Finite Group Databasebeta

← #26243885

Subgroups — #26243885

Order 12032.

1586 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251977C211yesmaximal, characteristic, complemented
2C2×#15689140C211yesmaximal, characteristic, complemented
3#20251907C211yesmaximal, characteristic, complemented
4C22×#13093941C211yesmaximal, characteristic, complemented
5#20245970C211yesmaximal, characteristic, complemented
6#20244595C218yesmaximal, complemented
7#20243097C211yesmaximal, characteristic, complemented
8C2×C47×#535C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13#15689166C2214yescomplemented
14#15689163—22no—
15#15689159—22no—
16#15689140—24no—
17C2×#13093951C2214yescomplemented
18#15689132—21no—
19#15689128—21no—
20#15689124C2214yescomplemented
21#15689119—22no—
22C2×#13093941—24no—
23C2×#13093941C2214yescomplemented
24#15689026—22no—
25#15689026C2214yescomplemented
26#15688979—22no—
27#15688882C2214yescomplemented
28C22×#41×C47C2211yescharacteristic
29C23×D4×C47C2211yescharacteristic
30C47×#535—22no—
31C47×#535—22no—
32C47×#535—22no—
33C47×#535C2214yes—
34C2×#127×C47—21no—
35C2×#124×C47—21no—
36C2×#121×C47—22no—
37C2×#120×C47—22no—
38C2×#117×C47C2211yescharacteristic
39C2×#10625329—21no—
40C2×#10625329—22no—
41C2×#10625329—21no—
42C2×#10625329—22no—
43C2×#10625329—22no—
44C2×#10625329D414yescomplemented
45C2×#10625329C2312yes—
46C2×#10625329C2312yes—
47C22×C47⋊C8—21no—
48C22×C47⋊C8—21no—
49#13093983—22no—
50#13093983—22no—
51#13093983—21no—
52#13093983—21no—
53#13093983—21no—
54#13093983—21no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47C2312yes—
57C2×C22⋊Dic47—21no—
58C2×C22⋊Dic47—21no—
59C2×C22⋊Dic47—28no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47C2312yes—
62#13093957—24no—
63#13093955—24no—
64#13093951—28no—
65#13093948—24no—
66#13093945—24no—
67#13093941—28no—
68C25×C47D412yes—
69C2×#41×C47—21no—
70C2×#41×C47—21no—
71C2×#41×C47—22no—
72C2×#41×C47—22no—
73C2×#41×C47—22no—
74C2×#41×C47—22no—
75C2×#41×C47D412yes—
76C2×#41×C47C2312yes—
77C22×Q8×C47D411yescharacteristic
78C22×D4×C47—21no—
79C22×D4×C47—24no—
80C22×D4×C47—21no—
81C22×D4×C47—21no—
82C22×D4×C47—21no—
83C22×D4×C47—21no—
84C22×D4×C47—22no—
85C22×D4×C47—22no—
86C22×D4×C47—22no—
87C22×D4×C47—22no—
88C22×D4×C47D412yes—
89C22×D4×C47D411yescharacteristic
90C22×D4×C47C2312yes—
91C23×C4×C47—21no—
92C23×C4×C47C2311yescharacteristic
93#127×C47—24no—
94#127×C47—24no—
95#124×C47—24no—
96#124×C47—24no—
97#121×C47—28no—
98#121×C47—28no—
99#120×C47—28no—
100#120×C47—28no—
101#117×C47—21no—
102#117×C47—21no—
103#117×C47D412yes—
104#117×C47C2312yes—
105C2×C4⋊C4×C47D412yes—
106C2×C22⋊C4×C47—24no—
107C2×C22⋊C4×C47—21no—
108C2×C42×C47—21no—
109C24×C47—22no—
110C24×C47—21no—
111C24×C47—22no—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—84no—
115C24×C47C2×D414yes—
116C24×C47C2×D411yescharacteristic
117#41×C47—24no—
118#41×C47—24no—
119#41×C47—22no—
120#41×C47—22no—
121#41×C47—22no—
122#41×C47—22no—
123#41×C47—42no—
124#41×C47—42no—
125#41×C47—42no—
126#41×C47—42no—
127C2×Q8×C47—21no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47C2×D412yes—
131C2×D4×C47—22no—
132C2×D4×C47—22no—
133C2×D4×C47—216no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—21no—
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—22no—
140C2×D4×C47—24no—
141C2×D4×C47—22no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47—22no—
145C2×D4×C47—22no—
146C2×D4×C47—22no—
147C2×D4×C47—42no—
148C2×D4×C47—42no—
149C2×D4×C47—42no—
150C2×D4×C47—42no—
151C2×D4×C47—42no—
152C2×D4×C47—42no—
153C2×D4×C47—42no—
154C2×D4×C47—42no—
155C2×D4×C47—42no—
156C2×D4×C47—42no—
157C2×D4×C47—88no—
158C2×D4×C47C2×D412yes—
159C22×C4×C47—22no—
160C22×C4×C47—21no—
161C22×C4×C47—22no—
162C22×C4×C47—22no—
163C22×C4×C47—42no—
164C22×C4×C47—42no—
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47C2×D411yescharacteristic
168C22×C4×C47C2×D411yescharacteristic
169C22×C4×C47C2×D411yescharacteristic
170C22×C4×C47C2×D412yes—
171C22×C4×C47#4112yes—
172C22×C4×C47#4112yes—
173C22×C4×C47#4112yes—
174C22×C4×C47C2411yescharacteristic
175C22×Dic47—22no—
176C22×Dic47—22no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47C2×D414yes—
