Finite Group Databasebeta

← #26219356

Subgroups — #26219356

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#20246040C211yesmaximal, characteristic, complemented
3#20245750C211yesmaximal, characteristic
4#20244030C211yesmaximal, characteristic, complemented
5#20244023C211yesmaximal, characteristic, complemented
6#20243981C211yesmaximal, characteristic, complemented
7#20243960C211yesmaximal, characteristic, complemented
8#20243721C211yesmaximal, characteristic
9#20243721C211yesmaximal, characteristic
10#20243652C211yesmaximal, characteristic
11#20243638C211yesmaximal, characteristic
12#20243285C211yesmaximal, characteristic, complemented
13C4×#13093949C211yesmaximal, characteristic, complemented
14C4×#13093946C211yesmaximal, characteristic
15C47×#1746C211yesmaximal, characteristic, complemented
16C22×#10625325C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21#15689225C418yescomplemented
22C2×#13093958C2211yescharacteristic
23C2×#13093958C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2211yescharacteristic
27C2×#13093949C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29C2×#13093946C2211yescharacteristic
30C2×#13093946C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689029C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689005—21no—
36#15689005—21no—
37#15688990—21no—
38#15688987—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43#15688970—21no—
44#15688970—21no—
45#15688970—21no—
46#95×D47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53C2×#97×C47C2211yescharacteristic
54C2×C4×C8×C47C2211yescharacteristic
55C47×#376C2211yescharacteristic
56C47×#336C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C47×#336C2211yescharacteristic
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—
68C4×Dic94—22no—
69C4×Dic94C2×C414yescomplemented
70C4×D188—22no—
71C4×D188—22no—
72C4×D188—22no—
73C4×D188—22no—
74C4×D188—22no—
75C4×D188C2×C414yes—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93#13093958—22no—
94#13093958C2×C414yescomplemented
95#13093956—22no—
96#13093949—22no—
97#13093949—22no—
98#13093949C2×C414yescomplemented
99#13093946—22no—
100#13093946—22no—
101#13093946C2×C414yes—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115C22×C8×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119#97×C47C2×C414yescomplemented
120C4×C8×C47—22no—
121#95×C47D411yescharacteristic
122#95×C47D411yescharacteristic
123#95×C47D411yescharacteristic
124#95×C47D411yescharacteristic
125C23×D47—21no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
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×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47C22×C412yes—
172C2×C8×C47C22×C412yes—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
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⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47C22×C412yes—
191C4⋊Dic47C22×C412yes—
192C4⋊Dic47#4112yes—
193C4⋊Dic47#4112yes—
194Dic47⋊C4—24no—
195Dic47⋊C4—81no—
196Dic47⋊C4—81no—
197C4×Dic47—24no—
198C4×Dic47—81no—
199C4×Dic47—81no—
200C4×Dic47—81no—
201C4×Dic47—81no—
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—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
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—22no—
239C2×Dic47—22no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
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—
269C8×C47—41no—
270C8×C47—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—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47—41no—
292C4×C47C2×#11711yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#52911yescharacteristic
295C4×C47#52911yescharacteristic
296Dic47—42no—
297Dic47—42no—
298Dic47—42no—
299Dic47—42no—
300Dic47—42no—
301Dic47—81no—
302Dic47—81no—
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#1776—471no—
315C4×#103—471no—
316C4×#102—471no—
317#1746D4711yescharacteristic, complemented
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×#107—471no—
334C2×#106—471no—
335C2×#103—471no—
336C2×#103—471no—
337C2×#103—471no—
338C2×#102—471no—
339C2×#102—471no—
340C2×#102—471no—
341C2×#97C2×D4711yescharacteristic
342C2×C4×C8C2×D4711yescharacteristic
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⋊C4—471no—
