Finite Group Databasebeta

← #26244188

Subgroups — #26244188

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689921C211yesmaximal, characteristic
2C2×#15689172C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic, complemented
4C2×#15689107C211yesmaximal, characteristic, complemented
5#20251832C211yesmaximal, characteristic, complemented
6#20244688C214yesmaximal, complemented
7#20244681C214yesmaximal, complemented
8C2×#15688892C211yesmaximal, characteristic, complemented
9C2×C47×#417C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11#15689921—24no—
12#15689921—24no—
13C2×#13094055C2212yes—
14C2×#13094054—21no—
15C2×#13094050C2212yes—
16C2×#13094049—21no—
17C2×#13094044—22no—
18C22×#10625329C2211yescharacteristic
19C22×#10625329C2211yescharacteristic
20C22×#10625329C2211yescharacteristic
21#15689172—24no—
22C2×#13093960C2214yescomplemented
23#15689122—22no—
24#15689122C2214yes—
25C2×#13093947—21no—
26C2×#13093944—21no—
27#15689107—24no—
28C2×#13093943C2212yescomplemented
29C2×#13093941C2212yescomplemented
30#15689091—22no—
31#15689089C2214yes—
32#15689086—22no—
33#15688892C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C22×M16×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#417—22no—
38C47×#417—22no—
39C2×#103×C47C2212yescomplemented
40C2×#102×C47C2212yescomplemented
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350D412yes—
48C2×#10625350C2312yes—
49C22×Dic94D411yescharacteristic
50C22×D188D411yescharacteristic
51C22×C22⋊D47—21no—
52C22×C4×D47—21no—
53#13094055—24no—
54#13094055—24no—
55#13094054—28no—
56#13094050—24no—
57#13094050—24no—
58#13094049—28no—
59#13094044—216no—
60C2×#10625337—22no—
61C2×#10625335—22no—
62C2×#10625336—22no—
63C2×#10625334—22no—
64C2×#10625329—21no—
65C2×#10625329—22no—
66C2×#10625329—21no—
67C2×#10625329—21no—
68C2×#10625329—22no—
69C2×#10625329—22no—
70C2×#10625329—22no—
71C2×#10625329—22no—
72C2×#10625329D414yes—
73C2×#10625329C2312yes—
74C2×#10625329C2312yes—
75C2×#10625329C2312yes—
76C22×C47⋊C8—21no—
77C22×C47⋊C8—21no—
78C22×C47⋊C8—21no—
79#13093960—28no—
80#13093947—24no—
81#13093944—24no—
82#13093943—24no—
83#13093941—24no—
84(C2×D47)⋊C8—28no—
85C2×#41×C47—22no—
86C2×#41×C47—21no—
87C2×#41×C47—21no—
88C2×#41×C47—22no—
89C2×#41×C47—21no—
90C2×#41×C47—21no—
91C2×#41×C47—22no—
92C2×#41×C47D414yes—
93C22×Q8×C47C2311yescharacteristic
94C22×D4×C47—21no—
95C22×D4×C47C2311yescharacteristic
96C23×C4×C47—21no—
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C2×M16×C47—22no—
100C2×M16×C47—22no—
101C2×M16×C47D412yes—
102C2×M16×C47D412yes—
103C22×C8×C47C2312yes—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47C2312yes—
110#103×C47—24no—
111#102×C47—24no—
112C24×C47—21no—
113C24×C47#4111yescharacteristic
114C23×D47—21no—
115#41×C47—24no—
116#41×C47—24no—
117#41×C47—24no—
118#41×C47—24no—
119#41×C47—42no—
120#41×C47—42no—
121#41×C47—42no—
122#41×C47—42no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47C2×D412yes—
129C2×D4×C47—24no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—41no—
136C2×D4×C47—41no—
137C2×D4×C47—42no—
138C2×D4×C47C2×D412yes—
139C22×C4×C47—21no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—41no—
143C22×C4×C47—41no—
144C22×C4×C47—42no—
145C22×C4×C47—41no—
146C22×C4×C47—41no—
