Finite Group Databasebeta

← #27080976

Subgroups — #27080976

Order 12032.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256986C211yesmaximal, characteristic
2#20252407C211yesmaximal, characteristic, complemented
3#20252395C213yesmaximal, complemented
4#20252392C213yesmaximal, complemented
5#20252366C213yesmaximal, complemented
6#20252357C213yesmaximal, complemented
7#41×C47⋊C8C211yesmaximal, characteristic, complemented
8C8×#10625354C211yesmaximal, characteristic, complemented
9Q8×#10625324C211yesmaximal, characteristic, complemented
10C8×Q8×D47C211yesmaximal, characteristic, complemented
11#20252133C213yesmaximal, complemented
12#20252097C213yesmaximal, complemented
13#20252055C213yesmaximal, complemented
14C4×#13094032C213yesmaximal, complemented
15C47×#2971C211yesmaximal, characteristic, complemented
16C2×#13094032C2213yes—
17C2×#13094032C2213yes—
18#15689856C413yes—
19#15689849C411yescharacteristic
20#15689841C413yes—
21#15689823C413yes—
22#15689817C413yes—
23#15689792C411yescharacteristic
24#15689785C411yescharacteristic
25#15689780C411yescharacteristic
26#15689670C2213yes—
27#15689657C2213yes—
28#15689655C2213yes—
29C4×#10625354C2211yescharacteristic
30C4×#10625350C2213yes—
31C4×Q8×D47C2211yescharacteristic
32#15689253C2213yes—
33#15689250C2213yes—
34Q8×C47⋊C8C2211yescharacteristic
35Q8×C47⋊C8C2211yescharacteristic
36C8×Dic94C2213yes—
37#15689247C2213yes—
38#15689202C2213yes—
39#15689200C2213yes—
40#15689199C2216yes—
41#15689192C2216yes—
42#15689191C2216yes—
43C8×D188C2213yes—
44D4×C47⋊C8C2213yes—
45D4×C47⋊C8C2213yes—
46#15689188C2213yes—
47(C22⋊D47)⋊C8C2216yes—
48#15689185C2216yes—
49#15689181C2216yes—
50#15689179C2213yes—
51#15689176C2213yes—
52#15689175C2213yes—
53#15689145C2213yes—
54C4⋊C8×D47C2213yes—
55#15689088C2216yes—
56#15689081C2213yes—
57#15689081C2213yes—
58#15689080C2213yes—
59#15689080C2211yescharacteristic
60#15689080C2213yes—
61#15689080C2213yes—
62C4×#10625324C2213yes—
63C4×#10625324C2213yes—
64C2×C4×C47⋊C8C2213yes—
65C4×C8×D47C2213yes—
66C4×C8×D47C2213yes—
67C4×#41×C47C2211yescharacteristic
68C8×Q8×C47C2211yescharacteristic
69C8×Q8×C47C2211yescharacteristic
70C47×#436C2213yes—
71C47×#436C2213yes—
72C47×#432C2213yes—
73C4×M16×C47C2213yes—
74C2×#10625354C2×C411yescharacteristic
75C2×#10625350C2313yes—
76C2×Q8×D47C2×C411yescharacteristic
77#13094032—212no—
78#13094032—212no—
79C2×#10625324C2313yes—
80C2×#10625324C2313yes—
81C2×#10625324C2313yes—
82C2×#10625324C2311yescharacteristic
83C22×C47⋊C8C2313yes—
84C22×C47⋊C8C2313yes—
85C2×C8×D47C2313yes—
86C2×C8×D47C2313yes—
87C2×C8×D47C2313yes—
88C2×C8×D47C2311yescharacteristic
89#13094023C2×C413yes—
90#13094020C2×C413yes—
91#13094018C2×C416yes—
92#13094017C2×C416yes—
93#13094016C2×C413yes—
94#13094015C2×C413yes—
95#13094012C2×C413yes—
96#13094011C2×C413yes—
97#13094010C2×C416yes—
98#13094007C2×C416yes—
99#13094006C2×C416yes—
100#13094004C2×C413yes—