179C22⋊C4×C47—216no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C42×C47—22no—
183C42×C47—22no—
184C22⋊Dic47—41no—
185C22⋊Dic47—41no—
186C22⋊Dic47—42no—
187C22⋊Dic47—88no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—42no—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47—41no—
194C4×Dic47—41no—
195#10625329—24no—
196#10625329—22no—
197#10625329—22no—
198#10625329—22no—
199#10625329—42no—
200#10625329C2×D414yescomplemented
201C2×C47⋊C8—24no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C2×C47⋊C8—42no—
205C23×C47—24no—
206C23×C47—22no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—42no—
210C23×C47—48no—
211C23×C47—44no—
212C23×C47—48no—
213C23×C47—42no—
214C23×C47—44no—
215C23×C47—44no—
216C23×C47—84no—
217C23×C47—88no—
218C23×C47—88no—
219C23×C47#12011yescharacteristic
220C23×C47#12011yescharacteristic
221C23×C47#12112yes—
222C23×C47#12312yes—
223C23×C47C22×D411yescharacteristic
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47—42no—
227D4×C47—22no—
228D4×C47—22no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—44no—
232D4×C47—42no—
233D4×C47—42no—
234D4×C47—42no—
235D4×C47—42no—
236D4×C47—44no—
237D4×C47—44no—
238D4×C47—42no—
239D4×C47—816no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—44no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—44no—
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47#12011yescharacteristic
249C2×C4×C47#12112yes—
250C2×C4×C47#12112yes—
251C2×C4×C47#12112yes—
252C2×C4×C47#12112yes—
253C2×C4×C47#12112yes—
254C2×C4×C47#12312yes—
255C2×C4×C47#12312yes—
256C2×C4×C47#12312yes—
257C2×C4×C47#12412yes—
258C2×C4×C47#12412yes—
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×Dic47—44no—
266C2×Dic47—44no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C47⋊C8—24no—
270C47⋊C8—42no—
271#23493—471nomaximal, Sylow
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—48no—
278C22×C47—42no—
279C22×C47—44no—
280C22×C47—44no—
281C22×C47—48no—
282C22×C47—84no—
283C22×C47—84no—
284C22×C47—88no—
285C22×C47—88no—
286C22×C47#39412yes—
287C22×C47#39412yes—
288C22×C47C2×#12011yescharacteristic
289C22×C47C2×#12111yescharacteristic
290C22×C47C2×#12311yescharacteristic
291C4×C47—42no—
292C4×C47—44no—
293C4×C47#39412yes—
294C4×C47#39412yes—
295C4×C47C2×#12111yescharacteristic
296C4×C47C2×#12111yescharacteristic
297C4×C47C2×#12311yescharacteristic
298C4×C47C2×#12411yescharacteristic
299Dic47—82no—
300Dic47—82no—
301C2×#535D4711yescharacteristic
302#2911—471no—
303C2×#421—471no—
304#2896—471no—
305C22×#100—471no—
306#2309—471no—
307#2037—478no—
308#1733—471no—
309C2×C47—21no—
310C2×C47—21no—
311C2×C47—22no—
312C2×C47—22no—
313C2×C47—84no—
314C2×C47—82no—
315C2×C47C2×#39411yescharacteristic
316C2×C47#301912yes—
317C22×#41C2×D4711yescharacteristic
318C23×D4C2×D4711yescharacteristic
319C22×M16—471no—
320C22×M16—471no—
321#535—22no—
322#535—22no—
323#535—22no—
324#535C2×D4714yes—
325C2×#127—21no—
326C2×#124—21no—
327C2×#121—22no—
328C2×#120—22no—
329C2×#117—471no—
330C2×#117—471no—
331C2×#117C2×D4711yescharacteristic
332#428—474no—
333#428—942no—
334#427—942no—
335#421—944no—
336C2×#104—474no—
337#419—941no—
338#418—941no—
339#417—474no—
340#416—942no—
341C2×#100—474no—
342C2×#100—944no—
343#390—474no—
344#390—942no—
345#379—942no—
346#339—474no—
347C47#2349311yescharacteristic, complemented, Sylow
348C25C22⋊D4712yes—
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C22⋊D4712yes—
356C2×#41C22×D4712yes—
357C22×Q8C22⋊D4711yescharacteristic
358C22×D4—21no—
359C22×D4—24no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—21no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C22⋊D4712yes—
369C22×D4C22⋊D4711yescharacteristic
370C22×D4C22×D4712yes—
371C23×C4—21no—
372C23×C4C22×D4711yescharacteristic
373C2×M16—474no—
374C2×M16—472no—
375C2×M16—472no—
376C2×M16—942no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—942no—
380C2×M16—942no—
381C22×C8—941no—
382C22×C8—941no—
383#127—24no—
384#127—24no—
385#124—24no—
386#124—24no—
387#121—28no—
388#121—28no—
389#120—28no—
390#120—28no—
391#117—21no—
392#117—21no—
393#117—942no—
394#117—942no—
395#117—941no—
396#117—941no—
397#117—941no—
398#117—941no—