147C22×C4×C47—82no—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47#4112yes—
155C22×C4×C47#4111yescharacteristic
156C22×C4×C47#4112yes—
157C22×C4×C47C2411yescharacteristic
158C2×C22⋊D47—22no—
159C2×C22⋊D47—22no—
160C2×C22⋊D47—42no—
161C2×C22⋊D47—42no—
162C22×Dic47—21no—
163#10625350—24no—
164#10625350—24no—
165#10625350—22no—
166#10625350—22no—
167#10625350—22no—
168#10625350—22no—
169#10625350—42no—
170#10625350—42no—
171#10625350—42no—
172#10625350—42no—
173C2×D188—22no—
174C2×D188—22no—
175C2×D188—21no—
176C2×D188C2×D412yes—
177C2×C4×D47—22no—
178C2×C4×D47—22no—
179C2×C4×D47—42no—
180C2×C4×D47—42no—
181C2×Dic94—22no—
182C2×Dic94—22no—
183C2×Dic94—21no—
184C2×Dic94C2×D412yes—
185M16×C47—24no—
186M16×C47—24no—
187C2×C8×C47—22no—
188C2×C8×C47—22no—
189C2×C8×C47C2×D414yes—
190C4⋊C4×C47—22no—
191C4⋊C4×C47—22no—
192C4⋊C4×C47—41no—
193C4⋊C4×C47—41no—
194#10625337—28no—
195#10625336—28no—
196#10625335—28no—
197#10625334—28no—
198#10625329—24no—
199#10625329—22no—
200#10625329—22no—
201#10625329—22no—
202#10625329—22no—
203#10625329—22no—
204#10625329—42no—
205#10625329—42no—
206#10625329C2×D414yes—
207C2×C47⋊C8—24no—
208C2×C47⋊C8—22no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—42no—
212C2×C47⋊C8—42no—
213C23×C47—22no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—81no—
219C23×C47—81no—
220C23×C47#12111yescharacteristic
221C23×C47#12111yescharacteristic
222C23×C47#12312yes—
223C23×C47#12412yes—
224C23×C47C2×#4111yescharacteristic
225C22×D47—22no—
226C22×D47—42no—
227C22×D47—81no—
228C22×D47—81no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—44no—
236D4×C47—44no—
237D4×C47—41no—
238D4×C47—41no—
239C2×C4×C47—22no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—44no—
243C2×C4×C47—44no—
244C2×C4×C47—42no—
245C2×C4×C47—82no—
246C2×C4×C47—82no—
247C2×C4×C47—82no—
248C2×C4×C47#12012yes—
249C2×C4×C47#12012yes—
250C2×C4×C47#12111yescharacteristic
251C2×C4×C47#12111yescharacteristic
252C2×C4×C47#12112yes—
253C2×C4×C47#12112yes—
254C2×C4×C47#12112yes—
255C2×C4×C47#12112yes—
256C2×C4×C47#12312yes—
257C2×C4×C47#12312yes—
258C2×C4×C47#12312yes—
259C2×C4×C47#12412yes—
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C22⋊D47—42no—
267C22⋊D47—42no—
268C22⋊D47—44no—
269C2×Dic47—22no—
270C2×Dic47—42no—
271D188—22no—
272D188—22no—
273D188—42no—
274C4×D47—42no—
275C4×D47—42no—
276C4×D47—44no—
277Dic94—22no—
278Dic94—22no—
279Dic94—42no—
280C8×C47—24no—
281C47⋊C8—24no—
282C47⋊C8—42no—
283C47⋊C8—42no—
284#23533—471nomaximal, Sylow
285C22×C47—21no—
286C22×C47—21no—
287C22×C47—21no—
288C22×C47—21no—
289C22×C47—21no—
290C22×C47—21no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—82no—
295C22×C47—82no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47#39412yes—
299C22×C47#39412yes—
300C22×C47C2×#12111yescharacteristic
301C22×C47C2×#12311yescharacteristic
302C22×C47C2×#12411yescharacteristic
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—84no—
307C4×C47—42no—
308C4×C47—82no—
309C4×C47#39412yes—
310C4×C47#39412yes—
311C4×C47C2×#12011yescharacteristic
312C4×C47C2×#12111yescharacteristic
313C4×C47C2×#12111yescharacteristic
314C4×C47C2×#12311yescharacteristic
315Dic47—42no—
316C2×#575—471no—
317C2×#430—471no—
318C2×#417—471no—
319C2×#417D4711yescharacteristic, complemented
