Finite Group Databasebeta

← #26226037

Subgroups — #26226037

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic
3#20245793C211yesmaximal, characteristic
4#20244359C212yesmaximal, complemented
5#20244206C212yesmaximal, complemented
6#20243699C212yesmaximal
7#20243697C212yesmaximal
8#20243394C211yesmaximal, characteristic, complemented
9#20243393C211yesmaximal, characteristic, complemented
10#20243390C211yesmaximal, characteristic, complemented
11C47×#1803C211yesmaximal, characteristic, complemented
12#15689677—24no—
13#15689677C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094004C2211yescharacteristic
18C2×#13094002—21no—
19C2×#13094000C2211yescharacteristic
20C2×C4×Dic94—21no—
21C2×C4×Dic94C2211yescharacteristic
22C2×#13093982C2212yes—
23C2×#13093981—21no—
24C2×C42×D47C2211yescharacteristic
25#15689179—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093949C2212yes—
28C2×#13093948C2212yes—
29C2×#13093947C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×#13093934C2212yes—
32#15688993C2211yescharacteristic
33#15688988C2212yes—
34#15688980—21no—
35#15688980—21no—
36C4⋊C4×Dic47—21no—
37C4⋊C4×Dic47—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#384C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094023—22no—
53#13094023C2×C414yes—
54#13094022—22no—
55#13094022C2×C414yes—
56#13094014—28no—
57#13094004—22no—
58#13094004C2×C414yes—
59#13094002—28no—
60#13094000—22no—
61#13094000C2×C414yes—
62C4×Dic94—22no—
63C4×Dic94—28no—
64C4×Dic94C2×C414yes—
65#13093982—42no—
66#13093982—42no—
67#13093981—28no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86C42×D47C2×C414yes—
87C42×D47D414yes—
88C4⋊(C47⋊C8)—22no—
89#13093949—42no—
90#13093948—42no—
91#13093947—42no—
92(C2×D47)⋊C8—24no—
93#13093934—24no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C4×Q8×C47—22no—
99C4×Q8×C47C2×C414yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108#103×C47—42no—
109C23×D47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47C22×C412yes—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2411yescharacteristic
122C22×Dic47—21no—
123C22×Dic47—21no—
124C22×Dic47—22no—
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×C4×D47—21no—
133C2×C4×D47—21no—
134C2×C4×D47—41no—
135C2×C4×D47—41no—
136C2×C4×D47C22⋊C414yes—
137C2×C4×D47C22⋊C414yes—
138C2×C4×D47C22×C412yes—
139C2×C4×D47C2×D412yes—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—24no—
146C2×Dic94—42no—
147C2×Dic94—42no—
148C2×Dic94C22×C412yes—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—42no—
157C4⋊C4×C47—42no—
158C4⋊C4×C47C22×C412yes—
159C42×C47—24no—
160C42×C47C22×C412yes—
161C42×C47C2×D412yes—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—44no—
165C4⋊Dic47—21no—
166C4⋊Dic47—24no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—42no—
170C4⋊Dic47—42no—
171C4⋊Dic47C22×C412yes—
172Dic47⋊C4—24no—
173Dic47⋊C4—24no—
174Dic47⋊C4—24no—
175Dic47⋊C4—24no—
176Dic47⋊C4—24no—
177C4×Dic47—24no—
178C4×Dic47—44no—
179C4×Dic47C22⋊C414yes—
180C4×Dic47C22⋊C414yes—
181C4×Dic47C22×C412yes—
182C4×Dic47C2×D412yes—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C23×C47C22×D411yescharacteristic
187C22×D47—21no—
188C22×D47—21no—
189C22×D47—21no—
190C22×D47—21no—
191C22×D47—21no—
192C22×D47—21no—
193C22×D47C2×C22⋊C412yes—
194Q8×C47—22no—
195Q8×C47—22no—
196Q8×C47—41no—
197Q8×C47—41no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47#13712yes—
212C2×C4×C47#13712yes—
213C2×C4×C47C23×C411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47H(5,𝔽2)11yescharacteristic
216C2×C4×C47H(5,𝔽2)11yescharacteristic
217C2×C4×C472−1+411yescharacteristic
218C2×C4×C472−1+411yescharacteristic
219C2×Dic47—21no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47C2×C22⋊C412yes—
235C2×Dic47C2×C22⋊C412yes—
236C2×Dic47C2×C22⋊C412yes—
237C4×D47—22no—
238C4×D47—22no—
239C4×D47—22no—
240C4×D47—22no—
241C4×D47—28no—
242C4×D47—44no—
243Dic94—22no—
244Dic94—22no—
245Dic94—48no—
246Dic94—41no—
247Dic94—41no—
248C8×C47—42no—
249C47⋊C8—42no—
250#21870—471nomaximal, Sylow
251C22×C47C22×C22⋊C411yescharacteristic
252C22×C47#53511yescharacteristic
253C22×C47#53611yescharacteristic
254C22×C47C2×#13711yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47#57511yescharacteristic
