Finite Group Databasebeta

← #26243777

Subgroups — #26243777

Order 12032.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689086C211yesmaximal, characteristic
5#20245767C211yesmaximal, characteristic
6#20244589C214yesmaximal, complemented
7#20244582C214yesmaximal, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C2×C47×#419C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13094000—22no—
13C2×#13094000—24no—
14C2×#13094000—24no—
15C2×#13094000C2214yes—
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C22×Dic47⋊C4—21no—
19C22×(C2×D47)⋊C4—21no—
20#15689166—24no—
21C2×#13093957C2212yescomplemented
22C2×#13093955C2212yescomplemented
23#15689132—24no—
24C2×#13093948C2214yescomplemented
25#15689086—24no—
26C2×(C2×D47)⋊C8C2214yes—
27#15688984—21no—
28#15688980C2212yes—
29#15688980C2212yes—
30#15688978—21no—
31C4×(C2×D47)⋊C4—22no—
32#15688886C2214yescomplemented
33C23×Q8×C47C2211yescharacteristic
34C22×M16×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#419—24no—
37C2×#103×C47C2212yescomplemented
38C2×#103×C47C2212yescomplemented
39C24×D47D411yescharacteristic
40C23×Dic47—21no—
41C23×Dic47D411yescharacteristic
42C22×C4×D47—22no—
43C22×C4×D47—22no—
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47D412yes—
49C22×C4×D47C2312yes—
50C2×#10625329—22no—
51C2×#10625329—22no—
52C2×#10625329D414yes—
53C22×C47⋊C8C2312yes—
54#13094000—216no—
55#13094000—28no—
56#13094000—28no—
57#13093983—22no—
58#13093983—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47D414yes—
62C2×C4⋊Dic47C2312yes—
63C2×C4⋊Dic47C2312yes—
64C2×Dic47⋊C4—28no—
65C2×Dic47⋊C4—24no—
66C2×C22⋊Dic47—21no—
67C2×(C2×D47)⋊C4—28no—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71C2×C4×Dic47—21no—
72#13093957—24no—
73#13093955—24no—
74#13093948—28no—
75(C2×D47)⋊C8—28no—
76C47⋊#95—22no—
77C47⋊#95—22no—
78C22×Q8×C47—22no—
79C22×Q8×C47—21no—
80C22×Q8×C47—22no—
81C22×Q8×C47—22no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47D414yes—
85C22×Q8×C47C2312yes—
86C23×C4×C47—21no—
87C23×C4×C47C2311yescharacteristic
88C2×M16×C47—22no—
89C2×M16×C47—22no—
90C2×M16×C47—22no—
91C22×C8×C47C2312yes—
92#117×C47—22no—
93#117×C47—22no—
94C2×C4⋊C4×C47C2311yescharacteristic
95C2×C4⋊C4×C47C2311yescharacteristic
96C2×C22⋊C4×C47—21no—
97C2×C42×C47—21no—
98#103×C47—24no—
99#103×C47—24no—
100C24×C47C2×D411yescharacteristic
101C23×D47—22no—
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—42no—
106C23×D47C2×D412yes—
107C2×Q8×C47—24no—
108C2×Q8×C47—22no—
109C2×Q8×C47—28no—
110C2×Q8×C47—28no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—42no—
116C2×Q8×C47—44no—
117C2×Q8×C47—42no—
118C2×Q8×C47—44no—
119C2×Q8×C47C2×D414yes—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47#4112yes—
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—44no—
141C22×Dic47C2×D412yes—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—44no—
152C2×C4×D47—44no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155M16×C47—44no—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C22⋊C4×C47—42no—
164C42×C47—42no—
165C22⋊Dic47—42no—
166(C2×D47)⋊C4—216no—
167(C2×D47)⋊C4—44no—
168(C2×D47)⋊C4—44no—
169C4⋊Dic47—24no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—22no—
173C4⋊Dic47C2×D414yes—
174Dic47⋊C4—216no—
175C4×Dic47—42no—
176#10625329—28no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8C2×D414yes—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47#12012yes—
183C23×C47#12112yes—
184C23×C47#12312yes—
185C23×C47C22×D411yescharacteristic
186C22×D47—28no—
