Finite Group Databasebeta

← #26218230

Subgroups — #26218230

Order 12032.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689921C211yesmaximal, characteristic
2C2×#15689889C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic, complemented
4C2×#15689107C211yesmaximal, characteristic, complemented
5#20243906C214yesmaximal, complemented
6#20243899C214yesmaximal, complemented
7C2×#15688892C211yesmaximal, characteristic, complemented
8#20243115C211yesmaximal, characteristic, complemented
9C2×C47×#417C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11#15689921—24no—
12#15689921C418yescomplemented
13C2×#13094055C2212yes—
14C2×#13094054—21no—
15C2×#13094050C2212yes—
16C2×#13094049—21no—
17C2×#13094044—22no—
18#15689889—24no—
19#15689889—24no—
20C2×#13094032—22no—
21C2×#13094030C2214yes—
22C22×#10625329C2211yescharacteristic
23C22×#10625329C2211yescharacteristic
24C22×#10625329C2211yescharacteristic
25C2×M16×D47—22no—
26#15689122—22no—
27#15689122C2214yes—
28C2×#13093947—21no—
29C2×#13093944—21no—
30#15689107—24no—
31C2×#13093943C2212yescomplemented
32C2×#13093941C2212yescomplemented
33#15688892C2214yescomplemented
34#15688888C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2211yescharacteristic
38C47×#417—22no—
39C47×#417—22no—
40C2×#103×C47C2212yescomplemented
41C2×#102×C47C2212yescomplemented
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—21no—
46C2×#10625350C2×C414yes—
47C2×#10625350D414yes—
48C2×#10625350D412yes—
49C2×#10625350C2312yes—
50C22×Dic94D411yescharacteristic
51C22×D188D411yescharacteristic
52C22×C22⋊D47—21no—
53C22×C4×D47—21no—
54#13094055—24no—
55#13094055C2×C418yescomplemented
56#13094054—28no—
57#13094050—24no—
58#13094050C2×C418yescomplemented
59#13094049—28no—
60#13094044—216no—
61C2×#10625337—22no—
62C2×#10625335—22no—
63C2×#10625336—22no—
64C2×#10625334—22no—
65#13094032—216no—
66#13094030—28no—
67#13094030—28no—
68C2×#10625329—21no—
69C2×#10625329—22no—
70C2×#10625329—21no—
71C2×#10625329—22no—
72C2×#10625329—21no—
73C2×#10625329—22no—
74C2×#10625329—22no—
75C2×#10625329C2×C414yes—
76C2×#10625329D414yes—
77C2×#10625329C2312yes—
78C2×#10625329C2312yes—
79C2×#10625329C2312yes—
80C2×#10625324—24no—
81M16×D47—216no—
82C22×C47⋊C8—21no—
83C22×C47⋊C8—21no—
84C22×C47⋊C8—21no—
85C2×C8×D47—24no—
86#13093947—24no—
87#13093944—24no—
88#13093943—24no—
89#13093941—24no—
90C2×#41×C47—22no—
91C2×#41×C47—21no—
92C2×#41×C47—21no—
93C2×#41×C47—22no—
94C2×#41×C47—21no—
95C2×#41×C47—21no—
96C2×#41×C47—22no—
97C2×#41×C47C2×C414yes—
98C22×Q8×C47C2311yescharacteristic
99C22×D4×C47—21no—
100C22×D4×C47C2311yescharacteristic
101C23×C4×C47—21no—
102C23×C4×C47—21no—
103C23×C4×C47C2311yescharacteristic
104C2×M16×C47—22no—
105C2×M16×C47—22no—
106C2×M16×C47D412yes—
107C2×M16×C47D412yes—
108C22×C8×C47C2312yes—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47C2312yes—
115#103×C47—24no—
116#102×C47—24no—
117C24×C47—21no—
118C24×C47#4111yescharacteristic
119C23×D47—21no—
120#41×C47—24no—
121#41×C47—24no—
122#41×C47—24no—
123#41×C47—24no—
124#41×C47—42no—
125#41×C47—42no—
126#41×C47—42no—
127#41×C47—42no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47C22×C412yes—
134C2×D4×C47—24no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—41no—
141C2×D4×C47—41no—
142C2×D4×C47—42no—
143C2×D4×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47—42no—
150C22×C4×C47—41no—
151C22×C4×C47—41no—
152C22×C4×C47—82no—
153C22×C4×C47C22×C412yes—
