Finite Group Databasebeta

← #26243143

Subgroups — #26243143

Order 12032.

2014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094054C211yesmaximal, characteristic, complemented
2C2×#15689694C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×#13093947C211yesmaximal, characteristic, complemented
5C2×#15689090C211yesmaximal, characteristic, complemented
6#20244396C214yesmaximal, complemented
7#20244391C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C2×C47×#537C211yesmaximal, characteristic
10C23×Dic94C2211yescharacteristic
11C2×#13094054—24no—
12C2×#13094054—24no—
13C2×#13094054—24no—
14C22×#10625337C2212yes—
15C22×#10625335C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18#15689694—24no—
19#15689694—24no—
20C2×#13094022—21no—
21C2×#13094011—21no—
22C2×#13094005—22no—
23C2×#13094003C2214yes—
24C2×#13093983C2211yescharacteristic
25#15689119—24no—
26C2×#13093948C2212yes—
27C2×#13093947—24no—
28C2×#13093947C2214yes—
29C2×#13093945C2212yes—
30#15689090—24no—
31C2×C22⋊(C47⋊C8)C2214yes—
32#15688885C2214yes—
33C22×#41×C47C2211yescharacteristic
34C22×#41×C47C2211yescharacteristic
35C47×#537—24no—
36C47×#537—24no—
37C2×C22⋊Q8×C47—21no—
38C2×C22⋊Q8×C47C2212yes—
39C2×#121×C47—21no—
40C2×#121×C47—22no—
41C2×#121×C47C2212yes—
42C22×C4⋊C4×C47C2211yescharacteristic
43C22×Dic94—22no—
44C22×Dic94—22no—
45C22×Dic94—22no—
46C22×Dic94—22no—
47C22×Dic94—21no—
48C22×Dic94—22no—
49C22×Dic94D414yescomplemented
50C22×Dic94C2312yes—
51C23×Dic47—21no—
52#13094054—232no—
53C2×#10625337—24no—
54C2×#10625337—24no—
55C2×#10625337—24no—
56C2×#10625335—24no—
57C2×#10625335—24no—
58C2×#10625335—24no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329D414yescomplemented
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8C2312yes—
66C22×C47⋊C8C2312yes—
67#13094022—28no—
68#13094011—28no—
69#13094005—216no—
70#13094003—28no—
71#13094003—28no—
72#13093983—22no—
73#13093983D414yescomplemented
74C2×C4⋊Dic47C2312yes—
75C2×Dic47⋊C4—24no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—24no—
78C2×C4×Dic47—21no—
79#13093948—24no—
80#13093947—28no—
81#13093945—24no—
82C22⋊(C47⋊C8)—28no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47D414yes—
95C22×Q8×C47D411yescharacteristic
96C22×Q8×C47C2311yescharacteristic
97C22×D4×C47—21no—
98C22×D4×C47—22no—
99C22×D4×C47D412yes—
100C22×D4×C47D411yescharacteristic
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C23×C4×C47—21no—
104C23×C4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106C22⋊Q8×C47—28no—
107C22⋊Q8×C47—24no—
108C22⋊Q8×C47—24no—
109#121×C47—216no—
110#121×C47—28no—
111#121×C47—24no—
112#121×C47—24no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47—22no—
116C2×C4⋊C4×C47D414yes—
117C2×C4⋊C4×C47C2312yes—
118C2×C22⋊C4×C47—22no—
119C2×C22⋊C4×C47—22no—
120C24×C47—21no—
121C24×C47C2×D412yes—
122C24×C47C2×D411yescharacteristic
123#41×C47—28no—
124#41×C47—28no—
125#41×C47—44no—
126#41×C47—44no—
127#41×C47—44no—
128#41×C47—44no—
129#41×C47—44no—
130#41×C47—44no—
131C2×Q8×C47—22no—
132C2×Q8×C47—22no—
133C2×Q8×C47—22no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47C2×D412yes—
140C2×D4×C47—24no—
141C2×D4×C47—24no—