399#117C22⋊D4712yes—
400#117C22×D4712yes—
401C2×C4⋊C4—472no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4C22⋊D4712yes—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—948no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42—21no—
410C2×C42—472no—
411C2×C42—941no—
412#107—944no—
413#106—944no—
414#104—948no—
415#103—944no—
416#102—944no—
417#100—948no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—24no—
422C24—24no—
423C24—84no—
424C24C2×C22⋊D4714yes—
425C24C2×C22⋊D4711yescharacteristic
426#41—24no—
427#41—24no—
428#41—22no—
429#41—22no—
430#41—22no—
431#41—22no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—42no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C2×C22⋊D4712yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—216no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—88no—
467C2×D4C2×C22⋊D4712yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4D4×D4712yes—
477C22×C4#1062535212yes—
478C22×C4#1062535212yes—
479C22×C4#1062535212yes—
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486M16—474no—
487M16—944no—
488M16—942no—
489M16—942no—
490M16—942no—
491M16—1882no—
492C2×C8—944no—
493C2×C8—942no—
494C2×C8—942no—
495C2×C8—1882no—
496C4⋊C4—22no—
497C4⋊C4—942no—
498C4⋊C4—942no—
499C4⋊C4—1882no—
500C4⋊C4C2×C22⋊D4714yes—
501C22⋊C4—216no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—3768no—
505C22⋊C4—1881no—
506C22⋊C4—1881no—
507C22⋊C4—1882no—
508C42—22no—
509C42—22no—
510C42—942no—
511C42—942no—
512C42—1881no—
513C42—1881no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—48no—
520C23—44no—
521C23—48no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—84no—
526C23—88no—
527C23—88no—
528C23#1309399311yescharacteristic
529C23#1309399311yescharacteristic
530C23#1309399612yes—
531C23#1309400512yes—
532C23C22×C22⋊D4711yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536D4—22no—
537D4—22no—
538D4—44no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—816no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—1884no—
557C2×C4—1884no—
558C2×C4—1882no—
559C2×C4—1882no—
560C2×C4#1309399311yescharacteristic
561C2×C4#1309399311yescharacteristic
562C2×C4#1309399512yes—
563C2×C4#1309399512yes—
564C2×C4#1309399612yes—
565C2×C4#1309399612yes—
566C2×C4#1309399612yes—
567C2×C4#1309400512yes—
568C2×C4#1309400512yes—
569C2×C4#1309400512yes—
570C2×C4#1309401112yes—
571C2×C4#1309401112yes—
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4C22×C22⋊D4711yescharacteristic
574C2×C4C22×C22⋊D4711yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C2×C4C2×#1062535211yescharacteristic
577C2×C4C2×#1062535211yescharacteristic
578C8—944no—
579C8—1882no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—48no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—84no—
591C22—88no—
592C22—84no—
593C22—88no—
594C22#1568903712yes—
595C22#1568903712yes—
596C22C2×#1309399311yescharacteristic
597C22C2×#1309399611yescharacteristic
598C22C2×#1309400511yescharacteristic
599C4—42no—
600C4—44no—
601C4—3762no—
602C4—3762no—
603C4#1568903712yes—
604C4#1568903712yes—
605C4C2×#1309399511yescharacteristic
606C4C2×#1309399611yescharacteristic
607C4C2×#1309400511yescharacteristic
608C4C2×#1309401111yescharacteristic
609C2—21no—
610C2—21no—
611C2—22no—
612C2—22no—
613C2—84no—
614C2—82no—
615C2C2×#1568903711yescharacteristic
616C2#2025274512yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#535C2
socleC22×C47C2×#120
hypercenterC2×#535D47
hypocenterC47#23493
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. #20251977C2semidirect
2. C2×#15689140C2semidirect
3. #20251907C2semidirect
4. C22×#13093941C2semidirect
5. #20245970C2semidirect
6. #20244595C2semidirect
7. #20243097C2semidirect
13. #15689166C22semidirect
17. C2×#13093951C22semidirect
20. #15689124C22semidirect
23. C2×#13093941C22semidirect
25. #15689026C22semidirect
27. #15688882C22semidirect
44. C2×#10625329D4semidirect
200. #10625329C2×D4semidirect
347. C47#23493semidirect

Maximal subgroups

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

subgroupindex
#202519772
C2×#156891402
#202519072
C22×#130939412
#202459702
#202445952
#202430972
C2×C47×#5352
#2349347