Finite Group Databasebeta

← #26107941

Subgroups — #26107941

Order 12032.

1518 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
1#20243376C211yesmaximal, characteristic, complemented
2#20243319C212yesmaximal, complemented
3#20241956C212yesmaximal, complemented
4#20241706C211yesmaximal, characteristic, complemented
5C2×D42×C47C211yesmaximal, characteristic
6C2×#13093945—24no—
7C2×#13093939—21no—
8C2×C22⋊(C47⋊C8)C2212yescomplemented
9#15688983C2212yescomplemented
10#15688890—21no—
11#15688857—24no—
12#15688849—21no—
13#15688847—21no—
14C23×D4×C47—21no—
15C23×D4×C47C411yescharacteristic, complemented
16C23×D4×C47C2211yescharacteristic
17D42×C47—24no—
18D42×C47—24no—
19C2×#127×C47C411yescharacteristic, complemented
20C2×#121×C47—21no—
21C2×#121×C47C411yescharacteristic, complemented
22C2×#121×C47C2211yescharacteristic
23C2×#120×C47—21no—
24C2×#120×C47—21no—
25C2×C4×D4×C47C411yescharacteristic, complemented
26C2×C4×D4×C47C2211yescharacteristic
27C22×C47⋊C8—22no—
28C2×C4⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30#13093945—48no—
31#13093939—42no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C25×C47—21no—
36C25×C47—21no—
37C22×D4×C47—28no—
38C22×D4×C47—21no—
39C22×D4×C47—21no—
40C22×D4×C47—22no—
41C22×D4×C47—22no—
42C22×D4×C47—21no—
43C22×D4×C47—21no—
44C22×D4×C47—21no—
45C22×D4×C47—21no—
46C22×D4×C47—22no—
47C22×D4×C47—44no—
48C22×D4×C47—44no—
49C22×D4×C47—44no—
50C22×D4×C47—42no—
51C22×D4×C47—42no—
52C22×D4×C47C2×C411yescharacteristic
53C22×D4×C47C2×C411yescharacteristic
54C22×D4×C47C2×C411yescharacteristic
55C22×D4×C47D411yescharacteristic
56C22×D4×C47D414yes—
57C23×C4×C47—21no—
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2311yescharacteristic
60#127×C47—22no—
61#127×C47—22no—
62#121×C47—28no—
63#121×C47—22no—
64#121×C47—22no—
65#121×C47—22no—
66#121×C47D414yes—
67#120×C47—28no—
68#120×C47—28no—
69C4×D4×C47—22no—
70C4×D4×C47—22no—
71C4×D4×C47—22no—
72C4×D4×C47—22no—
73C2×C4⋊C4×C47C2×C411yescharacteristic
74C2×C4⋊C4×C47D411yescharacteristic
75C2×C22⋊C4×C47—21no—
76C2×C22⋊C4×C47—21no—
77C2×C22⋊C4×C47—21no—
78C2×C22⋊C4×C47D411yescharacteristic
79C2×C22⋊C4×C47D411yescharacteristic
80C2×C42×C47C2×C411yescharacteristic
81C24×C47—21no—
82C24×C47—24no—
83C24×C47—21no—
84C24×C47—21no—
85C24×C47—24no—
86C24×C47—44no—
87C24×C47—44no—
88C24×C47—42no—
89C24×C47—42no—
90C24×C47—41no—
91C24×C47—41no—
92C24×C47—42no—
93C24×C47—82no—
94C24×C47—82no—
95C24×C47C22⋊C411yescharacteristic
96C24×C47C2×D411yescharacteristic
97C2×D4×C47—22no—
98C2×D4×C47—22no—
99C2×D4×C47—24no—
100C2×D4×C47—28no—
101C2×D4×C47—24no—
102C2×D4×C47—24no—
103C2×D4×C47—24no—
104C2×D4×C47—24no—
105C2×D4×C47—24no—
106C2×D4×C47—21no—
107C2×D4×C47—21no—
108C2×D4×C47—21no—
109C2×D4×C47—24no—
110C2×D4×C47—42no—
111C2×D4×C47—44no—
112C2×D4×C47—42no—
113C2×D4×C47—42no—
114C2×D4×C47—48no—
115C2×D4×C47—48no—
116C2×D4×C47—416no—
117C2×D4×C47—416no—
118C2×D4×C47—44no—
119C2×D4×C47—44no—
120C2×D4×C47—42no—
121C2×D4×C47—44no—
122C2×D4×C47—42no—
123C2×D4×C47—41no—
124C2×D4×C47—41no—
125C2×D4×C47—42no—
126C2×D4×C47—44no—
127C2×D4×C47—42no—
128C2×D4×C47—82no—
129C2×D4×C47C22⋊C412yes—
130C2×D4×C47C22⋊C412yes—
131C2×D4×C47C22⋊C412yes—
132C2×D4×C47D812yes—
133C2×D4×C47SD1612yes—
134C22×C4×C47—24no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—42no—
139C22×C4×C47—42no—
140C22×C4×C47—42no—
141C22×C4×C47—42no—
142C22×C4×C47C22⋊C411yescharacteristic
143C22×C4×C47C22⋊C411yescharacteristic
144C22×C4×C47C22⋊C411yescharacteristic
145C22×C4×C47C22×C411yescharacteristic
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×Dic47—42no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47C22⋊C412yes—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—22no—
156C22⋊C4×C47—24no—
157C22⋊C4×C47—22no—
158C42×C47—21no—
159C42×C47—21no—
160C22⋊Dic47—42no—