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D412yes—
160C22×C4×C47#4111yescharacteristic
161C22×C4×C47#4112yes—
162C22×C4×C47C2411yescharacteristic
163C2×C22⋊D47—24no—
164C2×C22⋊D47—22no—
165C2×C22⋊D47—24no—
166C2×C22⋊D47—22no—
167C22×Dic47—21no—
168#10625350—24no—
169#10625350—24no—
170#10625350—24no—
171#10625350—24no—
172#10625350—24no—
173#10625350—22no—
174#10625350—22no—
175#10625350C22⋊C418yes—
176#10625350C22×C414yes—
177#10625350C2×D414yes—
178C2×D188—22no—
179C2×D188—21no—
180C2×D188C22⋊C414yes—
181C2×D188C2×D412yes—
182C2×C4×D47—24no—
183C2×C4×D47—24no—
184C2×C4×D47—22no—
185C2×C4×D47—22no—
186C2×Dic94—22no—
187C2×Dic94—21no—
188C2×Dic94C22⋊C414yes—
189C2×Dic94C2×D412yes—
190M16×C47—24no—
191M16×C47—24no—
192C2×C8×C47—22no—
193C2×C8×C47—22no—
194C2×C8×C47C2×D414yes—
195C4⋊C4×C47—22no—
196C4⋊C4×C47—22no—
197C4⋊C4×C47—41no—
198C4⋊C4×C47—41no—
199#10625337—28no—
200#10625336—28no—
201#10625335—28no—
202#10625334—28no—
203#10625329—24no—
204#10625329—24no—
205#10625329—22no—
206#10625329—22no—
207#10625329—22no—
208#10625329—22no—
209#10625329—42no—
210#10625329C22×C414yes—
211#10625329C2×D414yes—
212C2×C47⋊C8—24no—
213C2×C47⋊C8—24no—
214C2×C47⋊C8—22no—
215C2×C47⋊C8—22no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—42no—
218#10625324—216no—
219C8×D47—216no—
220C23×C47—22no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47—81no—
226C23×C47—81no—
227C23×C47C4×D412yes—
228C23×C47#12111yescharacteristic
229C23×C47#12111yescharacteristic
230C23×C47#12312yes—
231C23×C47C2×#4111yescharacteristic
232C22×D47—22no—
233C22×D47—24no—
234C22×D47—81no—
235C22×D47—81no—
236Q8×C47—22no—
237Q8×C47—22no—
238Q8×C47—41no—
239Q8×C47—41no—
240D4×C47—22no—
241D4×C47—22no—
242D4×C47—44no—
243D4×C47—44no—
244D4×C47—41no—
245D4×C47—41no—
246C2×C4×C47—22no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—44no—
250C2×C4×C47—44no—
251C2×C4×C47—42no—
252C2×C4×C47—82no—
253C2×C4×C47—82no—
254C2×C4×C47—82no—
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C2×C22⋊C412yes—
257C2×C4×C47C4×D412yes—
258C2×C4×C47C4×D412yes—
259C2×C4×C47C4×D412yes—
260C2×C4×C47#12012yes—
261C2×C4×C47#12012yes—
262C2×C4×C47#12111yescharacteristic
263C2×C4×C47#12111yescharacteristic
264C2×C4×C47#12112yes—
265C2×C4×C47#12112yes—
266C2×C4×C47#12312yes—
267C2×C4×C47C23×C411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C22×D411yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C22⋊D47—24no—
274C22⋊D47—24no—
275C22⋊D47—28no—
276C2×Dic47—22no—
277C2×Dic47—24no—
278D188—22no—
279D188—24no—
280D188C2×C22⋊C414yes—
281C4×D47—24no—
282C4×D47—24no—
283C4×D47—28no—
284Dic94—22no—
285Dic94—24no—
286Dic94C2×C22⋊C414yes—
287C8×C47—24no—
288C47⋊C8—24no—
289C47⋊C8—24no—
290C47⋊C8—42no—
291#21004—471nomaximal, Sylow
292C22×C47—21no—
293C22×C47—21no—
294C22×C47—21no—
295C22×C47—21no—
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47—82no—
302C22×C47—82no—
303C22×C47—81no—
304C22×C47—81no—
305C22×C47#38612yes—
306C22×C47#38612yes—
307C22×C47C2×C4×D411yescharacteristic
308C22×C47C2×#12111yescharacteristic
309C22×C47C2×#12311yescharacteristic
310C2×D47—24no—
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—84no—
314C4×C47—42no—
315C4×C47—82no—
316C4×C47#38612yes—
317C4×C47#38612yes—
318C4×C47C22×C22⋊C411yescharacteristic
319C4×C47C2×C4×D411yescharacteristic
320C4×C47C2×#12011yescharacteristic
321C4×C47C2×#12111yescharacteristic
322Dic47—24no—
