Finite Group Databasebeta

← #26216081

Subgroups — #26216081

Order 12032.

1470 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15689086C211yesmaximal, characteristic
3#20245767C211yesmaximal, characteristic
4#20243482C214yesmaximal, complemented
5#20243481C214yesmaximal, complemented
6#20243109C211yesmaximal, characteristic, complemented
7#20243108C211yesmaximal, characteristic, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093982—24no—
14C2×#13093982—21no—
15C2×#13093982—21no—
16C2×#13093982C418yescomplemented
17C2×#13093982C2212yes—
18C2×#13093982C2212yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×(C2×D47)⋊C4—21no—
21C22×C4×Dic47—21no—
22#15689086—24no—
23C2×(C2×D47)⋊C8C2212yes—
24C2×(C2×D47)⋊C8C2212yes—
25#15688984—21no—
26#15688980C2212yes—
27#15688980C2212yes—
28#15688978—21no—
29C4×(C2×D47)⋊C4—21no—
30C4×(C2×D47)⋊C4—21no—
31#15688886C2214yescomplemented
32#15688885C2214yescomplemented
33#15688885C2214yescomplemented
34C22×M16×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#340C2214yescomplemented
38C24×D47D411yescharacteristic
39C23×Dic47—21no—
40C23×Dic47D411yescharacteristic
41C22×C4×D47—22no—
42C22×C4×D47—22no—
43C22×C4×D47—22no—
44C22×C4×D47—22no—
45C22×C4×D47C2×C414yes—
46C22×C4×D47D411yescharacteristic
47C22×C4×D47D411yescharacteristic
48C22×C4×D47D414yes—
49C22×C4×D47C2311yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C22×C47⋊C8C2312yes—
55#13093983—22no—
56#13093983—22no—
57#13093982—28no—
58#13093982—28no—
59#13093982—24no—
60#13093982—24no—
61#13093982C2×C418yes—
62#13093982C2×C418yes—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47C2×C414yes—
66C2×C4⋊Dic47C2312yes—
67C2×C4⋊Dic47C2312yes—
68C2×C22⋊Dic47—21no—
69C2×(C2×D47)⋊C4—28no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—22no—
74C2×C4×Dic47—28no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47—22no—
77C2×C4×Dic47—21no—
78(C2×D47)⋊C8—24no—
79(C2×D47)⋊C8—24no—
80C47⋊#95—22no—
81C47⋊#95—22no—
82C23×C4×C47—21no—
83C23×C4×C47C2311yescharacteristic
84C2×M16×C47—22no—
85C2×M16×C47—22no—
86C2×M16×C47—22no—
87C22×C8×C47C2312yes—
88#117×C47—22no—
89#117×C47—22no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47C2×C414yes—
93C2×C4⋊C4×C47C2312yes—
94C2×C4⋊C4×C47C2312yes—
95C2×C22⋊C4×C47—21no—
96C2×C42×C47—21no—
97C24×C47C2×D411yescharacteristic
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—42no—
101C23×D47—42no—
102C23×D47C22⋊C414yes—
103C23×D47C2×D411yescharacteristic
104C23×D47C2×D411yescharacteristic
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—44no—
109C22×C4×C47C22×C412yes—
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D412yes—
112C22×C4×C47#4112yes—
113C22×C4×C47#4112yes—
114C22×C4×C47#4112yes—
115C22×C4×C47#4112yes—
116C22×C4×C47C2411yescharacteristic
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—24no—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—44no—
123C22×Dic47C22⋊C414yes—
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C2×C4×D47—28no—
127C2×C4×D47—28no—
128C2×C4×D47—28no—
129C2×C4×D47—28no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—21no—
135C2×C4×D47—21no—
136C2×C4×D47—21no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47C22⋊C414yes—
143C2×C4×D47C22⋊C414yes—
144C2×C4×D47C22×C412yes—
145C2×C4×D47C22×C412yes—
146C2×C4×D47C2×D412yes—
147C2×C4×D47C2×D412yes—
148M16×C47—44no—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47C22×C414yes—
157C22⋊C4×C47—42no—
158C42×C47—42no—
159C22⋊Dic47—42no—
160(C2×D47)⋊C4—28no—
161(C2×D47)⋊C4—28no—
162(C2×D47)⋊C4—44no—
163(C2×D47)⋊C4—44no—
164C4⋊Dic47—24no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47C22×C414yes—
169C4×Dic47—28no—
170C4×Dic47—28no—
171C4×Dic47—42no—
172#10625329—44no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47C2×C22⋊C412yes—
179C23×C47#12312yes—
180C23×C47#12312yes—