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—48no—
194C22×D47—48no—
195C22×D47—44no—
196C22×D47—44no—
197C22×D47#13714yes—
198Q8×C47—24no—
199Q8×C47—24no—
200Q8×C47—42no—
201Q8×C47—42no—
202Q8×C47—48no—
203C2×C4×C47—24no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—48no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47#12012yes—
212C2×C4×C47#12114yes—
213C2×C4×C47#12114yes—
214C2×C4×C47#12112yes—
215C2×C4×C47#12312yes—
216C2×C4×C47#12314yes—
217C2×C4×C47#12414yes—
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—48no—
229C2×Dic47—44no—
230C2×Dic47—44no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C4×D47—42no—
235C4×D47—42no—
236C4×D47—42no—
237C4×D47—42no—
238C4×D47—48no—
239C8×C47—42no—
240C47⋊C8—24no—
241#23481—471nomaximal, Sylow
242C22×C47—24no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—24no—
246C22×C47#39414yes—
247C22×C47C2×#12011yescharacteristic
248C22×C47C2×#12111yescharacteristic
249C22×C47C2×#12311yescharacteristic
250C22×C47C2×#13712yes—
251C22×C47#57512yes—
252C2×D47—24no—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—416no—
256C2×D47—42no—
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—42no—
260C4×C47—44no—
261C4×C47—81no—
262C4×C47—81no—
263C4×C47#39414yes—
264C4×C47C2×#12111yescharacteristic
265C4×C47C2×#12111yescharacteristic
266C4×C47C2×#12311yescharacteristic
267C4×C47C2×#12411yescharacteristic
268Dic47—42no—
269Dic47—44no—
270Dic47—82no—
271C22×C22⋊Q8—471no—
272C2×#428—471no—
273C2×#419—471no—
274C2×#419D4711yescharacteristic, complemented
275C2×#407—471no—
276#2280—471no—
277#2036—474no—
278#2035—474no—
279#1736—471no—
280C2×C47—24no—
281C2×C47C2×#39411yescharacteristic
282C2×C47#300411yescharacteristic
283C2×C47#300511yescharacteristic
284C2×C47#305612yes—
285C2×C47#309512yes—
286D47—44no—
287C23×Q8C2×D4711yescharacteristic
288C24×C4—471no—
289C22×M16—471no—
290C22×M16C2×D4711yescharacteristic
291C2×C22⋊Q8—474no—
292C2×C22⋊Q8—944no—
293C2×C22⋊Q8—942no—
294C2×C22⋊Q8—944no—
295C2×#117—471no—
296C2×#117C2×D4711yescharacteristic
297C22×C4⋊C4—471no—
298C22×C4⋊C4—941no—
299C22×C22⋊C4—941no—
300#428—944no—
301C2×#107—472no—
302C2×#106—472no—
303#419—24no—
304#419—944no—
305C2×#103—474no—
306C2×#103C2×D4712yescomplemented
307C2×#103C2×D4712yescomplemented
308#407—944no—
309C2×C22⋊C8—474no—
310#381—941no—
311#380—472no—
312#380—472no—
313#379—941no—
314C4×C22⋊C4—942no—
315#340—474no—
316C47#2348111yescharacteristic, complemented, Sylow
317C25—471no—
318C22×Q8—22no—
319C22×Q8—21no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8C22⋊D4714yes—
325C22×Q8C22×D4712yes—
326C23×C4—21no—
327C23×C4—471no—
328C23×C4—472no—
329C23×C4—472no—
330C23×C4—941no—
331C23×C4—942no—
332C23×C4—942no—
333C23×C4—942no—
334C23×C4—942no—
335C23×C4—942no—
336C23×C4—942no—
337C23×C4C22×D4711yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—474no—
342C2×M16—942no—
343C2×M16—942no—
344C22×C8—472no—
345C22×C8C22×D4712yes—
346C22⋊Q8—948no—
347C22⋊Q8—9416no—
348C22⋊Q8—948no—
349#117—22no—
350#117—22no—
351#117—942no—
352#117—942no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—944no—
357C2×C4⋊C4—948no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C4⋊C4C22×D4711yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—948no—
365C2×C22⋊C4—942no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—941no—
368C2×C42—21no—
369C2×C42—941no—
370#107—944no—
371#106—944no—
372#103—24no—
373#103—24no—
374#103—948no—
375C22⋊C8—948no—
376#95—942no—
377#95—942no—
378C24—472no—
379C24—944no—
380C24—942no—
381C24—942no—
382C24—942no—
