Finite Group Databasebeta

← #26217702

Subgroups — #26217702

Order 12032.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094052C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic, complemented
3C2×#15689117C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic, complemented
5#20245968C211yesmaximal, characteristic, complemented
6#20243744C214yesmaximal, complemented
7#20243737C214yesmaximal, complemented
8#20243086C211yesmaximal, characteristic, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C23×D188C2211yescharacteristic
12C2×#13094052—24no—
13C2×#13094052—24no—
14C2×#13094052C418yescomplemented
15C22×#10625325C2212yes—
16C22×D376C2212yes—
17C23×C47⋊C8C2211yescharacteristic
18#15689604—24no—
19#15689604—24no—
20C2×#13093989C2212yescomplemented
21C2×#13093986C2212yescomplemented
22C2×C4×C22⋊D47—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093982—21no—
25C2×C4⋊C4×D47—21no—
26#15689117—24no—
27C2×#13093947C2212yescomplemented
28C2×#13093944—24no—
29C2×#13093944C2214yes—
30C2×#13093944C2212yescomplemented
31#15689025—21no—
32#15689025—21no—
33#15689024C2214yes—
34#15688978—22no—
35#15688877C2214yescomplemented
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C22×C4⋊C4×C47C2211yescharacteristic
39C47×#340C2214yescomplemented
40C24×D47—21no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D414yes—
46C2×#10625350D414yes—
47C22×Dic94C2311yescharacteristic
48C22×D188—22no—
49C22×D188—22no—
50C22×D188—22no—
51C22×D188—22no—
52C22×D188—21no—
53C22×D188—22no—
54C22×D188C2×C414yescomplemented
55C22×D188C2312yes—
56C22×D188C2311yescharacteristic
57C22×C22⋊D47—21no—
58C22×C4×D47—21no—
59C22×C4×D47—22no—
60#13094052—232no—
61C2×#10625325—24no—
62C2×#10625325—24no—
63C2×#10625325C2×C418yescomplemented
64C2×D376—24no—
65C2×D376—24no—
66C2×D376C2×C418yescomplemented
67C22×C47⋊C8—22no—
68C22×C47⋊C8—22no—
69C22×C47⋊C8—22no—
70C22×C47⋊C8C2312yes—
71#13093989—24no—
72#13093989—24no—
73#13093986—24no—
74#13093986—24no—
75C4×C22⋊D47—216no—
76#13093983—22no—
77#13093983—22no—
78#13093982—28no—
79C2×C4⋊Dic47D411yescharacteristic
80C2×C4⋊Dic47D411yescharacteristic
81C2×Dic47⋊C4—22no—
82C4⋊C4×D47—28no—
83C2×C22⋊Dic47D412yes—
84C2×(C2×D47)⋊C4—22no—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4—24no—
87C2×C4×Dic47—22no—
88C2×C4×Dic47C2312yes—
89#13093947—24no—
90#13093944—24no—
91#13093944—28no—
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94C2×M16×C47—22no—
95C2×M16×C47—22no—
96C2×M16×C47C2×C414yescomplemented
97C22×C8×C47C2312yes—
98#117×C47—22no—
99#117×C47—22no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—22no—
102C2×C4⋊C4×C47D414yes—
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2312yes—
105C2×C22⋊C4×C47—21no—
106C2×C42×C47—21no—
107C24×C47C2×D411yescharacteristic
108C23×D47—21no—
109C23×D47—22no—
110C23×D47—24no—
111C23×D47—42no—
112C23×D47—42no—
113C23×D47—82no—
114C23×D47—82no—
115C22×C4×C47—22no—
116C22×C4×C47—24no—
117C22×C4×C47—22no—
118C22×C4×C47—82no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D412yes—
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D412yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×C4×C47C2411yescharacteristic
