Finite Group Databasebeta

← #26219352

Subgroups — #26219352

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689225C211yesmaximal, characteristic
2#20246034C211yesmaximal, characteristic, complemented
3#20245748C211yesmaximal, characteristic
4#20244028C211yesmaximal, characteristic, complemented
5#20244021C211yesmaximal, characteristic, complemented
6#20243979C211yesmaximal, characteristic, complemented
7#20243958C211yesmaximal, characteristic, complemented
8#20243723C211yesmaximal, characteristic
9#20243723C211yesmaximal, characteristic
10#20243646C211yesmaximal, characteristic
11#20243632C211yesmaximal, characteristic
12C4×#13093947C211yesmaximal, characteristic, complemented
13C4×#13093944C211yesmaximal, characteristic
14#20243141C211yesmaximal, characteristic, complemented
15C47×#1776C211yesmaximal, characteristic, complemented
16C22×#10625325C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×#13093986C2211yescharacteristic
20C2×#13093986C2211yescharacteristic
21#15689225C418yescomplemented
22C2×#13093958C2211yescharacteristic
23C2×#13093949C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093946C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689028C2211yescharacteristic
31#15689028C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689007—21no—
35#15689007—21no—
36#15688990—21no—
37#15688987—21no—
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4×(C2×D47)⋊C4—21no—
41C4×(C2×D47)⋊C4—21no—
42#15688970—21no—
43#15688969C2211yescharacteristic
44#95×D47—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C2×#10625325—22no—
66C2×#10625325—22no—
67C2×#10625325C2×C414yes—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C4×Dic94—22no—
71C4×Dic94C2×C414yescomplemented
72C4×D188—22no—
73C4×D188C2×C414yes—
74#13093986—22no—
75#13093986—22no—
76#13093986—22no—
77#13093986—22no—
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093958C2×C414yescomplemented
92#13093949C2×C414yescomplemented
93#13093947—22no—
94#13093947—22no—
95#13093946C2×C414yes—
96#13093944—22no—
97#13093944—22no—
98C4×C47⋊C8—22no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C47⋊#95D411yescharacteristic
110C47⋊#95D411yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C42×C47—21no—
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#107×C47—22no—
121#106×C47—22no—
122#97×C47C2×C414yescomplemented
123#95×C47—21no—
124#95×C47—21no—
125C23×D47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188C22×C412yes—
144C2×D188#4112yes—
145C2×D188#4112yes—
146C2×C4×D47—24no—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—81no—
150C2×C4×D47—81no—
151C2×C4×D47—81no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C2×C4×D47—81no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94C22×C412yes—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47C22×C412yes—
166C2×C8×C47C22×C412yes—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47#4112yes—
174C4⋊C4×C47#4112yes—
175C42×C47—81no—
176C42×C47—81no—
177C42×C47C22×C412yes—
178C42×C47#4112yes—
179(C2×D47)⋊C4—24no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—24no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47C22×C412yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—81no—
188Dic47⋊C4—81no—
189C4×Dic47—24no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—81no—
195C4×Dic47—81no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202#10625325—28no—
203C23×C47C22×D411yescharacteristic
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47C4×D412yes—
228C2×C4×C47C4×D412yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256D188—22no—
257D188—22no—
258D188—22no—
259D188#11714yes—
260C4×D47—84no—
261C4×D47—84no—
262C4×D47—84no—
263Dic94—22no—
264Dic94—22no—
265Dic94—41no—
266Dic94—41no—
267C8×C47—22no—
268C8×C47—22no—
269C47⋊C8—41no—
270C47⋊C8—41no—
271#21125—471nomaximal, Sylow
272C22×C47C2×C4×D411yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47#52511yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57811yescharacteristic
279C2×D47—24no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—42no—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—42no—
289C4×C47—42no—
290C4×C47—81no—
291C4×C47—81no—
292C4×C47C2×#11711yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#52911yescharacteristic
295C4×C47#52911yescharacteristic
296Dic47—41no—
297Dic47—41no—
298Dic47—41no—
299Dic47—41no—
300Dic47—42no—
301Dic47—42no—
302Dic47—42no—
303C2×#440—471no—
304#2320—471no—
305#2275—471no—
306#1928—471no—
307#1927—471no—
308#1921—471no—
309#1918—471no—
310#1877—471no—
311#1877—471no—
312#1864—471no—
313#1862—471no—
314#1776D4711yescharacteristic, complemented
315C4×#103—471no—
316C4×#102—471no—
317#1746—471no—
318C2×C47#229011yescharacteristic
319C2×C47#294611yescharacteristic
320C2×C47#294711yescharacteristic
321C2×C47#311811yescharacteristic
322C2×C47#312011yescharacteristic
323C2×C47#312311yescharacteristic
324C2×C47#312411yescharacteristic
325D47—44no—
326C22×SD16—471no—