101#13094003C2×C413yes—
102#13093999C2×C413yes—
103#13093998C2×C413yes—
104#13093997C2×C416yes—
105#13093995C2×C413yes—
106#13093995C2×C413yes—
107Q8×Dic47C2×C411yescharacteristic
108Q8×Dic47C2×C411yescharacteristic
109C4×Dic94C2×C413yes—
110C4×Dic94C2×C413yes—
111C4×Dic94C2313yes—
112#13093989C2×C413yes—
113C4×D188C2×C413yes—
114C4×D188C2×C413yes—
115C4×D188C2313yes—
116#13093986C2×C413yes—
117C4×C22⋊D47C2×C413yes—
118C4×C22⋊D47C2×C413yes—
119C4×C22⋊D47C2×C413yes—
120C4×C22⋊D47C2×C413yes—
121C4×C22⋊D47C2316yes—
122#13093983C2313yes—
123#13093982C2×C413yes—
124#13093981C2311yescharacteristic
125#13093981C2313yes—
126C4⋊C4×D47C2×C413yes—
127C42×D47C2313yes—
128C42×D47C2313yes—
129C4⋊(C47⋊C8)C2313yes—
130C4⋊(C47⋊C8)C2313yes—
131Dic47⋊C8C2316yes—
132Dic47⋊C8C2316yes—
133C22⋊(C47⋊C8)C2316yes—
134(C2×D47)⋊C8C2316yes—
135(C2×D47)⋊C8C2316yes—
136#13093934C2313yes—
137#13093934C2313yes—
138#13093934C2311yescharacteristic
139#13093934C2313yes—
140C4×C47⋊C8C2313yes—
141C4×C47⋊C8C2313yes—
142C4×C47⋊C8C2311yescharacteristic
143C4×C47⋊C8C2313yes—
144C8×Dic47C2313yes—
145C8×Dic47C2313yes—
146C8×Dic47C2311yescharacteristic
147C8×Dic47C2313yes—
148C2×#41×C47C2×C411yescharacteristic
149C2×M16×C47C2313yes—
150C2×M16×C47C2313yes—
151C4×Q8×C47C2×C411yescharacteristic
152C4×Q8×C47C2311yescharacteristic
153C4×D4×C47C2×C413yes—
154C4×D4×C47C2313yes—
155#117×C47C2×C413yes—
156C2×C42×C47C2313yes—
157C4⋊C8×C47C2313yes—
158C4⋊C8×C47C2313yes—
159C22⋊C8×C47C2316yes—
160#97×C47C2311yescharacteristic
161#97×C47C2313yes—
162C4×C8×C47C2313yes—
163C4×C8×C47C2313yes—
164#41×C47—24no—
165C2×Q8×C47C22×C411yescharacteristic
166C2×D4×C47C22×C413yes—
167C22×C4×C47C22×C413yes—
168C22×C4×C47C22×C413yes—
169C22×C4×C47C2413yes—
170C2×C22⋊D47C22×C413yes—
171C2×C22⋊D47C22×C413yes—
172C2×C22⋊D47C22×C416yes—
173#10625354—24no—
174Q8×D47—24no—
175#10625350—26no—
176#10625350—26no—
177C2×D188C22×C413yes—
178C2×D188C22×C413yes—
179C2×C4×D47C22×C413yes—
180C2×C4×D47C22×C413yes—
181C2×C4×D47C22×C413yes—
182C2×C4×D47C22×C413yes—
183C2×C4×D47C2413yes—
184C2×C4×D47C2411yescharacteristic
185C2×Dic94C22×C413yes—
186C2×Dic94C22×C413yes—
187M16×C47—26no—
188M16×C47—26no—
189C2×C8×C47C2413yes—
190C2×C8×C47C2413yes—
191C2×C8×C47C2411yescharacteristic
192C2×C8×C47C2411yescharacteristic
193C4⋊C4×C47C22×C413yes—
194C4⋊C4×C47C22×C413yes—
195C22⋊C4×C47C22×C416yes—
196C42×C47C22×C413yes—
197C42×C47C22×C413yes—
198C42×C47C2413yes—
199C42×C47C2411yescharacteristic
200C22⋊Dic47C22×C413yes—
201C22⋊Dic47C22×C413yes—
202C22⋊Dic47C22×C416yes—
203(C2×D47)⋊C4C22×C411yescharacteristic
204(C2×D47)⋊C4C22×C411yescharacteristic
205(C2×D47)⋊C4C22×C413yes—
206(C2×D47)⋊C4C22×C413yes—
207(C2×D47)⋊C4C22×C416yes—
208(C2×D47)⋊C4C22×C416yes—
209C4⋊Dic47C22×C413yes—
210C4⋊Dic47C22×C413yes—
211C4⋊Dic47C22×C413yes—
212C4⋊Dic47C22×C413yes—
213Dic47⋊C4C22×C416yes—
214Dic47⋊C4C22×C411yescharacteristic
215Dic47⋊C4C22×C411yescharacteristic