354C4×C4⋊C4—471no—
355C4×C22⋊C4—941no—
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358#376C2×D4711yescharacteristic
359C2×#95—941no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#336C2×D4711yescharacteristic
367#336C2×D4711yescharacteristic
368#336C2×D4711yescharacteristic
369#336C2×D4711yescharacteristic
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×C8C22×D4711yescharacteristic
380C22×C8C22×D4711yescharacteristic
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—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42—941no—
408C2×C42C22×D4711yescharacteristic
409C2×C42C22×D4711yescharacteristic
410C2×C42C22×D4711yescharacteristic
411#107—474no—
412#107—942no—
413#106—942no—
414#103—474no—
415#103—942no—
416#103—942no—
417#102—474no—
418#102—942no—
419#102—942no—
420#97C4×D4714yescomplemented
421C4×C8—22no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95D18811yescharacteristic
433#95D18811yescharacteristic
434#95D18811yescharacteristic
435#95D18811yescharacteristic
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—944no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—944no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—3761no—
460C22×C4—3761no—
461C22×C4—3761no—
462C22×C4—3761no—
463C22×C4—3761no—
464C22×C4—3761no—
465C22×C4C2×D18811yescharacteristic
466C22×C4C2×D18811yescharacteristic
467C22×C4C2×D18811yescharacteristic
468C22×C4C2×D18811yescharacteristic
469C22×C4C2×D18811yescharacteristic
470C22×C4C2×D18811yescharacteristic
471C22×C4C23×D4711yescharacteristic
472SD16—948no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8C2×C4×D4712yes—
484C2×C8C2×C4×D4712yes—
485C4⋊C4—472no—
486C4⋊C4—472no—
487C4⋊C4—472no—
488C4⋊C4—472no—
489C4⋊C4—944no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—3761no—
499C4⋊C4—3761no—
500C22⋊C4—944no—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—944no—
508C42—3761no—
509C42—3761no—
510C42—3761no—
511C42—3761no—
512C42C2×C4×D4712yes—
513C42#1062535012yes—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23C22×D18811yescharacteristic
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—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—942no—
550C2×C4—942no—
551C2×C4—3764no—
552C2×C4—3764no—
553C2×C4—3764no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4C4×D18812yes—
571C2×C4C4×D18812yes—
572C2×C4C22×C4×D4711yescharacteristic
573C2×C4#41×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4#41×D4711yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4C2×#1062535011yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—944no—
583C22—1882no—
584C22—1882no—
585C22—1882no—
586C22—1882no—
587C22—1882no—
588C22—1882no—
589C22C2×C4×D18811yescharacteristic
590C22#1568963511yescharacteristic
591C22#1568963511yescharacteristic
592C22#1568992811yescharacteristic
593C22#1568993511yescharacteristic
594C22#1568993511yescharacteristic
595C22#1568994011yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—3761no—
603C4—3761no—
604C4—1882no—
605C4—1882no—
606C4—1882no—
607C4—1882no—
608C4—1882no—
609C4#117×D4711yescharacteristic
610C4#1568959111yescharacteristic
611C4#1568965111yescharacteristic
612C4#1568965111yescharacteristic
613C2—1884no—
614C2#2024583211yescharacteristic
615C2#2025222211yescharacteristic
616C2#2025223311yescharacteristic
617C2#2025357411yescharacteristic
618C2#2025360411yescharacteristic
619C2#2025364111yescharacteristic
620C2#2025365211yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1746C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1746D47
hypocenterC47#21125
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #20246040C2semidirect
4. #20244030C2semidirect
5. #20244023C2semidirect
6. #20243981C2semidirect
7. #20243960C2semidirect
12. #20243285C2semidirect
13. C4×#13093949C2semidirect
15. C47×#1746C2semidirect
21. #15689225C4semidirect
69. C4×Dic94C2×C4semidirect
94. #13093958C2×C4semidirect
98. #13093949C2×C4semidirect
119. #97×C47C2×C4semidirect
317. #1746D47semidirect
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
#202460402
#202457502
#202440302
#202440232
#202439812
#202439602
#202437212
#202436522
#202436382
#202432852
C4×#130939492
C4×#130939462
C47×#17462
#2112547