127C2×C22⋊D47—24no—
128C2×C22⋊D47—24no—
129C2×C22⋊D47—24no—
130C22×Dic47—21no—
131C22×Dic47C2×D412yes—
132C22×Dic47C2×D412yes—
133#10625350—28no—
134#10625350—28no—
135#10625350—28no—
136#10625350C22⋊C4116yescomplemented
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—28no—
141C2×D188—28no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—22no—
146C2×D188—24no—
147C2×D188—22no—
148C2×D188—42no—
149C2×D188—44no—
150C2×D188—42no—
151C2×D188—44no—
152C2×D188C22×C414yes—
153C2×D188C22×C412yes—
154C2×D188C2×D412yes—
155C2×D188C2×D412yes—
156C2×C4×D47—24no—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—28no—
160C2×C4×D47—84no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C22×C412yes—
165C2×Dic94C2×D412yes—
166C2×Dic94C2×D412yes—
167M16×C47—28no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47C22×C414yes—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47C2×D414yes—
176C22⋊C4×C47—42no—
177C42×C47—42no—
178C22⋊Dic47—24no—
179(C2×D47)⋊C4—28no—
180(C2×D47)⋊C4—84no—
181(C2×D47)⋊C4—84no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184Dic47⋊C4—28no—
185C4×Dic47—28no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—44no—
192D376—216no—
193#10625325—216no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47C4×D412yes—
197C23×C47#12012yes—
198C23×C47#12112yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—24no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—44no—
209C22×D47—44no—
210C22×D47—48no—
211C22×D47—88no—
212C22×D47—84no—
213C22×D47—84no—
214C22×D47—88no—
215C2×C4×C47—24no—
216C2×C4×C47—44no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—84no—
221C2×C4×C47C2×C22⋊C414yes—
222C2×C4×C47C4×D412yes—
223C2×C4×C47C4×D414yes—
224C2×C4×C47#12012yes—
225C2×C4×C47#12112yes—
226C2×C4×C47#12114yes—
227C2×C4×C47#12314yes—
228C2×C4×C47C23×C411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C22⋊D47—216no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242D188—24no—
243D188—22no—
244D188—24no—
245D188—22no—
246D188—22no—
247D188—42no—
248D188—42no—
249D188—48no—
250D188C2×C22⋊C414yes—
251C4×D47—216no—
252C4×D47—88no—
253Dic94—22no—
254Dic94—22no—
255Dic94—22no—
256Dic94C2×C22⋊C414yes—
257C8×C47—24no—
258C47⋊C8—42no—
259#20932—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—24no—
264C22×C47#38614yes—
265C22×C47C2×C4×D411yescharacteristic
266C22×C47C2×#12011yescharacteristic
267C22×C47C2×#12111yescharacteristic
268C22×C47#57612yes—
269C22×C47#57712yes—
270C2×D47—24no—
271C2×D47—44no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—44no—
275C2×D47—48no—
276C2×D47—44no—
277C2×D47—816no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47—84no—
281C4×C47—44no—
282C4×C47—82no—
283C4×C47#38614yes—
284C4×C47C22×C22⋊C411yescharacteristic
285C4×C47C2×C4×D411yescharacteristic
286C4×C47C2×#12111yescharacteristic
287C4×C47C2×#12311yescharacteristic
288Dic47—24no—
289Dic47—42no—
290Dic47—42no—
291C22×AGL(1,ℤ/8)—471no—
292C2×#520—471no—
293C2×#416—471no—
294C22×#102—471no—
295#2309—471no—
296#1880—474no—
297#1879—474no—
298#1736D4711yescharacteristic, complemented
299#1729—471no—
300C2×C47—24no—
301C2×C47C2×#38611yescharacteristic
302C2×C47#294612yes—