323C2×#575—471no—
324C2×#568—471no—
325C2×#417—471no—
326C2×#417D4711yescharacteristic, complemented
327C2×#413—471no—
328#1908—474no—
329#1907—474no—
330C2×#343—471no—
331#1738—471no—
332C2×C47—22no—
333C2×C47—21no—
334C2×C47—21no—
335C2×C47—22no—
336C2×C47—82no—
337C2×C47C2×#38611yescharacteristic
338C2×C47#306811yescharacteristic
339C2×C47#306911yescharacteristic
340D47—44no—
341C22×#41—471no—
342C22×#41C2×D4711yescharacteristic
343#575—478no—
344#575—944no—
345C2×#137—472no—
346C2×#137—941no—
347C2×AGL(1,ℤ/8)—472no—
348C2×AGL(1,ℤ/8)—941no—
349C2×#135—942no—
350#568—944no—
351#568—944no—
352C2×#131—474no—
353C2×#131—942no—
354C22×M16—471no—
355C22×M16—471no—
356C22×M16—471no—
357C22×M16—942no—
358C22×M16C2×D4711yescharacteristic
359C22×C4⋊C4C2×D4711yescharacteristic
360#417—22no—
361#417—22no—
362#417—474no—
363#417—942no—
364C2×#103—941no—
365C2×#103C2×D4712yescomplemented
366C2×#102—941no—
367C2×#102C2×D4712yescomplemented
368#413—944no—
369C2×#101—472no—
370C2×#100—472no—
371#343—474no—
372#341—474no—
373C47#2100411yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—22no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—474no—
382C2×#41—474no—
383C2×#41—472no—
384C2×#41—472no—
385C2×#41—942no—
386C2×#41—942no—
387C2×#41—941no—
388C2×#41—941no—
389C2×#41C4×D4714yes—
390C22×Q8—471no—
391C22×Q8C22×D4711yescharacteristic
392C22×D4—21no—
393C22×D4—471no—
394C22×D4—941no—
395C22×D4C22×D4711yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—941no—
399C23×C4C22×D4711yescharacteristic
400#137—478no—
401#137—944no—
402#137—948no—
403AGL(1,ℤ/8)—478no—
404AGL(1,ℤ/8)—944no—
405AGL(1,ℤ/8)—948no—
406#135—9416no—
407C2×Q16—942no—
408C2×SD16—942no—
409C2×SD16—942no—
410C2×D8—942no—
411#131—948no—
412#131—948no—
413#131—9416no—
414C2×M16—22no—
415C2×M16—22no—
416C2×M16—474no—
417C2×M16—474no—
418C2×M16—472no—
419C2×M16—472no—
420C2×M16—472no—
421C2×M16—9416no—
422C2×M16—942no—
423C2×M16—941no—
424C2×M16—942no—
425C2×M16—941no—
426C2×M16—941no—
427C2×M16—944no—
428C2×M16—942no—
429C2×M16—942no—
430C2×M16C22⋊D4712yes—
431C2×M16C22⋊D4712yes—
432C22×C8—941no—
433C22×C8—941no—
434C22×C8—944no—
435C22×C8—941no—
436C22×C8C22×D4712yes—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4C22×D4712yes—
443#103—24no—
444#103—944no—
445#102—24no—
446#102—944no—
447#101—944no—
448#100—944no—
449C24—21no—
450C24—941no—
451C24#1062535211yescharacteristic
452#41—24no—
453#41—24no—
454#41—24no—
455#41—24no—
456#41—42no—
457#41—42no—
458#41—42no—
459#41—42no—
460#41—478no—
461#41—474no—
462#41—474no—
463#41—944no—
464#41—944no—
465#41—944no—
466#41—944no—
467#41—944no—
468#41—942no—
469#41—942no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—474no—
476C2×Q8—472no—
477C2×Q8—941no—
478C2×Q8—942no—
479C2×Q8C2×C4×D4712yes—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—41no—
487C2×D4—41no—
488C2×D4—42no—
489C2×D4—474no—
490C2×D4—472no—
491C2×D4—944no—
492C2×D4—942no—
493C2×D4—944no—
494C2×D4—942no—
495C2×D4—941no—
496C2×D4—942no—
497C2×D4C2×C4×D4712yes—
498C22×C4—21no—
499C22×C4—24no—
500C22×C4—22no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—42no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—82no—
507C22×C4—941no—
508C22×C4—944no—
509C22×C4—944no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4C2×C4×D4712yes—
513C22×C4#1062535012yes—
514C22×C4D4×D4712yes—
515C22×C4D4×D4711yescharacteristic
516C22×C4D4×D4711yescharacteristic