327C2×C4×Q8—471no—
328C2×C4×D4—471no—
329C2×C4×D4—471no—
330C2×C4×D4—471no—
331#440—478no—
332C2×#107—471no—
333C2×#107C2×D4711yescharacteristic
334C2×#106C2×D4711yescharacteristic
335C2×#103—471no—
336C2×#103—471no—
337C2×#103—471no—
338C2×#102—471no—
339C2×#102—471no—
340C2×#102—471no—
341C2×#97C2×D4711yescharacteristic
342C2×C4×C8—471no—
343#391—471no—
344#391—471no—
345#389—471no—
346#389—471no—
347#387—941no—
348#387—941no—
349#383—941no—
350#382—941no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4C2×D4711yescharacteristic
354C4×C4⋊C4C2×D4711yescharacteristic
355C4×C22⋊C4—941no—
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358#376—471no—
359C2×#95—941no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340C2×D4711yescharacteristic
365#340C2×D4711yescharacteristic
366#336—471no—
367#336—471no—
368#336—471no—
369#336—471no—
370C47#2112511yescharacteristic, complemented, Sylow
371C22×Q8—471no—
372C22×D4—471no—
373C23×C4—941no—
374C23×C4—941no—
375C23×C4—941no—
376C2×SD16—474no—
377C2×SD16—942no—
378C2×SD16—942no—
379C22×C8—471no—
380C22×C8—471no—
381C22×C8C22×D4711yescharacteristic
382C22×C8C22×D4711yescharacteristic
383C4×Q8—474no—
384C4×Q8—942no—
385C4×D4—474no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4—942no—
390C4×D4—942no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—471no—
406C2×C42—471no—
407C2×C42—941no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42C22×D4711yescharacteristic
411#107—22no—
412#107—474no—
413#106—22no—
414#103—474no—
415#103—942no—
416#103—942no—
417#102—474no—
418#102—942no—
419#102—942no—
420#97C4×D4714yescomplemented
421C4×C8—942no—
422#95—21no—
423#95—21no—
424#95—471no—
425#95—471no—
426#95—471no—
427#95—471no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436C24—941no—
437C2×Q8—472no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C2×Q8—941no—
443C2×D4—472no—
444C2×D4—472no—
445C2×D4—472no—
446C2×D4—941no—
447C2×D4—941no—
448C2×D4—941no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—944no—
463C22×C4—3761no—
464C22×C4—3761no—
465C22×C4—3761no—
466C22×C4—3761no—
467C22×C4—3761no—
468C22×C4—3761no—
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4C23×D4711yescharacteristic
472SD16—948no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8C2×C4×D4712yes—
484C2×C8C2×C4×D4712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—472no—
492C4⋊C4—472no—
493C4⋊C4—944no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—3761no—
497C4⋊C4—3761no—
498C4⋊C4#1062535012yes—
499C4⋊C4#1062535012yes—
500C22⋊C4—944no—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C42—81no—
504C42—81no—
505C42—944no—
506C42—941no—
507C42—941no—
508C42—941no—
509C42—941no—
510C42—3761no—
511C42—3761no—
512C42C2×C4×D4712yes—
513C42#1062535412yes—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23C2×D4×D4711yescharacteristic
522Q8—942no—
523Q8—942no—
524Q8—1881no—
525Q8—1881no—
526D4—474no—
527D4—942no—
528D4—942no—
529D4—942no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—3764no—
562C2×C4—3764no—
563C2×C4—3764no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4#1309398612yes—
571C2×C4#1309398612yes—
572C2×C4C22×C4×D4711yescharacteristic
573C2×C4#41×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C2×C4C2×#1062535011yescharacteristic
577C2×C4C2×#1062535411yescharacteristic
578C8—22no—
579C8—22no—
580C8—1881no—
581C8—1881no—
582C22—944no—
583C22—1882no—
584C22—1882no—
585C22—1882no—
586C22—1882no—
587C22—1882no—
588C22—1882no—
589C22C2×#1309398611yescharacteristic
590C22#1568963211yescharacteristic
591C22#1568963211yescharacteristic
592C22#1568992711yescharacteristic
593C22#1568993211yescharacteristic
594C22#1568993311yescharacteristic
595C22#1568993911yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—1882no—
603C4—1882no—
604C4—1882no—
605C4—1881no—
606C4—1881no—
607C4—1881no—
608C4—1881no—
609C4#117×D4711yescharacteristic
610C4C4×#1062535011yescharacteristic
611C4#1568966011yescharacteristic
612C4#1568966011yescharacteristic
613C2—1884no—
614C2#2024582911yescharacteristic
615C2#2025221811yescharacteristic
616C2#2025222911yescharacteristic
617C2#2025357011yescharacteristic
618C2#2025360011yescharacteristic
619C2#2025363911yescharacteristic
620C2#2025364811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#21125
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. #20246034C2semidirect
4. #20244028C2semidirect
5. #20244021C2semidirect
6. #20243979C2semidirect
7. #20243958C2semidirect
12. C4×#13093947C2semidirect
14. #20243141C2semidirect
15. C47×#1776C2semidirect
21. #15689225C4semidirect
71. C4×Dic94C2×C4semidirect
91. #13093958C2×C4semidirect
92. #13093949C2×C4semidirect
122. #97×C47C2×C4semidirect
314. #1776D47semidirect
370. C47#21125semidirect
420. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892252
#202460342
#202457482
#202440282
#202440212
#202439792
#202439582
#202437232
#202436462
#202436322
C4×#130939472
C4×#130939442
#202431412
C47×#17762
#2112547