303C2×C47#300811yescharacteristic
304C2×C47#301011yescharacteristic
305C2×C47#304512yes—
306D47—44no—
307D47—84no—
308C22×#41—471no—
309C23×D4—471no—
310C2×AGL(1,ℤ/8)—478no—
311C2×AGL(1,ℤ/8)—944no—
312C2×AGL(1,ℤ/8)—944no—
313C22×SD16—472no—
314C22×D8—472no—
315C22×M16C2×D4711yescharacteristic
316C23×C8—471no—
317#520—944no—
318#520—944no—
319C2×C4×Q8—472no—
320C2×C4×D4—472no—
321C2×C4×D4—942no—
322C2×#117—471no—
323C2×#117—941no—
324C2×#117C2×D4711yescharacteristic
325C22×C4⋊C4—941no—
326C22×C4⋊C4C2×D4711yescharacteristic
327#416—944no—
328C2×#103—472no—
329C2×#102—474no—
330C2×#102—472no—
331C2×#102—944no—
332#390—474no—
333#390—941no—
334#390—941no—
335#379—942no—
336#340C2×D4714yescomplemented
337#336—474no—
338C47#2093211yescharacteristic, complemented, Sylow
339C25—941no—
340C2×#41—474no—
341C2×#41—474no—
342C2×#41—474no—
343C2×#41—942no—
344C2×#41—942no—
345C2×#41—942no—
346C22×Q8—471no—
347C22×D4—474no—
348C22×D4—471no—
349C22×D4—472no—
350C22×D4—941no—
351C22×D4—942no—
352C22×D4—942no—
353C22×D4—942no—
354C22×D4—942no—
355C22×D4—941no—
356C22×D4—942no—
357C23×C4—21no—
358C23×C4—942no—
359C23×C4—941no—
360C23×C4C22×D4711yescharacteristic
361AGL(1,ℤ/8)—9432no—
362C2×SD16—478no—
363C2×SD16—944no—
364C2×SD16—944no—
365C2×D8—478no—
366C2×D8—944no—
367C2×D8—944no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16C4×D4714yescomplemented
371C22×C8—472no—
372C22×C8—942no—
373C22×C8—942no—
374C22×C8—942no—
375C22×C8C22×D4712yes—
376C4×Q8—944no—
377C4×Q8—944no—
378C4×D4—944no—
379C4×D4—9416no—
380C4×D4—944no—
381#117—22no—
382#117—22no—
383#117—948no—
384#117—942no—
385#117—942no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—948no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4D18814yes—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—472no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—944no—
400C2×C42—21no—
401C2×C42—472no—
402C2×C42—942no—
403#103—944no—
404#102—944no—
405#102—948no—
406C24—941no—
407C24—944no—
408C24—942no—
409C24—3762no—
410C24—3762no—
411C24—1882no—
412C24—1882no—
413C24D4×D4711yescharacteristic
414#41—4716no—
415#41—948no—
416#41—948no—
417#41—948no—
418C2×Q8—472no—
419C2×Q8—472no—
420C2×Q8—472no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×D4—472no—
425C2×D4—474no—
426C2×D4—472no—
427C2×D4—472no—
428C2×D4—944no—
429C2×D4—944no—
430C2×D4—944no—
431C2×D4—948no—
432C2×D4—944no—
433C2×D4—948no—
434C2×D4—942no—
435C2×D4—942no—
436C2×D4—941no—
437C2×D4—942no—
438C2×D4—941no—
439C2×D4—942no—
440C2×D4—941no—
441C2×D4—942no—
442C2×D4—1882no—
443C2×D4—1882no—
444C2×D4—1884no—
445C2×D4—1884no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—82no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—948no—
453C22×C4—941no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—3764no—
458C22×C4C2×C4×D4712yes—
459C22×C4C2×D18812yes—
460C22×C4#1062535012yes—
461C22×C4D4×D4712yes—
462C22×C4D4×D4712yes—
463C22×C4D4×D4711yescharacteristic
464C22×C4#1062535412yes—
465C22×C4C23×D4711yescharacteristic
466SD16—9416no—
467D8—9416no—
468M16—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—1884no—