216Dic47⋊C4C22×C413yes—
217Dic47⋊C4C22×C413yes—
218Dic47⋊C4C22×C416yes—
219C4×Dic47C22×C413yes—
220C4×Dic47C22×C413yes—
221C4×Dic47C22×C413yes—
222C4×Dic47C22×C413yes—
223C4×Dic47C2411yescharacteristic
224C4×Dic47C2413yes—
225C2×C47⋊C8—26no—
226C2×C47⋊C8—26no—
227C2×C47⋊C8C2413yes—
228C2×C47⋊C8C2413yes—
229C2×C47⋊C8C2411yescharacteristic
230C2×C47⋊C8C2411yescharacteristic
231#10625324—26no—
232#10625324—26no—
233#10625324—26no—
234#10625324#4114yescomplemented
235C8×D47—26no—
236C8×D47—26no—
237C8×D47—26no—
238C8×D47#4114yescomplemented
239C23×C47C23×C413yes—
240C22×D47C23×C413yes—
241C22×D47C23×C411yescharacteristic
242Q8×C47#13114yescomplemented
243D4×C47—26no—
244C2×C4×C47—26no—
245C2×C4×C47—23no—
246C2×C4×C47—23no—
247C2×C4×C47—43no—
248C2×C4×C47C23×C413yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C23×C413yes—
252C2×C4×C47C23×C413yes—
253C2×C4×C47C2511yescharacteristic
254C22⋊D47—26no—
255C22⋊D47—26no—
256C22⋊D47—46no—
257C2×Dic47C23×C411yescharacteristic
258C2×Dic47C23×C411yescharacteristic
259C2×Dic47C23×C413yes—
260C2×Dic47C23×C413yes—
261C2×Dic47C23×C413yes—
262C2×Dic47C23×C411yescharacteristic
263D188—26no—
264D188—26no—
265C4×D47—26no—
266C4×D47—26no—
267C4×D47—23no—
268C4×D47—23no—
269C4×D47—43no—
270C4×D47—43no—
271C4×D47C2×#4112yes—
272C4×D47C2×#4112yes—
273Dic94—26no—
274Dic94—26no—
275C8×C47—23no—
276C8×C47—23no—
277C8×C47C2×#4112yes—
278C8×C47C2×#4112yes—
279C47⋊C8—23no—
280C47⋊C8—23no—
281C47⋊C8C2×#4112yes—
282C47⋊C8C2×#4112yes—
283#60218—471nomaximal, Sylow
284C22×C47—23no—
285C22×C47—23no—
286C22×C47—23no—
287C22×C47C24×C411yescharacteristic
288C2×D47—21no—
289C2×D47—21no—
290C2×D47—23no—
291C2×D47—23no—
292C2×D47—23no—
293C2×D47C4×#4112yes—
294C4×C47—23no—
295C4×C47—21no—
296C4×C47—23no—
297C4×C47—21no—
298C4×C47C2×#13116yes—
299C4×C47C22×#4111yescharacteristic
300C4×C47C22×#4111yescharacteristic
301Dic47—23no—
302Dic47—23no—
303Dic47—21no—
304Dic47—21no—
305Dic47—23no—
306Dic47C4×#4112yes—
307#3480—471no—
308#2971—471no—
309#2971D4711yescharacteristic, complemented
310#2969—473no—
311#2969—473no—
312#2966—473no—
313#2966—473no—
314C8×#41—471no—
315C8×#41—471no—
316Q8×M16—471no—
317C2×C8×Q8—471no—
318#2935—473no—
319#2931—473no—
320#2925—473no—
321C4×#131—473no—
322C2×C47—43no—
323C2×C47C2×C4×#4111yescharacteristic
324C2×C47C22×#13111yescharacteristic
325C2×C47#357311yescharacteristic
326D47—22no—
327D47—43no—
328C2×#131—473no—
329C2×#131—473no—
330#559—473no—
331#557—471no—
332#556—473no—
333#553—473no—
334#553—473no—
335#550—471no—
336#548—471no—
337#548—471no—
338#532—473no—
339#529—473no—
340#529—473no—
341C4×#41—471no—
342C4×#41—473no—
343C4×#41C2×D4711yescharacteristic
344C2×C4×Q8—471no—
345#446—473no—
346#446—473no—
347C8×Q8—471no—
348C8×Q8—471no—
349C8×Q8—473no—
350C8×Q8—473no—
351C8×Q8C2×D4711yescharacteristic
352C8×Q8C2×D4711yescharacteristic
353#436—476no—
354#436—473no—
355#436—473no—