383C24—1882no—
384C24D4×D4711yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D4714yes—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—472no—
405C22×C4—944no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—944no—
411C22×C4—944no—
412C22×C4—944no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—1884no—
421C22×C4—1884no—
422C22×C4—1884no—
423C22×C4—1884no—
424C22×C4—1884no—
425C22×C4#1062535012yes—
426C22×C4#1062535012yes—
427C22×C4D4×D4711yescharacteristic
428C22×C4D4×D4712yes—
429C22×C4D4×D4712yes—
430C22×C4#1062535212yes—
431C22×C4C2×C22⋊D4712yes—
432C22×C4C23×D4711yescharacteristic
433M16—44no—
434M16—948no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—474no—
439C2×C8—942no—
440C2×C8—942no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—474no—
446C4⋊C4—9416no—
447C4⋊C4—944no—
448C4⋊C4—942no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C22⋊C4—42no—
452C22⋊C4—9416no—
453C22⋊C4—1884no—
454C22⋊C4—1884no—
455C22⋊C4—1882no—
456C42—42no—
457C42—1882no—
458C23—22no—
459C23—22no—
460C23—474no—
461C23—948no—
462C23—942no—
463C23—942no—
464C23—942no—
465C23—942no—
466C23—942no—
467C23—942no—
468C23—1888no—
469C23—1888no—
470C23—1884no—
471C23—1884no—
472C23#1309399212yes—
473C23#1309399712yes—
474C23#1309400612yes—
475C23C2×D4×D4711yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481C2×C4—24no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—944no—
490C2×C4—942no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1888no—
497C2×C4—1884no—
498C2×C4—1884no—
499C2×C4—1888no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4#1309399212yes—
505C2×C4#1309399614yes—
506C2×C4#1309399712yes—
507C2×C4#1309399714yes—
508C2×C4#1309400612yes—
509C2×C4#1309400714yes—
510C2×C4#1309401014yes—
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4C2×D4×D4711yescharacteristic
513C2×C4C22×C22⋊D4711yescharacteristic
514C2×C4C2×#1062535211yescharacteristic
515C2×C4C2×#1062535011yescharacteristic
516C2×C4C2×#1062535011yescharacteristic
517C8—42no—
518C8—944no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—944no—
524C22—944no—
525C22—944no—
526C22—18816no—
527C22—1882no—
528C22—1882no—
529C22—1882no—
530C22—1882no—
531C22#1568903814yes—
532C22C2×#1309399211yescharacteristic
533C22C2×#1309399711yescharacteristic
534C22C2×#1309400611yescharacteristic
535C22#137×D4712yes—
536C22#1568992212yes—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—3762no—
541C4—1882no—
542C4—1884no—
543C4#1568903814yes—
544C4C2×#1309399611yescharacteristic
545C4C2×#1309399711yescharacteristic
546C4C2×#1309400711yescharacteristic
547C4C2×#1309401011yescharacteristic
548C2—24no—
549C2—1884no—
550C2C2×#1568903811yescharacteristic
551C2#2025262011yescharacteristic
552C2#2025262911yescharacteristic
553C2#2025301612yes—
554C2#2025333412yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC24×D47D4
hypercenterC2×#419D47
hypocenterC47#23481
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689132C2semidirect
6. #20244589C2semidirect
7. #20244582C2semidirect
8. #20243108C2semidirect
9. C2×C47×#419C2semidirect
21. C2×#13093957C22semidirect
22. C2×#13093955C22semidirect
24. C2×#13093948C22semidirect
32. #15688886C22semidirect
37. C2×#103×C47C22semidirect
38. C2×#103×C47C22semidirect
274. C2×#419D47semidirect
306. C2×#103C2×D47semidirect
307. C2×#103C2×D47semidirect
316. C47#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#130940002
C2×#156891662
C2×#156891322
C2×#156890862
#202457672
#202445892
#202445822
#202431082
C2×C47×#4192
#2348147