Finite Group Databasebeta

← #26319509

Subgroups — #26319509

Order 12032.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246040C211yesmaximal, characteristic, complemented
2#20244890C211yesmaximal, characteristic, complemented
3#20244766C211yesmaximal, characteristic, complemented
4#20244762C211yesmaximal, characteristic
5#20244705C211yesmaximal, characteristic
6#20244241C211yesmaximal, characteristic
7#20244113C211yesmaximal, characteristic, complemented
8#20244088C211yesmaximal, characteristic
9#20244077C211yesmaximal, characteristic, complemented
10#20244012C211yesmaximal, characteristic
11#20243970C211yesmaximal, characteristic, complemented
12#20243746C211yesmaximal, characteristic
13#20243655C211yesmaximal, characteristic, complemented
14#20243245C211yesmaximal, characteristic, complemented
15C47×#2644C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×#13093955C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689029C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689005—21no—
36#15689005—21no—
37#15688990—21no—
38#15688987—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C47×#397—21no—
49C47×#397—21no—
50C47×#397—21no—
51C47×#396—21no—
52C47×#390C2211yescharacteristic
53C47×#390C2211yescharacteristic
54C47×#390C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#389C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×Dic94C2311yescharacteristic
60C2×#10625335—22no—
61C2×#10625335—22no—
62C2×#10625335—41no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C4×Dic94—22no—
68C4×Dic94—22no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47—21no—
77#13093955—22no—
78#13093955—22no—
79#13093955—41no—
80C4⋊(C47⋊C8)—22no—
81C4⋊(C47⋊C8)—22no—
82#13093947—22no—
83#13093947—22no—
84#13093947—41no—
85#13093945—22no—
86#13093945—22no—
87#13093945—41no—
88#13093935—41no—
89C4×C47⋊C8—41no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×D4×C47C2311yescharacteristic
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47C2311yescharacteristic
111C2×C42×C47C2311yescharacteristic
112C2×C42×C47C2311yescharacteristic
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47D411yescharacteristic
117#95×C47D411yescharacteristic
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2411yescharacteristic
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94#4112yes—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—81no—
154C4⋊C4×C47—81no—
155C4⋊C4×C47—81no—
156C22⋊C4×C47—81no—
157C22⋊C4×C47—81no—
158C22⋊C4×C47—81no—
159C22⋊C4×C47—81no—
160C22⋊C4×C47—81no—
161C22⋊C4×C47—81no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168#10625335—44no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47#4112yes—
175Dic47⋊C4—24no—
176Dic47⋊C4—81no—
177Dic47⋊C4—81no—
178C4×Dic47—24no—
179C4×Dic47—81no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8#4112yes—
191C2×C47⋊C8#4112yes—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47C22×D411yescharacteristic
200D4×C47—41no—
201D4×C47—41no—
202D4×C47—41no—
203D4×C47—41no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
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—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47H(5,𝔽2)11yescharacteristic
236C2×C4×C472−1+411yescharacteristic
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—41no—
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—
249Dic94—22no—
250Dic94—22no—
251Dic94—41no—
252Dic94—41no—
253C47⋊C8—22no—
254C47⋊C8—22no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257#29442—471nomaximal, Sylow
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47#52511yescharacteristic
266C22×C47#53211yescharacteristic
267C22×C47#53711yescharacteristic
268C22×C47#57711yescharacteristic
269C22×C47#57711yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57711yescharacteristic
272C4×C47—41no—
273C4×C47—41no—
274C4×C47—41no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47#55111yescharacteristic
282C4×C47#55211yescharacteristic
283C4×C47#55311yescharacteristic
284C4×C47#55611yescharacteristic
285Dic47—42no—
286Dic47—42no—
287Dic47—81no—
288Dic47—81no—
289#2644D4711yescharacteristic
290#2320—471no—
291#2088—471no—
292#2066—471no—
293#2065—471no—
294#2055—471no—
295#1968—471no—
296#1941—471no—
297#1936—471no—
298#1935—471no—
299#1926—471no—
