Finite Group Databasebeta

← #26319477

Subgroups — #26319477

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249353C211yesmaximal, 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#20244022C211yesmaximal, characteristic, complemented
11#20243966C211yesmaximal, characteristic, complemented
12#20243722C211yesmaximal, characteristic
13#20243649C211yesmaximal, characteristic
14#20243245C211yesmaximal, characteristic, complemented
15C47×#2319C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×#13093955C2211yescharacteristic
18C2×#13093955C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093947C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C47⋊#401—21no—
31#15689064—21no—
32#15689064—21no—
33#15689064—21no—
34#15689029C2211yescharacteristic
35#15689029C2211yescharacteristic
36#15689029C2211yescharacteristic
37#15689021C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4×D4×C47C2211yescharacteristic
49C47×#390C2211yescharacteristic
50C47×#390C2211yescharacteristic
51C47×#389C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#388—21no—
54C47×#388—21no—
55C47×#381—21no—
56C47×#380—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
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
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73#13093955—22no—
74#13093955—22no—
75#13093955—41no—
76C4⋊(C47⋊C8)—22no—
77C4⋊(C47⋊C8)—22no—
78#13093947—22no—
79#13093947—22no—
80#13093947—41no—
81#13093945—22no—
82#13093945—22no—
83#13093945—41no—
84#13093935—41no—
85C4×C47⋊C8—41no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×D4×C47C2311yescharacteristic
96C23×C4×C47—21no—
97C4×D4×C47—22no—
98C4×D4×C47—22no—
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×C42×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#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—81no—
132C22×C4×C47—81no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—81no—
156C4⋊C4×C47—81no—
157C4⋊C4×C47#4112yes—
158C22⋊C4×C47—24no—
159C22⋊C4×C47—81no—
160C22⋊C4×C47—81no—
161C22⋊C4×C47—81no—
162C22⋊C4×C47—81no—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47—21no—
169C42×C47—81no—
170#10625335—44no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177Dic47⋊C4—81no—
178Dic47⋊C4—81no—
179Dic47⋊C4—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—22no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47C22×D411yescharacteristic
200D4×C47—22no—
201D4×C47—22no—
202D4×C47—41no—
203D4×C47—41no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
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—21no—
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×C47—84no—
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47H(5,𝔽2)11yescharacteristic
237C2×C4×C472−1+411yescharacteristic
238C2×Dic47—41no—
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—
249C2×Dic47—41no—
250Dic94—41no—
251Dic94—41no—
252Dic94—41no—
253Dic94—41no—
254C47⋊C8—22no—
255C47⋊C8—22no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258#29440—471nomaximal, Sylow
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47—81no—
266C22×C47#52511yescharacteristic
267C22×C47#53211yescharacteristic
268C22×C47#53711yescharacteristic
269C22×C47#57711yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57711yescharacteristic
273C4×C47—41no—
274C4×C47—42no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—41no—
279C4×C47—41no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47#55111yescharacteristic
283C4×C47#55211yescharacteristic
284C4×C47#55311yescharacteristic
285C4×C47#55611yescharacteristic
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289Dic47—81no—
290#2645—471no—
291#2319D4711yescharacteristic
292#2088—471no—
293#2066—471no—
294#2065—471no—
295#2055—471no—
296#1968—471no—
297#1941—471no—
298#1936—471no—
299#1935—471no—
300#1927—471no—
301#1919—471no—
302#1877—471no—
303#1864—471no—
304#1768—471no—
305C2×C47—82no—
306C2×C47#260111yescharacteristic
307C2×C47#312311yescharacteristic
308C2×C47#312511yescharacteristic
309C2×C47#317311yescharacteristic
310C2×C47#317411yescharacteristic
311C2×C47#326011yescharacteristic
312C2×C47#326311yescharacteristic
313C22×SD16—471no—