356#436C2×D4713yes—
357#436C2×D4713yes—
358#435—476no—
359#435—476no—
360C8×D4—473no—
361C8×D4—473no—
362C8×D4—473no—
363C8×D4—476no—
364C8×D4—473no—
365#433—476no—
366#433—476no—
367#432—473no—
368#432C2×D4713yes—
369#431—473no—
370#431—473no—
371#423—473no—
372C2×C4⋊C8—473no—
373#408—476no—
374#405—473no—
375#405—473no—
376#405—471no—
377#405—473no—
378#405—473no—
379#405—473no—
380C4×M16—473no—
381C4×M16—473no—
382C4×M16C2×D4713yes—
383C2×C4×C8—473no—
384C2×C4×C8—473no—
385C2×C4×C8—473no—
386C47#6021811yescharacteristic, complemented, Sylow
387C2×#41—471no—
388C2×#41—473no—
389C2×#41C4×D4711yescharacteristic
390C22×Q8—471no—
391#131—9412no—
392#131—9412no—
393C2×M16—471no—
394C2×M16—473no—
395C2×M16—473no—
396C2×M16—473no—
397C2×M16C22×D4713yes—
398C2×M16C22×D4713yes—
399C22×C8—471no—
400C22×C8—473no—
401C22×C8—473no—
402C22×C8—473no—
403C22×C8—473no—
404C22×C8—473no—
405C4⋊Q8—473no—
406#127—473no—
407#126—476no—
408#126—476no—
409#125—473no—
410#125—473no—
411#124—473no—
412#124—476no—
413#124—473no—
414#123—476no—
415#123—476no—
416C22⋊Q8—473no—
417C22⋊Q8—473no—
418#121—473no—
419#121—473no—
420#121—473no—
421#121—473no—
422#121—476no—
423C4×Q8—471no—
424C4×Q8—471no—
425C4×Q8—473no—
426C4×Q8—473no—
427C4×Q8—473no—
428C4×Q8—473no—
429C4×Q8C4×D4711yescharacteristic
430C4×Q8C22×D4711yescharacteristic
431C4×D4—473no—
432C4×D4—473no—
433C4×D4—473no—
434C4×D4—473no—
435C4×D4—476no—
436C4×D4—473no—
437C4×D4—473no—
438C4×D4—473no—
439C4×D4—473no—
440C4×D4C4×D4713yes—
441C4×D4C22×D4713yes—
442#117—473no—
443#117—473no—
444#117—471no—
445#117—473no—
446#117C4×D4713yes—
447C2×C4⋊C4—473no—
448C2×C42—473no—
449C2×C42—473no—
450C2×C42C22×D4713yes—
451C4⋊C8—476no—
452C4⋊C8—476no—
453C4⋊C8—473no—
454C4⋊C8—473no—
455C4⋊C8C22×D4713yes—
456C4⋊C8C22×D4713yes—
457C22⋊C8—476no—
458C22⋊C8—476no—
459C22⋊C8—476no—
460C22⋊C8C22×D4716yes—
461#97—471no—
462#97—473no—
463#97—473no—
464#97—473no—
465#97C22×D4711yescharacteristic
466#97C22×D4713yes—
467C4×C8—471no—
468C4×C8—473no—
469C4×C8—473no—
470C4×C8—473no—
471C4×C8—471no—
472C4×C8—473no—
473C4×C8—473no—
474C4×C8—473no—
475C4×C8C22×D4713yes—
476C4×C8C22×D4713yes—
477#41—24no—
478#41—946no—
479#41—946no—
480#41—944no—
481C2×Q8—473no—
482C2×Q8—473no—
483C2×Q8—944no—
484C2×Q8C2×C4×D4711yescharacteristic
485C2×D4—473no—
486C2×D4—473no—
487C2×D4—476no—
488C2×D4—473no—
489C2×D4—473no—
490C2×D4C2×C4×D4713yes—
491C22×C4—471no—
492C22×C4—473no—
493C22×C4—473no—
494C22×C4—473no—
495C22×C4—473no—
496C22×C4—473no—
497C22×C4C2×C4×D4713yes—
498C22×C4C2×C4×D4713yes—
499C22×C4C23×D4713yes—
500M16—26no—
501M16—26no—
502M16—474no—
503M16—946no—
504M16—946no—
505M16—946no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—474no—
509C2×C8—473no—
510C2×C8—473no—
511C2×C8—946no—
512C2×C8—946no—
513C2×C8—946no—
514C2×C8—946no—
515C2×C8—946no—
516C2×C8C23×D4713yes—
517C2×C8C23×D4713yes—
518C2×C8C23×D4711yescharacteristic