300#1920—471no—
301#1880—471no—
302#1865—471no—
303#1768—471no—
304C2×C47—161no—
305C2×C47#260111yescharacteristic
306C2×C47#312511yescharacteristic
307C2×C47#312611yescharacteristic
308C2×C47#317311yescharacteristic
309C2×C47#317411yescharacteristic
310C2×C47#326011yescharacteristic
311C2×C47#326311yescharacteristic
312C22×SD16—471no—
313C2×C4×Q8—471no—
314C2×#106—471no—
315C2×#106—471no—
316C2×#106—471no—
317C2×C4⋊C8—471no—
318C2×C4⋊C8—471no—
319C2×#103—471no—
320C2×#103—471no—
321C2×#103—471no—
322C2×#102—471no—
323C2×#102—471no—
324C2×#102—471no—
325C2×#97—471no—
326C2×C4×C8—471no—
327#397—21no—
328#397—21no—
329#397—21no—
330#396—21no—
331#391—471no—
332#391—471no—
333#390C2×D4711yescharacteristic
334#390C2×D4711yescharacteristic
335#390C2×D4711yescharacteristic
336#389—471no—
337#389—471no—
338#389C2×D4711yescharacteristic
339#389C2×D4711yescharacteristic
340#389C2×D4711yescharacteristic
341#387—941no—
342#387—941no—
343#383—941no—
344#382—941no—
345C4×C4⋊C4—471no—
346#376C2×D4711yescharacteristic
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#336—471no—
354#336—471no—
355C47#2944211yescharacteristic, complemented, Sylow
356C22×Q8—471no—
357C22×D4C22×D4711yescharacteristic
358C2×SD16—942no—
359C2×SD16—942no—
360C2×SD16—1881no—
361C22×C8—471no—
362C22×C8—471no—
363C22×C8—471no—
364C22×C8—471no—
365C4×Q8—942no—
366C4×Q8—942no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—941no—
386C2×C42—941no—
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389C2×C42C22×D4711yescharacteristic
390#106—942no—
391#106—942no—
392#106—1881no—
393C4⋊C8—942no—
394C4⋊C8—942no—
395#103—942no—
396#103—942no—
397#103—1881no—
398#102—942no—
399#102—942no—
400#102—1881no—
401#97—1881no—
402C4×C8—1881no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95C22⋊D4711yescharacteristic
414#95C22⋊D4711yescharacteristic
415#95C22⋊D4711yescharacteristic
416#95C22⋊D4711yescharacteristic
417C24—21no—
418C2×Q8—472no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C23×D4711yescharacteristic
444SD16—1884no—
445C2×C8—472no—
446C2×C8—472no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—472no—
467C4⋊C4—944no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—3761no—
474C4⋊C4—3761no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—944no—
488C42—3761no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×C22⋊D4711yescharacteristic
497Q8—942no—
498Q8—942no—
499Q8—1881no—
500Q8—1881no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4#41×D4711yescharacteristic
545C2×C4#41×D4711yescharacteristic
546C2×C4#41×D4711yescharacteristic
547C2×C4#41×D4711yescharacteristic
548C2×C4#1309407011yescharacteristic
549C2×C4#1309407111yescharacteristic
550C8—942no—
551C8—942no—
552C8—1881no—
553C8—1881no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22#1568963011yescharacteristic
562C22#1568966911yescharacteristic
563C22#1568970111yescharacteristic
564C22#1568993111yescharacteristic
565C22#1568993411yescharacteristic
566C22#1568993411yescharacteristic
567C22#1568993411yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—3761no—
578C4—3761no—
579C4—1882no—
580C4—1882no—
581C4#1568979811yescharacteristic
582C4#1568980911yescharacteristic
583C4#1568982411yescharacteristic
584C4#1568984011yescharacteristic
585C2—161no—
586C2#2024898711yescharacteristic
587C2#2025365911yescharacteristic
588C2#2025367411yescharacteristic
589C2#2025399911yescharacteristic
590C2#2025401011yescharacteristic
591C2#2025483411yescharacteristic
592C2#2025485311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2644C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2644D47
hypocenterC47#29442
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2644

Complements

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

NH ≅ G/Ntype
1. #20246040C2semidirect
2. #20244890C2semidirect
3. #20244766C2semidirect
7. #20244113C2semidirect
9. #20244077C2semidirect
11. #20243970C2semidirect
13. #20243655C2semidirect
14. #20243245C2semidirect
355. C47#29442semidirect

Maximal subgroups

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

subgroupindex
#202460402
#202448902
#202447662
#202447622
#202447052
#202442412
#202441132
#202440882
#202440772
#202440122
#202439702
#202437462
#202436552
#202432452
C47×#26442
#2944247