314C2×C4×D4C2×D4711yescharacteristic
315C2×#106—471no—
316C2×#106—471no—
317C2×#106—471no—
318C2×C4⋊C8—471no—
319C2×C4⋊C8—471no—
320C2×#103—471no—
321C2×#103—471no—
322C2×#103—471no—
323C2×#102—471no—
324C2×#102—471no—
325C2×#102—471no—
326C2×#97—471no—
327C2×C4×C8—471no—
328#401—941no—
329#400—941no—
330#400—941no—
331#400—941no—
332#391—471no—
333#391—471no—
334#391—471no—
335#390C2×D4711yescharacteristic
336#390C2×D4711yescharacteristic
337#389—471no—
338#389—471no—
339#389—471no—
340#389C2×D4711yescharacteristic
341#389C2×D4711yescharacteristic
342#388—21no—
343#388—21no—
344#381—21no—
345#380—21no—
346C4×C4⋊C4C2×D4711yescharacteristic
347#376C2×D4711yescharacteristic
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#336—471no—
355#336—471no—
356C47#2944011yescharacteristic, complemented, Sylow
357C22×Q8—471no—
358C22×D4C22×D4711yescharacteristic
359C23×C4—21no—
360C2×SD16—942no—
361C2×SD16—942no—
362C2×SD16—1881no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—471no—
367C4×D4—22no—
368C4×D4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42C22×D4711yescharacteristic
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389#106—942no—
390#106—942no—
391#106—1881no—
392C4⋊C8—942no—
393C4⋊C8—942no—
394#103—942no—
395#103—942no—
396#103—1881no—
397#102—942no—
398#102—942no—
399#102—1881no—
400#97—1881no—
401C4×C8—1881no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95C22⋊D4711yescharacteristic
415#95C22⋊D4711yescharacteristic
416#95C22⋊D4711yescharacteristic
417#95C22⋊D4711yescharacteristic
418C24—21no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4#1062535012yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C2×C22⋊D4711yescharacteristic
445C22×C4C2×C22⋊D4711yescharacteristic
446C22×C4C2×C22⋊D4711yescharacteristic
447C22×C4C23×D4711yescharacteristic
448SD16—1884no—
449C2×C8—472no—
450C2×C8—472no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—3761no—
475C4⋊C4—3761no—
476C4⋊C4—3761no—
477C4⋊C4#1062535012yes—
478C22⋊C4—24no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—81no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D4711yescharacteristic
498Q8—1881no—
499Q8—1881no—
500Q8—1881no—
501Q8—1881no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
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—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
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—1881no—
545C2×C4—1881no—
546C2×C4#41×D4711yescharacteristic
547C2×C4#41×D4711yescharacteristic
548C2×C4C2×#1062535011yescharacteristic
549C2×C4C2×#1062535011yescharacteristic
550C2×C4#1309407011yescharacteristic
551C2×C4#1309407111yescharacteristic
552C8—942no—
553C8—942no—
554C8—1881no—
555C8—1881no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1568963411yescharacteristic
564C22#1568966911yescharacteristic
565C22#1568970111yescharacteristic
566C22#1568993111yescharacteristic
567C22#1568993411yescharacteristic
568C22#1568993411yescharacteristic
569C22#1568993411yescharacteristic
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—3761no—
580C4—3761no—
581C4—3761no—
582C4—3761no—
583C4#1568979711yescharacteristic
584C4#1568980611yescharacteristic
585C4#1568981711yescharacteristic
586C4#1568984111yescharacteristic
587C2—82no—
588C2#2024898511yescharacteristic
589C2#2025364011yescharacteristic
590C2#2025366511yescharacteristic
591C2#2025399911yescharacteristic
592C2#2025401011yescharacteristic
593C2#2025483411yescharacteristic
594C2#2025485311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2319C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2319D47
hypocenterC47#29440
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. #20249353C2semidirect
2. #20244890C2semidirect
3. #20244766C2semidirect
7. #20244113C2semidirect
9. #20244077C2semidirect
10. #20244022C2semidirect
11. #20243966C2semidirect
14. #20243245C2semidirect
356. C47#29440semidirect

Maximal subgroups

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

subgroupindex
#202493532
#202448902
#202447662
#202447622
#202447052
#202442412
#202441132
#202440882
#202440772
#202440222
#202439662
#202437222
#202436492
#202432452
C47×#23192
#2944047