475C2×C8C2×C4×D4714yes—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—948no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4C2×D18814yes—
484C22⋊C4—42no—
485C22⋊C4—948no—
486C22⋊C4—944no—
487C22⋊C4—3764no—
488C22⋊C4—3764no—
489C42—42no—
490C42—948no—
491C42—942no—
492C42—942no—
493C23—22no—
494C23—22no—
495C23—942no—
496C23—942no—
497C23—942no—
498C23—944no—
499C23—3764no—
500C23—3764no—
501C23—3768no—
502C23—3768no—
503C23—1888no—
504C23—1882no—
505C23—1882no—
506C23—1884no—
507C23—1884no—
508C23—1882no—
509C23—1882no—
510C23#1309398612yes—
511C23#1309399412yes—
512C23#1309399712yes—
513C23C2×D4×D4711yescharacteristic
514Q8—474no—
515Q8—942no—
516Q8—942no—
517Q8—942no—
518D4—474no—
519D4—9416no—
520D4—944no—
521D4—942no—
522D4—944no—
523D4—942no—
524D4—942no—
525D4—1888no—
526D4—1882no—
527D4—1882no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—942no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—9416no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—3768no—
543C2×C4C22⋊C4×D4714yes—
544C2×C4#1309398612yes—
545C2×C4C4×D18814yes—
546C2×C4#1309399412yes—
547C2×C4#1309399712yes—
548C2×C4#1309399914yes—
549C2×C4#1309400914yes—
550C2×C4C22×C4×D4711yescharacteristic
551C2×C4C2×D4×D4711yescharacteristic
552C2×C4C2×D4×D4711yescharacteristic
553C2×C4C22×D18811yescharacteristic
554C2×C4C2×#1062535011yescharacteristic
555C2×C4C2×#1062535411yescharacteristic
556C8—24no—
557C8—1882no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—944no—
563C22—37616no—
564C22—3762no—
565C22—3762no—
566C22—3764no—
567C22—1884no—
568C22—1888no—
569C22—1882no—
570C22—1884no—
571C22—1882no—
572C22—1884no—
573C22#1568900114yes—
574C22C2×#1309398611yescharacteristic
575C22C2×#1309399411yescharacteristic
576C22C2×#1309399711yescharacteristic
577C22#1568992712yes—
578C22#1568993212yes—
579C4—44no—
580C4—82no—
581C4—944no—
582C4—1882no—
583C4—1882no—
584C4#1568900114yes—
585C4C2×C22⋊C4×D4711yescharacteristic
586C4C2×C4×D18811yescharacteristic
587C4C2×#1309399911yescharacteristic
588C4C2×#1309400911yescharacteristic
589C2—24no—
590C2—3764no—
591C2—1884no—
592C2C2×#1568900111yescharacteristic
593C2#2025221812yes—
594C2#2025265711yescharacteristic
595C2#2025267511yescharacteristic
596C2#2025292612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC22×#13094052C2
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20932
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#15689604C2semidirect
3. C2×#15689117C2semidirect
4. C22×#13093944C2semidirect
5. #20245968C2semidirect
6. #20243744C2semidirect
7. #20243737C2semidirect
8. #20243086C2semidirect
9. C47×#1736C2semidirect
14. C2×#13094052C4semidirect
20. C2×#13093989C22semidirect
21. C2×#13093986C22semidirect
27. C2×#13093947C22semidirect
30. C2×#13093944C22semidirect
35. #15688877C22semidirect
39. C47×#340C22semidirect
54. C22×D188C2×C4semidirect
63. C2×#10625325C2×C4semidirect
66. C2×D376C2×C4semidirect
96. C2×M16×C47C2×C4semidirect
136. #10625350C22⋊C4semidirect
298. #1736D47semidirect
336. #340C2×D47semidirect
338. C47#20932semidirect
370. C2×M16C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940522
C2×#156896042
C2×#156891172
C22×#130939442
#202459682
#202437442
#202437372
#202430862
C47×#17362
#2093247