142C2×D4×C47—24no—
143C2×D4×C47—24no—
144C2×D4×C47—28no—
145C2×D4×C47—22no—
146C2×D4×C47—22no—
147C2×D4×C47—22no—
148C2×D4×C47—21no—
149C2×D4×C47—21no—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—21no—
153C2×D4×C47—42no—
154C2×D4×C47—42no—
155C2×D4×C47—84no—
156C2×D4×C47C2×D412yes—
157C22×C4×C47—22no—
158C22×C4×C47—21no—
159C22×C4×C47—24no—
160C22×C4×C47—24no—
161C22×C4×C47—22no—
162C22×C4×C47—22no—
163C22×C4×C47—22no—
164C22×C4×C47—22no—
165C22×C4×C47—22no—
166C22×C4×C47—22no—
167C22×C4×C47—42no—
168C22×C4×C47—42no—
169C22×C4×C47—82no—
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2×D411yescharacteristic
172C22×C4×C47C2×D412yes—
173C22×C4×C47C2×D412yes—
174C22×C4×C47C2×D412yes—
175C22×C4×C47C2×D412yes—
176C22×C4×C47C2×D412yes—
177C22×C4×C47#4112yes—
178C22×C4×C47C2411yescharacteristic
179C22×Dic47—24no—
180C22×Dic47—22no—
181C22×Dic47—22no—
182C22×Dic47—42no—
183C22×Dic47—42no—
184C2×Dic94—28no—
185C2×Dic94—28no—
186C2×Dic94—22no—
187C2×Dic94—22no—
188C2×Dic94—22no—
189C2×Dic94—22no—
190C2×Dic94—24no—
191C2×Dic94—22no—
192C2×Dic94—42no—
193C2×Dic94—44no—
194C2×Dic94—42no—
195C2×Dic94—44no—
196C2×Dic94C2×D414yes—
197C4⋊C4×C47—28no—
198C4⋊C4×C47—24no—
199C4⋊C4×C47—22no—
200C4⋊C4×C47C2×D414yes—
201C22⋊C4×C47—28no—
202C22⋊C4×C47—28no—
203C22⋊Dic47—216no—
204C22⋊Dic47—24no—
205#10625337—216no—
206#10625335—216no—
207C4⋊Dic47—22no—
208C4⋊Dic47C2×D414yes—
209Dic47⋊C4—216no—
210C4×Dic47—24no—
211#10625329—28no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8—22no—
215C2×C47⋊C8—22no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—44no—
218C2×C47⋊C8C2×D414yes—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—24no—
226C23×C47—24no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47—42no—
230C23×C47—42no—
231C23×C47#12012yes—
232C23×C47#12012yes—
233C23×C47#12112yes—
234C23×C47C22×D411yescharacteristic
235Q8×C47—22no—
236Q8×C47—22no—
237Q8×C47—41no—
238Q8×C47—41no—
239Q8×C47—44no—
240D4×C47—22no—
241D4×C47—22no—
242D4×C47—48no—
243D4×C47—48no—
244D4×C47—41no—
245D4×C47—41no—
246D4×C47—44no—
247D4×C47—88no—
248C2×C4×C47—22no—
249C2×C4×C47—24no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—44no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47—42no—
259C2×C4×C47—42no—
260C2×C4×C47—48no—
261C2×C4×C47—84no—
262C2×C4×C47#12014yes—
263C2×C4×C47#12012yes—
264C2×C4×C47#12012yes—
265C2×C4×C47#12114yes—
266C2×C4×C47#12114yes—
267C2×C4×C47#12112yes—
268C2×C4×C47#12714yes—
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C22×D411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C2×Dic47—24no—
276C2×Dic47—24no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—48no—
282Dic94—24no—
283Dic94—24no—
284Dic94—42no—
285Dic94—42no—
286Dic94—48no—
287C47⋊C8—24no—
288C47⋊C8—42no—
289#23394—471nomaximal, Sylow
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—24no—
293C22×C47—24no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—44no—
301C22×C47—44no—
302C22×C47—82no—
303C22×C47—82no—
304C22×C47#39214yes—
305C22×C47C2×#12011yescharacteristic
306C22×C47C2×#12011yescharacteristic
307C22×C47C2×#12111yescharacteristic
308C22×C47#57712yes—
309C22×C47#57812yes—
