C2×C4—3764no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4#1309399212yes—
613C2×C4#1309399614yes—
614C2×C4#1309399712yes—
615C2×C4#1309399714yes—
616C2×C4#1309400612yes—
617C2×C4#1309400714yes—
618C2×C4#1309401014yes—
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4C2×D4×D4711yescharacteristic
621C2×C4C22×C22⋊D4711yescharacteristic
622C2×C4C2×#1062535211yescharacteristic
623C2×C4C2×#1062535011yescharacteristic
624C2×C4C2×#1062535011yescharacteristic
625C8—41no—
626C8—41no—
627C8—944no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—1882no—
642C22—1882no—
643C22—1882no—
644C22—1882no—
645C22—1882no—
646C22—1882no—
647C22—1882no—
648C22#1568903814yes—
649C22C2×#1309399211yescharacteristic
650C22C2×#1309399711yescharacteristic
651C22C2×#1309400611yescharacteristic
652C22#1568992511yescharacteristic
653C22#1568993011yescharacteristic
654C22#1568993011yescharacteristic
655C22#1568993711yescharacteristic
656C4—42no—
657C4—81no—
658C4—81no—
659C4—3762no—
660C4—1884no—
661C4—1882no—
662C4#1568903814yes—
663C4C2×#1309399611yescharacteristic
664C4C2×#1309399711yescharacteristic
665C4C2×#1309400711yescharacteristic
666C4C2×#1309401011yescharacteristic
667C2—24no—
668C2—82no—
669C2—3762no—
670C2C2×#1568903811yescharacteristic
671C2#2025268811yescharacteristic
672C2#2025269911yescharacteristic
673C2#2025307111yescharacteristic
674C2#2025308211yescharacteristic
675C2#2025336411yescharacteristic
676C2#2025337511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#416C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
hypercenterC2×#416D47
hypocenterC47#23485
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689089C2semidirect
5. #20245976C2semidirect
6. #20244589C2semidirect
7. #20244582C2semidirect
8. #20244575C2semidirect
9. #20244568C2semidirect
10. C2×#15688886C2semidirect
11. C2×C47×#416C2semidirect
62. C2×#10625329D4semidirect
332. C2×#416D47semidirect
385. C47#23485semidirect

Maximal subgroups

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

subgroupindex
C2×#156897012
C2×#156891662
C2×#156891242
C2×#156890892
#202459762
#202445892
#202445822
#202445752
#202445682
C2×#156888862
C2×C47×#4162
#2348547