320C2×#413—471no—
321#2879—471no—
322#2052—474no—
323#2051—474no—
324C2×#343—471no—
325C2×C47—22no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—22no—
329C2×C47—82no—
330C2×C47C2×#39411yescharacteristic
331C2×C47#306811yescharacteristic
332C2×C47#306911yescharacteristic
333D47—82no—
334C22×#41—471no—
335C22×#41C2×D4711yescharacteristic
336#575—944no—
337#575—944no—
338C2×#137—472no—
339C2×#137—941no—
340C2×AGL(1,ℤ/8)—472no—
341C2×AGL(1,ℤ/8)—941no—
342C2×#135—942no—
343C22×M16—471no—
344C22×M16—471no—
345C22×M16—471no—
346C22×M16C2×D4711yescharacteristic
347C22×C4⋊C4C2×D4711yescharacteristic
348#430—944no—
349C2×#108—474no—
350#417—22no—
351#417—22no—
352#417—474no—
353#417—942no—
354C2×#103—941no—
355C2×#103C2×D4712yescomplemented
356C2×#102—941no—
357C2×#102C2×D4712yescomplemented
358#413—944no—
359C2×#101—472no—
360C2×#100—472no—
361#408—474no—
362#408—942no—
363#407—942no—
364#343—474no—
365C47#2353311yescharacteristic, complemented, Sylow
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—22no—
371C2×#41—21no—
372C2×#41—21no—
373C2×#41—472no—
374C2×#41—472no—
375C2×#41—941no—
376C2×#41—941no—
377C2×#41—942no—
378C2×#41—942no—
379C2×#41—942no—
380C2×#41—942no—
381C2×#41D18814yes—
382C22×Q8—471no—
383C22×Q8C22×D4711yescharacteristic
384C22×D4—21no—
385C22×D4—471no—
386C22×D4—941no—
387C22×D4C22×D4711yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—941no—
391C23×C4C22×D4711yescharacteristic
392#137—944no—
393#137—948no—
394#137—944no—
395AGL(1,ℤ/8)—944no—
396AGL(1,ℤ/8)—948no—
397AGL(1,ℤ/8)—944no—
398#135—9416no—
399C2×Q16—942no—
400C2×SD16—942no—
401C2×SD16—942no—
402C2×D8—942no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—474no—
406C2×M16—472no—
407C2×M16—472no—
408C2×M16—472no—
409C2×M16—942no—
410C2×M16—941no—
411C2×M16—941no—
412C2×M16—941no—
413C2×M16—942no—
414C2×M16—942no—
415C2×M16—942no—
416C2×M16—942no—
417C2×M16C22⋊D4712yes—
418C2×M16C22⋊D4712yes—
419C22×C8—941no—
420C22×C8—941no—
421C22×C8—941no—
422C22×C8C22×D4712yes—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4C22×D4712yes—
429#108—948no—
430#103—24no—
431#103—944no—
432#102—24no—
433#102—944no—
434#101—944no—
435#100—944no—
436C22⋊C8—948no—
437C24—21no—
438C24—941no—
439C24#1062535211yescharacteristic
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—944no—
449#41—944no—
450#41—942no—
451#41—942no—
452#41—942no—
453#41—942no—
454#41—1882no—
455#41—1882no—
456#41—1882no—
457#41—1882no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—472no—
464C2×Q8—941no—
465C2×Q8—942no—
466C2×Q8—942no—
467C2×Q8C2×D18812yes—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—42no—
477C2×D4—472no—
478C2×D4—942no—
479C2×D4—942no—
480C2×D4—942no—
481C2×D4—941no—
482C2×D4—942no—
483C2×D4—1882no—
484C2×D4—1882no—
485C2×D4C2×D18812yes—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—82no—
495C22×C4—941no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—1882no—
499C22×C4—1882no—
500C22×C4C2×D18812yes—
501C22×C4#1062535012yes—
502C22×C4D4×D4711yescharacteristic
503C22×C4D4×D4711yescharacteristic
504C22×C4D4×D4712yes—
505C22×C4#1062535211yescharacteristic
506C22×C4#1062535212yes—
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C23×D4711yescharacteristic
510Q16—948no—