161C22⋊Dic47—42no—
162C4⋊Dic47—44no—
163C2×C47⋊C8—44no—
164C2×C47⋊C8—42no—
165C23×C47—24no—
166C23×C47—21no—
167C23×C47—22no—
168C23×C47—21no—
169C23×C47—22no—
170C23×C47—22no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—48no—
174C23×C47—41no—
175C23×C47—41no—
176C23×C47—42no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—48no—
180C23×C47—48no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—41no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—44no—
190C23×C47—44no—
191C23×C47—84no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47—84no—
195C23×C47—88no—
196C23×C47—88no—
197C23×C47—88no—
198C23×C47#10212yescomplemented
199C23×C47C2×C22⋊C411yescharacteristic
200C23×C47#12012yes—
201C23×C47C2×D811yescharacteristic
202C23×C47C2×SD1611yescharacteristic
203C23×C47AGL(1,ℤ/8)12yes—
204D4×C47—24no—
205D4×C47—24no—
206D4×C47—24no—
207D4×C47—22no—
208D4×C47—24no—
209D4×C47—48no—
210D4×C47—42no—
211D4×C47—42no—
212D4×C47—48no—
213D4×C47—48no—
214D4×C47—48no—
215D4×C47—44no—
216D4×C47—44no—
217D4×C47—42no—
218D4×C47—84no—
219D4×C47#9914yescomplemented
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—22no—
227C2×C4×C47—41no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—48no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—42no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47#10212yescomplemented
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C2×C22⋊C411yescharacteristic
239C2×C4×C47#12012yes—
240C2×Dic47—82no—
241C2×Dic47—82no—
242C2×Dic47—82no—
243C47⋊C8—82no—
244#8375—471nomaximal, Sylow
245C22×C47—22no—
246C22×C47—21no—
247C22×C47—22no—
248C22×C47—21no—
249C22×C47—24no—
250C22×C47—22no—
251C22×C47—41no—
252C22×C47—41no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—88no—
269C22×C47—88no—
270C22×C47—82no—
271C22×C47—82no—
272C22×C47—82no—
273C22×C47—82no—
274C22×C47—816no—
275C22×C47#37911yescharacteristic
276C22×C47C2×#10211yescharacteristic
277C22×C47#41811yescharacteristic
278C22×C47#44712yes—
279C22×C47#45012yes—
280C4×C47—22no—
281C4×C47—21no—
282C4×C47—22no—
283C4×C47—41no—
284C4×C47—41no—
285C4×C47—42no—
286C4×C47C2×#9912yes—
287Dic47—162no—
288C2×D42D4711yescharacteristic
289#1799—471no—
290#1786—472no—
291#1519—472no—
292#1459—471no—
293C2×C47—22no—
294C2×C47—21no—
295C2×C47—21no—
296C2×C47—22no—
297C2×C47—44no—
298C2×C47—42no—
299C2×C47—82no—
300C2×C47—82no—
301C2×C47#178111yescharacteristic
302C2×C47#178611yescharacteristic
303C2×C47#179511yescharacteristic
304C23×D4—21no—
305C23×D4Dic4711yescharacteristic, complemented
306C23×D4C2×D4711yescharacteristic
307D42—24no—
308D42—24no—
309C2×#127Dic4711yescharacteristic, complemented
310C2×#121—21no—
311C2×#121Dic4711yescharacteristic, complemented
312C2×#121C2×D4711yescharacteristic
313C2×#120—21no—
314C2×#120—21no—
315C2×C4×D4Dic4711yescharacteristic, complemented
316C2×C4×D4C2×D4711yescharacteristic
317C2×#102—944no—
318C2×#99—941no—
319C2×C22⋊C8—472no—
320#380—472no—
321#342—941no—
322#327—944no—
323#324—941no—
324#323—941no—
325C47#837511yescharacteristic, complemented, Sylow
326C25—21no—
327C25—21no—
328C22×D4—28no—
329C22×D4—21no—
330C22×D4—21no—
331C22×D4—22no—
332C22×D4—22no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4—22no—
338C22×D4—44no—
339C22×D4—44no—
340C22×D4—44no—
341C22×D4—42no—
342C22×D4—42no—
343C22×D4C2×Dic4711yescharacteristic
344C22×D4C2×Dic4711yescharacteristic
345C22×D4C2×Dic4711yescharacteristic
346C22×D4C22⋊D4711yescharacteristic
347C22×D4C22⋊D4714yes—
348C23×C4—21no—
349C23×C4C2×Dic4711yescharacteristic
350C23×C4C22×D4711yescharacteristic
351C22×C8—942no—
352#127—22no—
353#127—22no—
354#121—28no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121C22⋊D4714yes—
359#120—28no—
360#120—28no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4C2×Dic4711yescharacteristic