181C23×C47C22×D411yescharacteristic
182C22×D47—21no—
183C22×D47—21no—
184C22×D47—21no—
185C22×D47—21no—
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—28no—
191C22×D47—44no—
192C22×D47—44no—
193C22×D47—48no—
194C22×D47—48no—
195C22×D47C2×C22⋊C412yes—
196C22×D47C2×C22⋊C412yes—
197C2×C4×C47—24no—
198C2×C4×C47—48no—
199C2×C4×C47—44no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47C2×C22⋊C412yes—
204C2×C4×C47#11714yes—
205C2×C4×C47#11714yes—
206C2×C4×C47#12314yes—
207C2×C4×C47#12312yes—
208C2×C4×C47#12314yes—
209C2×C4×C47#12312yes—
210C2×C4×C47C23×C411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C2×#4111yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—28no—
221C2×Dic47—24no—
222C2×Dic47—48no—
223C2×Dic47—44no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47C2×C22⋊C412yes—
228C2×Dic47C2×C22⋊C412yes—
229C4×D47—22no—
230C4×D47—22no—
231C4×D47—22no—
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—28no—
238C4×D47—28no—
239C8×C47—42no—
240C47⋊C8—42no—
241#20701—471nomaximal, Sylow
242C22×C47—24no—
243C22×C47—24no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47#38114yes—
247C22×C47C22×C22⋊C411yescharacteristic
248C22×C47C2×#12311yescharacteristic
249C22×C47C2×#12311yescharacteristic
250C22×C47#57512yes—
251C22×C47#57512yes—
252C2×D47—22no—
253C2×D47—22no—
254C2×D47—22no—
255C2×D47—22no—
256C2×D47—22no—
257C2×D47—22no—
258C2×D47—416no—
259C2×D47—42no—
260C2×D47—42no—
261C2×D47—42no—
262C2×D47—42no—
263C4×C47—44no—
264C4×C47—82no—
265C4×C47#38114yes—
266C4×C47C2×#11711yescharacteristic
267C4×C47C2×#11711yescharacteristic
268C4×C47C2×#12311yescharacteristic
269C4×C47C2×#12311yescharacteristic
270Dic47—22no—
271Dic47—22no—
272Dic47—44no—
273Dic47—82no—
274C22×#117—471no—
275C2×#407—471no—
276#2280—471no—
277#1825—474no—
278#1825—474no—
279#1736—471no—
280#1736—471no—
281#1736—471no—
282#1736D4711yescharacteristic, complemented
283C2×C47—24no—
284C2×C47C2×#38111yescharacteristic
285C2×C47#289611yescharacteristic
286C2×C47#289611yescharacteristic
287C2×C47#309312yes—
288C2×C47#309312yes—
289D47—42no—
290D47—42no—
291C24×C4—471no—
292C22×M16—471no—
293C22×M16C2×D4711yescharacteristic
294C2×#117—471no—
295C2×#117—478no—
296C2×#117—472no—
297C2×#117—472no—
298C2×#117—941no—
299C2×#117—941no—
300C2×#117—944no—
301C2×#117C2×D4711yescharacteristic
302C22×C4⋊C4—471no—
303C22×C4⋊C4C2×D4711yescharacteristic
304C22×C22⋊C4—941no—
305C22×C42—941no—
306#407—944no—
307C2×C22⋊C8—472no—
308C2×C22⋊C8—472no—
309#381—941no—
310#380—472no—
311#380—472no—
312#379—941no—
313C4×C22⋊C4—941no—
314C4×C22⋊C4—941no—
315#340—474no—
316#340—474no—
317#340—474no—
318#340C2×D4714yescomplemented
319C47#2070111yescharacteristic, complemented, Sylow
320C25—471no—
321C23×C4—21no—
322C23×C4—471no—
323C23×C4—474no—
324C23×C4—471no—
325C23×C4—471no—
326C23×C4—474no—
327C23×C4—471no—
328C23×C4—471no—
329C23×C4—942no—
330C23×C4—942no—
331C23×C4—942no—
332C23×C4—942no—
333C23×C4—941no—
334C23×C4C22×D4711yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—942no—
339C2×M16—942no—
340C2×M16—942no—
341C22×C8—472no—
342C22×C8C22×D4712yes—
343#117—22no—
344#117—22no—
345#117—478no—
346#117—478no—
347#117—948no—
348#117—948no—
349#117—944no—
350#117—944no—
351#117—942no—
352#117—942no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—474no—
356C2×C4⋊C4—472no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4C4×D4714yes—
361C2×C4⋊C4C22×D4712yes—
362C2×C4⋊C4C22×D4712yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—948no—
365C2×C22⋊C4—942no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—941no—
370C2×C42—21no—
371C2×C42—948no—
372C2×C42—942no—
373C2×C42—942no—
374C2×C42—941no—
375C22⋊C8—944no—
376C22⋊C8—944no—
377#95—942no—
378#95—942no—
379C24—471no—
380C24—471no—
381C24—474no—
382C24—942no—
383C24—942no—
384C24—1882no—
385C24—1882no—