310C4×C47—42no—
311C4×C47—42no—
312C4×C47—44no—
313C4×C47#39214yes—
314C4×C47C2×#12011yescharacteristic
315C4×C47C2×#12111yescharacteristic
316C4×C47C2×#12111yescharacteristic
317C4×C47C2×#12711yescharacteristic
318Dic47—44no—
319Dic47—44no—
320C22×#137—471no—
321C2×#537D4711yescharacteristic
322C2×#536—471no—
323C2×#416—471no—
324C22×#103—471no—
325C2×#408—471no—
326#2001—474no—
327#2000—474no—
328#1736—471no—
329C2×C47—24no—
330C2×C47—82no—
331C2×C47—82no—
332C2×C47C2×#39211yescharacteristic
333C2×C47#300911yescharacteristic
334C2×C47#301111yescharacteristic
335C2×C47#301312yes—
336C2×C47#306312yes—
337C22×#41C2×D4711yescharacteristic
338C22×#41C2×D4711yescharacteristic
339C23×Q8—471no—
340C2×#137—944no—
341C2×#137—944no—
342C2×#137—944no—
343C22×Q16—472no—
344C22×SD16—472no—
345C22×M16—471no—
346C23×C8—471no—
347#537—24no—
348#537—24no—
349#536—944no—
350#536—944no—
351C2×C4⋊Q8—941no—
352C2×#124—941no—
353C2×#123—942no—
354C2×C22⋊Q8—21no—
355C2×C22⋊Q8—474no—
356C2×C22⋊Q8C2×D4712yes—
357C2×#121—21no—
358C2×#121—22no—
359C2×#121C2×D4712yes—
360C2×#117—471no—
361C22×C4⋊C4C2×D4711yescharacteristic
362#416—944no—
363C2×#103—472no—
364C2×#103—474no—
365C2×#103—944no—
366C2×#102—472no—
367#408—944no—
368C2×C22⋊C8—474no—
369#340—474no—
370C47#2339411yescharacteristic, complemented, Sylow
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—22no—
381C2×#41—22no—
382C2×#41C22⋊D4714yes—
383C22×Q8—474no—
384C22×Q8—472no—
385C22×Q8—942no—
386C22×Q8—941no—
387C22×Q8—942no—
388C22×Q8—942no—
389C22×Q8—942no—
390C22×Q8—942no—
391C22×Q8C22⋊D4711yescharacteristic
392C22×Q8C22×D4711yescharacteristic
393C22×D4—21no—
394C22×D4—22no—
395C22×D4C22⋊D4712yes—
396C22×D4C22⋊D4711yescharacteristic
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—941no—
401C23×C4C22×D4712yes—
402C23×C4C22×D4711yescharacteristic
403#137—9432no—
404C2×Q16—944no—
405C2×Q16—944no—
406C2×Q16—944no—
407C2×SD16—944no—
408C2×SD16—944no—
409C2×SD16—944no—
410C2×M16—474no—
411C2×M16—942no—
412C2×M16—942no—
413C22×C8—472no—
414C22×C8—472no—
415C22×C8—942no—
416C22×C8—942no—
417C22×C8—942no—
418C4⋊Q8—948no—
419#124—948no—
420#123—9416no—
421C22⋊Q8—28no—
422C22⋊Q8—24no—
423C22⋊Q8—24no—
424C22⋊Q8—948no—
425C22⋊Q8—948no—
426#121—216no—
427#121—28no—
428#121—24no—
429#121—24no—
430#117—474no—
431#117—942no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—472no—
436C2×C4⋊C4—944no—
437C2×C4⋊C4C22⋊D4714yes—
438C2×C4⋊C4C22×D4712yes—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—941no—
442C2×C22⋊C4—944no—
443C2×C42—941no—
444#103—944no—
445#103—948no—
446#102—944no—
447C22⋊C8—948no—
448C24—21no—
449C24C2×C22⋊D4712yes—
450C24C2×C22⋊D4711yescharacteristic
451#41—28no—
452#41—28no—
453#41—44no—
454#41—44no—
455#41—44no—
456#41—44no—
457#41—44no—
458#41—44no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—474no—
468C2×Q8—944no—
469C2×Q8—942no—
470C2×Q8—948no—
471C2×Q8—948no—
472C2×Q8—942no—
473C2×Q8—942no—
474C2×Q8—942no—
475C2×Q8—942no—
476C2×Q8—1882no—
477C2×Q8—1884no—
478C2×Q8—1882no—
479C2×Q8—1884no—
480C2×Q8C2×C22⋊D4712yes—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—24no—