519C2×C8C23×D4711yescharacteristic
520C4⋊C4—476no—
521C4⋊C4—471no—
522C4⋊C4—471no—
523C4⋊C4—473no—
524C4⋊C4—473no—
525C4⋊C4—476no—
526C4⋊C4—473no—
527C4⋊C4—473no—
528C4⋊C4—473no—
529C4⋊C4—473no—
530C4⋊C4C2×C4×D4713yes—
531C4⋊C4C2×C4×D4713yes—
532C22⋊C4—473no—
533C22⋊C4—473no—
534C22⋊C4—476no—
535C22⋊C4—471no—
536C22⋊C4—471no—
537C22⋊C4—473no—
538C22⋊C4—476no—
539C22⋊C4—473no—
540C22⋊C4—476no—
541C22⋊C4C2×C4×D4716yes—
542C42—473no—
543C42—473no—
544C42—473no—
545C42—473no—
546C42—471no—
547C42—473no—
548C42C2×C4×D4713yes—
549C42C2×C4×D4713yes—
550C42C23×D4711yescharacteristic
551C42C23×D4713yes—
552C23—471no—
553C23—473no—
554C23C22×C4×D4713yes—
555Q8—946no—
556Q8—946no—
557Q8#1309403214yescomplemented
558D4—26no—
559D4—946no—
560D4—946no—
561D4—946no—
562D4—946no—
563D4—1886no—
564C2×C4—26no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—43no—
568C2×C4—471no—
569C2×C4—472no—
570C2×C4—472no—
571C2×C4—471no—
572C2×C4—471no—
573C2×C4—473no—
574C2×C4—473no—
575C2×C4—473no—
576C2×C4—946no—
577C2×C4—946no—
578C2×C4—943no—
579C2×C4—943no—
580C2×C4—1883no—
581C2×C4—1883no—
582C2×C4C22×C4×D4713yes—
583C2×C4C22×C4×D4711yescharacteristic
584C2×C4C22×C4×D4711yescharacteristic
585C2×C4C22×C4×D4713yes—
586C2×C4C22×C4×D4713yes—
587C2×C4C24×D4711yescharacteristic
588C8—23no—
589C8—23no—
590C8—472no—
591C8—472no—
592C8—943no—
593C8—943no—
594C8#41×D4712yes—
595C8#41×D4712yes—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—472no—
600C22—941no—
601C22—941no—
602C22—943no—
603C22—943no—
604C22—943no—
605C22C23×C4×D4711yescharacteristic
606C4—21no—
607C4—23no—
608C4—21no—
609C4—23no—
610C4—472no—
611C4—941no—
612C4—943no—
613C4—943no—
614C4—941no—
615C4—943no—
616C4C2×#1309403216yes—
617C4C2×#41×D4711yescharacteristic
618C4C2×#41×D4711yescharacteristic
619C2—43no—
620C2—942no—
621C2—1883no—
622C2C4×#41×D4711yescharacteristic
623C2C22×#1309403211yescharacteristic
624C2#2025810111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2971C2
prime radical#20256986C2
cosocleC2×C4×C47C25
hypercenter#2971D47
hypocenterC47#60218
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #20252407C2semidirect
3. #20252395C2semidirect
4. #20252392C2semidirect
5. #20252366C2semidirect
6. #20252357C2semidirect
7. #41×C47⋊C8C2semidirect
8. C8×#10625354C2semidirect
9. Q8×#10625324C2semidirect
10. C8×Q8×D47C2semidirect
11. #20252133C2semidirect
12. #20252097C2semidirect
13. #20252055C2semidirect
14. C4×#13094032C2semidirect
15. C47×#2971C2semidirect
234. #10625324#41semidirect
238. C8×D47#41semidirect
242. Q8×C47#131semidirect
309. #2971D47semidirect
386. C47#60218semidirect
557. Q8#13094032semidirect

Maximal subgroups

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

subgroupindex
#202569862
#202524072
#202523952
#202523922
#202523662
#202523572
#41×C47⋊C82
C8×#106253542
Q8×#106253242
C8×Q8×D472
#202521332
#202520972
#202520552
C4×#130940322
C47×#29712
#6021847