386C24D4×D4711yescharacteristic
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—474no—
392C22×C4—471no—
393C22×C4—471no—
394C22×C4—474no—
395C22×C4—474no—
396C22×C4—472no—
397C22×C4—472no—
398C22×C4—472no—
399C22×C4—472no—
400C22×C4—942no—
401C22×C4—942no—
402C22×C4—948no—
403C22×C4—948no—
404C22×C4—948no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—948no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—941no—
411C22×C4—941no—
412C22×C4—941no—
413C22×C4—941no—
414C22×C4—941no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—941no—
418C22×C4—944no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—1884no—
422C22×C4C2×C4×D4712yes—
423C22×C4#1062535012yes—
424C22×C4#1062535012yes—
425C22×C4D4×D4711yescharacteristic
426C22×C4D4×D4712yes—
427C22×C4#1062535212yes—
428C22×C4#1062535412yes—
429C22×C4C23×D4711yescharacteristic
430M16—44no—
431M16—1884no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—942no—
436C2×C8—942no—
437C2×C8—942no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—474no—
443C4⋊C4—944no—
444C4⋊C4—942no—
445C4⋊C4—942no—
446C4⋊C4—942no—
447C4⋊C4C2×C4×D4714yes—
448C22⋊C4—42no—
449C22⋊C4—948no—
450C22⋊C4—948no—
451C22⋊C4—1884no—
452C22⋊C4—1884no—
453C22⋊C4—1882no—
454C42—42no—
455C42—948no—
456C42—948no—
457C42—1882no—
458C23—22no—
459C23—22no—
460C23—472no—
461C23—472no—
462C23—948no—
463C23—941no—
464C23—941no—
465C23—941no—
466C23—941no—
467C23—942no—
468C23—942no—
469C23—942no—
470C23—942no—
471C23—1884no—
472C23—1884no—
473C23—1888no—
474C23—1888no—
475C23C22⋊C4×D4712yes—
476C23#1309400612yes—
477C23#1309400912yes—
478C23C2×D4×D4711yescharacteristic
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—472no—
486C2×C4—472no—
487C2×C4—948no—
488C2×C4—941no—
489C2×C4—941no—
490C2×C4—941no—
491C2×C4—941no—
492C2×C4—948no—
493C2×C4—948no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—942no—
502C2×C4—944no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1888no—
507C2×C4—1884no—
508C2×C4C22⋊C4×D4712yes—
509C2×C4#1309398114yes—
510C2×C4#1309398214yes—
511C2×C4#1309400612yes—
512C2×C4#1309400614yes—
513C2×C4#1309400912yes—
514C2×C4#1309400914yes—
515C2×C4C22×C4×D4711yescharacteristic
516C2×C4C2×D4×D4711yescharacteristic
517C2×C4C2×#1062535211yescharacteristic
518C2×C4C2×#1062535011yescharacteristic
519C2×C4C2×#1062535011yescharacteristic
520C2×C4C2×#1062535411yescharacteristic
521C8—42no—
522C8—1882no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—942no—
528C22—942no—
529C22—942no—
530C22—942no—
531C22—942no—
532C22—942no—
533C22—18816no—
534C22—1882no—
535C22—1882no—
536C22—1882no—
537C22—1882no—
538C22#1568898514yes—
539C22C2×C22⋊C4×D4711yescharacteristic
540C22C2×#1309400611yescharacteristic
541C22C2×#1309400911yescharacteristic
542C22#1568991912yes—
543C22#1568992212yes—
544C4—44no—
545C4—82no—
546C4—942no—
547C4—942no—
548C4—3762no—
549C4—1884no—
550C4#1568898514yes—
551C4C2×#1309398111yescharacteristic
552C4C2×#1309398211yescharacteristic
553C4C2×#1309400611yescharacteristic
554C4C2×#1309400911yescharacteristic
555C2—24no—
556C2—1882no—
557C2—1882no—
558C2C2×#1568898511yescharacteristic
559C2#2025190211yescharacteristic
560C2#2025190611yescharacteristic
561C2#2025331812yes—
562C2#2025332212yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20701
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
4. #20243482C2semidirect
5. #20243481C2semidirect
6. #20243109C2semidirect
7. #20243108C2semidirect
8. #20243107C2semidirect
9. C47×#1736C2semidirect
16. C2×#13093982C4semidirect
31. #15688886C22semidirect
32. #15688885C22semidirect
33. #15688885C22semidirect
37. C47×#340C22semidirect
282. #1736D47semidirect
318. #340C2×D47semidirect
319. C47#20701semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156890862
#202457672
#202434822
#202434812
#202431092
#202431082
#202431072
C47×#17362
#2070147