367C2×C4⋊C4C22⋊D4711yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—942no—
372C2×C22⋊C4C22⋊D4711yescharacteristic
373C2×C22⋊C4C22⋊D4711yescharacteristic
374C2×C42C2×Dic4711yescharacteristic
375#102—1888no—
376#99—1882no—
377C22⋊C8—942no—
378C22⋊C8—942no—
379#95—942no—
380C24—21no—
381C24—24no—
382C24—21no—
383C24—21no—
384C24—24no—
385C24—44no—
386C24—44no—
387C24—42no—
388C24—42no—
389C24—41no—
390C24—41no—
391C24—42no—
392C24—82no—
393C24—82no—
394C24C22⋊Dic4711yescharacteristic
395C24C2×C22⋊D4711yescharacteristic
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—24no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—416no—
416C2×D4—416no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—82no—
428C2×D4#1062533412yes—
429C2×D4#1062533512yes—
430C2×D4C22⋊Dic4712yes—
431C2×D4C22⋊Dic4712yes—
432C2×D4C22⋊Dic4712yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1882no—
442C22×C4C22⋊Dic4711yescharacteristic
443C22×C4C22⋊Dic4711yescharacteristic
444C22×C4C22⋊Dic4711yescharacteristic
445C22×C4C22×Dic4711yescharacteristic
446C22×C4C2×C22⋊D4712yes—
447C22×C4C2×C22⋊D4712yes—
448C22×C4C2×C22⋊D4711yescharacteristic
449C2×C8—1884no—
450C2×C8—1882no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—1884no—
454C4⋊C4C22⋊Dic4712yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—1882no—
461C22⋊C4—1882no—
462C42—21no—
463C42—21no—
464C23—24no—
465C23—21no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—48no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—48no—
479C23—48no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—44no—
485C23—44no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—88no—
495C23—88no—
496C23—88no—
497C23#1309394512yescomplemented
498C23C2×C22⋊Dic4711yescharacteristic
499C23#1309399312yes—
500C23C2×#1062533411yescharacteristic
501C23C2×#1062533511yescharacteristic
502C23#1309405012yes—
503D4—24no—
504D4—24no—
505D4—24no—
506D4—22no—
507D4—24no—
508D4—48no—
509D4—42no—
510D4—42no—
511D4—48no—
512D4—48no—
513D4—48no—
514D4—44no—
515D4—44no—
516D4—42no—
517D4—84no—
518D4#1309393914yescomplemented
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3762no—
536C2×C4—3762no—
537C2×C4—3762no—
538C2×C4#1309394512yescomplemented
539C2×C4C2×C22⋊Dic4711yescharacteristic
540C2×C4C2×C22⋊Dic4711yescharacteristic
541C2×C4#1309399312yes—
542C8—3762no—
543C22—21no—
544C22—22no—
545C22—21no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—88no—
567C22—88no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—816no—
573C22#1568897911yescharacteristic
574C22C2×#1309394511yescharacteristic
575C22#1568912811yescharacteristic
576C22#1568925712yes—
577C22#1568927412yes—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—7522no—
585C4C2×#1309393912yes—
586C2—22no—
587C2—21no—
588C2—21no—
589C2—22no—
590C2—44no—
591C2—42no—
592C2—82no—
593C2—82no—
594C2#2024330311yescharacteristic
595C2#2024331911yescharacteristic
596C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×D42×C47C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×D42D47
hypocenterC47#8375
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #20243376C2semidirect
2. #20243319C2semidirect
3. #20241956C2semidirect
4. #20241706C2semidirect
8. C2×C22⋊(C47⋊C8)C22semidirect
9. #15688983C22semidirect
15. C23×D4×C47C4semidirect
19. C2×#127×C47C4semidirect
21. C2×#121×C47C4semidirect
25. C2×C4×D4×C47C4semidirect
198. C23×C47#102semidirect
219. D4×C47#99semidirect
236. C2×C4×C47#102semidirect
305. C23×D4Dic47semidirect
309. C2×#127Dic47semidirect
311. C2×#121Dic47semidirect
315. C2×C4×D4Dic47semidirect
325. C47#8375semidirect
497. C23#13093945semidirect
518. D4#13093939semidirect
538. C2×C4#13093945semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202433192
#202419562
#202417062
C2×D42×C472
#837547