511SD16—948no—
512SD16—948no—
513D8—948no—
514M16—24no—
515M16—24no—
516M16—474no—
517M16—944no—
518M16—942no—
519M16—942no—
520M16—942no—
521M16—942no—
522M16—942no—
523M16—1882no—
524M16—1882no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—944no—
528C2×C8—942no—
529C2×C8—942no—
530C2×C8—942no—
531C2×C8—1882no—
532C2×C8—1882no—
533C2×C8C2×C22⋊D4714yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—81no—
544C23—81no—
545C23—942no—
546C23—3761no—
547C23—3761no—
548C23—1882no—
549C23#1309399511yescharacteristic
550C23#1309399511yescharacteristic
551C23#1309400812yes—
552C23#1309401012yes—
553C23C2×#1062535211yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—41no—
557Q8—41no—
558Q8—942no—
559Q8—942no—
560Q8—1882no—
561D4—22no—
562D4—22no—
563D4—44no—
564D4—44no—
565D4—41no—
566D4—41no—
567D4—942no—
568D4—942no—
569D4—1884no—
570D4—1882no—
571D4—1882no—
572D4—1882no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—942no—
583C2×C4—1882no—
584C2×C4—1884no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4#1309399412yes—
588C2×C4#1309399412yes—
589C2×C4#1309399511yescharacteristic
590C2×C4#1309399511yescharacteristic
591C2×C4#1309399612yes—
592C2×C4#1309399612yes—
593C2×C4#1309399812yes—
594C2×C4#1309399812yes—
595C2×C4#1309400612yes—
596C2×C4#1309400612yes—
597C2×C4#1309400812yes—
598C2×C4#1309401012yes—
599C2×C4C2×D4×D4711yescharacteristic
600C2×C4C2×D4×D4711yescharacteristic
601C2×C4C22×C22⋊D4711yescharacteristic
602C2×C4C22×D18811yescharacteristic
603C2×C4C2×#1062535211yescharacteristic
604C2×C4C2×#1062535011yescharacteristic
605C8—24no—
606C8—944no—
607C8—1882no—
608C8—1882no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—82no—
619C22—82no—
620C22—81no—
621C22—81no—
622C22—3764no—
623C22—1882no—
624C22—1882no—
625C22—1882no—
626C22#1568903912yes—
627C22#1568903912yes—
628C22C2×#1309399511yescharacteristic
629C22C2×#1309400811yescharacteristic
630C22C2×#1309401011yescharacteristic
631C4—42no—
632C4—82no—
633C4—1882no—
634C4#1568903912yes—
635C4#1568903912yes—
636C4C2×#1309399411yescharacteristic
637C4C2×#1309399611yescharacteristic
638C4C2×#1309399811yescharacteristic
639C4C2×#1309400611yescharacteristic
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—82no—
645C2—3762no—
646C2C2×#1568903911yescharacteristic
647C2#2025311211yescharacteristic
648C2#2025312111yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC22×C47C2×#121
prime radicalC2×#15689921C2
hypercenterC2×#417D47
hypocenterC47#23533
LCSG ▷ C22×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×#15689172C2semidirect
3. C2×#15689122C2semidirect
4. C2×#15689107C2semidirect
5. #20251832C2semidirect
6. #20244688C2semidirect
7. #20244681C2semidirect
8. C2×#15688892C2semidirect
9. C2×C47×#417C2semidirect
22. C2×#13093960C22semidirect
28. C2×#13093943C22semidirect
29. C2×#13093941C22semidirect
33. #15688892C22semidirect
39. C2×#103×C47C22semidirect
40. C2×#102×C47C22semidirect
319. C2×#417D47semidirect
355. C2×#103C2×D47semidirect
357. C2×#102C2×D47semidirect
365. C47#23533semidirect

Maximal subgroups

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

subgroupindex
C2×#156899212
C2×#156891722
C2×#156891222
C2×#156891072
#202518322
#202446882
#202446812
C2×#156888922
C2×C47×#4172
#2353347