517C22×C4D4×D4712yes—
518C22×C4#1062535211yescharacteristic
519C22×C4C2×C22⋊D4711yescharacteristic
520C22×C4C2×C22⋊D4711yescharacteristic
521C22×C4C23×D4711yescharacteristic
522Q16—948no—
523SD16—948no—
524SD16—948no—
525D8—948no—
526M16—24no—
527M16—24no—
528M16—474no—
529M16—474no—
530M16—9416no—
531M16—944no—
532M16—944no—
533M16—942no—
534M16—942no—
535M16—942no—
536M16—942no—
537M16—1882no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—9416no—
541C2×C8—944no—
542C2×C8—944no—
543C2×C8—942no—
544C2×C8—942no—
545C2×C8—942no—
546C2×C8—1882no—
547C2×C8C2×C22⋊D4714yes—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—81no—
559C23—944no—
560C23—942no—
561C23—3761no—
562C23—3761no—
563C23(C22⋊D47)⋊C412yes—
564C23#1309399511yescharacteristic
565C23#1309399511yescharacteristic
566C23#1309400612yes—
567C23C2×#1062535211yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—474no—
573Q8—944no—
574Q8—942no—
575D4—22no—
576D4—22no—
577D4—44no—
578D4—44no—
579D4—41no—
580D4—41no—
581D4—474no—
582D4—948no—
583D4—944no—
584D4—944no—
585D4—944no—
586D4—942no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—944no—
597C2×C4—942no—
598C2×C4—948no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4C22⋊C4×D4712yes—
602C2×C4C22⋊C4×D4712yes—
603C2×C4(C22⋊D47)⋊C412yes—
604C2×C4C4×C22⋊D4712yes—
605C2×C4C4×C22⋊D4712yes—
606C2×C4#1309399212yes—
607C2×C4#1309399212yes—
608C2×C4#1309399511yescharacteristic
609C2×C4#1309399511yescharacteristic
610C2×C4#1309399712yes—
611C2×C4#1309399712yes—
612C2×C4#1309400612yes—
613C2×C4C22×C4×D4711yescharacteristic
614C2×C4C2×D4×D4711yescharacteristic
615C2×C4C2×D4×D4711yescharacteristic
616C2×C4C2×D4×D4711yescharacteristic
617C2×C4C22×C22⋊D4711yescharacteristic
618C2×C4C2×#1062535011yescharacteristic
619C8—24no—
620C8—944no—
621C8—944no—
622C8—1882no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—944no—
637C22—3764no—
638C22—1882no—
639C22—1882no—
640C22#1568899912yes—
641C22#1568899912yes—
642C22C2×(C22⋊D47)⋊C411yescharacteristic
643C22C2×#1309399511yescharacteristic
644C22C2×#1309400611yescharacteristic
645C4—42no—
646C4—82no—
647C4—944no—
648C4#1568899912yes—
649C4#1568899912yes—
650C4C2×C22⋊C4×D4711yescharacteristic
651C4C2×C4×C22⋊D4711yescharacteristic
652C4C2×#1309399211yescharacteristic
653C4C2×#1309399711yescharacteristic
654C2—21no—
655C2—21no—
656C2—22no—
657C2—22no—
658C2—82no—
659C2—1884no—
660C2C2×#1568899911yescharacteristic
661C2#2025311211yescharacteristic
662C2#2025312111yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC22×C47C2×#121
prime radicalC2×#15689921C2
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#21004
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#15689889C2semidirect
3. C2×#15689122C2semidirect
4. C2×#15689107C2semidirect
5. #20243906C2semidirect
6. #20243899C2semidirect
7. C2×#15688892C2semidirect
8. #20243115C2semidirect
9. C2×C47×#417C2semidirect
12. #15689921C4semidirect
31. C2×#13093943C22semidirect
32. C2×#13093941C22semidirect
33. #15688892C22semidirect
34. #15688888C22semidirect
40. C2×#103×C47C22semidirect
41. C2×#102×C47C22semidirect
55. #13094055C2×C4semidirect
58. #13094050C2×C4semidirect
326. C2×#417D47semidirect
365. C2×#103C2×D47semidirect
367. C2×#102C2×D47semidirect
373. C47#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#156899212
C2×#156898892
C2×#156891222
C2×#156891072
#202439062
#202438992
C2×#156888922
#202431152
C2×C47×#4172
#2100447