485C2×D4—28no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—84no—
497C2×D4C2×C22⋊D4712yes—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—82no—
511C22×C4—944no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—1882no—
515C22×C4—1882no—
516C22×C4D4×D4712yes—
517C22×C4D4×D4712yes—
518C22×C4D4×D4712yes—
519C22×C4#1062535212yes—
520C22×C4C2×C22⋊D4712yes—
521C22×C4C2×C22⋊D4711yescharacteristic
522C22×C4C2×C22⋊D4712yes—
523C22×C4C2×C22⋊D4712yes—
524C22×C4C23×D4711yescharacteristic
525Q16—9416no—
526SD16—9416no—
527M16—948no—
528C2×C8—474no—
529C2×C8—942no—
530C2×C8—942no—
531C2×C8—942no—
532C2×C8—942no—
533C2×C8—942no—
534C2×C8—1884no—
535C4⋊C4—28no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—474no—
539C4⋊C4—9416no—
540C4⋊C4—942no—
541C4⋊C4C2×C22⋊D4714yes—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—9416no—
545C22⋊C4—944no—
546C42—944no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23#1309399212yes—
560C23#1309399312yes—
561C23#1309399612yes—
562C23C22×C22⋊D4711yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—44no—
566Q8—41no—
567Q8—41no—
568Q8—944no—
569Q8—944no—
570Q8—1888no—
571Q8—1882no—
572Q8—1882no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—48no—
577D4—44no—
578D4—41no—
579D4—41no—
580D4—88no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—1888no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4#1309399212yes—
603C2×C4#1309399214yes—
604C2×C4#1309399312yes—
605C2×C4#1309399514yes—
606C2×C4#1309399612yes—
607C2×C4#1309399614yes—
608C2×C4#1309402014yes—
609C2×C4C2×D4×D4711yescharacteristic
610C2×C4C2×D4×D4711yescharacteristic
611C2×C4C2×D4×D4711yescharacteristic
612C2×C4C22×C22⋊D4711yescharacteristic
613C2×C4C22×C22⋊D4711yescharacteristic
614C2×C4C2×#1062535211yescharacteristic
615C8—944no—
616C8—1882no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—82no—
631C22#1568903114yes—
632C22C2×#1309399211yescharacteristic
633C22C2×#1309399311yescharacteristic
634C22C2×#1309399611yescharacteristic
635C22#1568993412yes—
636C22#1568993812yes—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—1884no—
641C4—1884no—
642C4#1568903114yes—
643C4C2×#1309399211yescharacteristic
644C4C2×#1309399511yescharacteristic
645C4C2×#1309399611yescharacteristic
646C4C2×#1309402011yescharacteristic
647C2—24no—
648C2—82no—
649C2—82no—
650C2C2×#1568903111yescharacteristic
651C2#2025266511yescharacteristic
652C2#2025268311yescharacteristic
653C2#2025270012yes—
654C2#2025308112yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#537C2
socleC23×C47C22×D4
hypercenterC2×#537D47
hypocenterC47#23394
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C22×#13094054C2semidirect
2. C2×#15689694C2semidirect
3. C2×#15689119C2semidirect
4. C22×#13093947C2semidirect
5. C2×#15689090C2semidirect
6. #20244396C2semidirect
7. #20244391C2semidirect
8. #20243107C2semidirect
49. C22×Dic94D4semidirect
61. C2×#10625329D4semidirect
73. #13093983D4semidirect
370. C47#23394semidirect

Maximal subgroups

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

subgroupindex
C22×#130940542
C2×#156896942
C2×#156891192
C22×#130939472
C2×#156890902
#202443962
#202443912
#202431072
C2×C47×#5372
#2339447