257C22×C47#57511yescharacteristic
258C2×D47—22no—
259C2×D47—22no—
260C2×D47—22no—
261C2×D47—41no—
262C2×D47—41no—
263C2×D47—41no—
264C2×D47—41no—
265C4×C47—22no—
266C4×C47—42no—
267C4×C47—42no—
268C4×C47—82no—
269C4×C47#41914yes—
270C4×C47#52011yescharacteristic
271C4×C47#52011yescharacteristic
272C4×C47#53711yescharacteristic
273C4×C47#53711yescharacteristic
274Dic47—22no—
275Dic47—22no—
276Dic47—22no—
277Dic47—42no—
278Dic47—42no—
279Dic47—82no—
280C2×#533—471no—
281C2×#432—471no—
282#2285—471no—
283#1993—472no—
284#1960—472no—
285#1873—472no—
286#1873—472no—
287#1803—471no—
288#1803—471no—
289#1803—471no—
290#1803D4711yescharacteristic, complemented
291C2×C47#231111yescharacteristic
292C2×C47C2×#41911yescharacteristic
293C2×C47#289611yescharacteristic
294C2×C47#321611yescharacteristic
295C2×C47#321611yescharacteristic
296C2×C47#323711yescharacteristic
297C2×C47#323711yescharacteristic
298D47—42no—
299#533—478no—
300#533—944no—
301C2×C4⋊Q8—471no—
302C2×C4⋊Q8—471no—
303C2×#125—941no—
304C2×C22⋊Q8—471no—
305C2×C22⋊Q8—471no—
306C2×C22⋊Q8—941no—
307C2×C4×Q8—471no—
308C2×C4×Q8—941no—
309C2×C4×Q8C2×D4711yescharacteristic
310C2×#117—472no—
311C2×#117—941no—
312C22×C42—471no—
313#432—944no—
314C2×C4⋊C8—471no—
315C2×C4⋊C8C2×D4711yescharacteristic
316C2×#103—472no—
317C2×#103—472no—
318C2×#103—472no—
319C2×#103C2×D4712yes—
320C2×C22⋊C8—472no—
321C2×#97—472no—
322#384—471no—
323#384C2×D4711yescharacteristic
324#382—472no—
325#380—941no—
326#380—941no—
327C4×C4⋊C4—941no—
328C4×C4⋊C4—941no—
329#340—472no—
330#340—472no—
331#340—472no—
332#340C2×D4712yes—
333C47#2187011yescharacteristic, complemented, Sylow
334C22×Q8—471no—
335C22×Q8—941no—
336C22×Q8C22×D4711yescharacteristic
337C23×C4—471no—
338C23×C4—471no—
339C23×C4—471no—
340C22×C8—472no—
341C22×C8C22×D4712yes—
342C4⋊Q8—474no—
343C4⋊Q8—474no—
344C4⋊Q8—942no—
345C4⋊Q8—942no—
346#125—948no—
347C22⋊Q8—474no—
348C22⋊Q8—474no—
349C22⋊Q8—942no—
350C22⋊Q8—948no—
351C22⋊Q8—942no—
352C4×Q8—22no—
353C4×Q8—474no—
354C4×Q8—948no—
355C4×Q8—942no—
356C4×Q8C4×D4714yes—
357#117—948no—
358#117—1882no—
359#117—1882no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—941no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—942no—
378C2×C42—21no—
379C2×C42—471no—
380C2×C42—471no—
381C2×C42—471no—
382C2×C42—474no—
383C2×C42—474no—
384C2×C42—941no—
385C2×C42—942no—
386C2×C42C22×D4711yescharacteristic
387C4⋊C8—22no—
388C4⋊C8—942no—
389#103—42no—
390#103—1882no—
391#103—1882no—
392#103—1882no—
393C22⋊C8—944no—
394#97—944no—
395#95—942no—
396#95—942no—
397C24—471no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—472no—
404C2×Q8—944no—
405C2×Q8—941no—
406C2×Q8—941no—
407C2×Q8—941no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×Q8—1882no—
411C2×Q8—1882no—
412C2×Q8C2×C4×D4712yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—471no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—474no—
420C22×C4—472no—
421C22×C4—472no—
422C22×C4—474no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—942no—
432C22×C4—1881no—
433C22×C4—1881no—
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4711yescharacteristic
436C22×C4C23×D4711yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—472no—
450C4⋊C4—944no—
451C4⋊C4—944no—
452C4⋊C4—944no—
453C4⋊C4—944no—
454C4⋊C4—944no—
455C4⋊C4—944no—
456C4⋊C4—941no—
457C4⋊C4—942no—
458C4⋊C4—942no—
459C4⋊C4—1882no—
460C4⋊C4—1882no—
461C4⋊C4C2×C4×D4712yes—
462C22⋊C4—944no—
463C22⋊C4—944no—
464C22⋊C4—1884no—
465C42—24no—
466C42—474no—
467C42—474no—
468C42—472no—
469C42—472no—
470C42—944no—
471C42—1884no—
472C42C2×C4×D4712yes—
473C42D4×D4712yes—
474C23—472no—
475C23—941no—
476C23—941no—
477C23—941no—
478C23—941no—
479C23—941no—
480C23—941no—
481C23C2×D4×D4711yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—942no—
487Q8—942no—
488Q8—1888no—
489Q8—1881no—
490Q8—1881no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—472no—
503C2×C4—472no—
504C2×C4—472no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—941no—
509C2×C4—941no—
510C2×C4—941no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—948no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—1884no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4C22⋊C4×D4712yes—
527C2×C4C22⋊C4×D4712yes—
528C2×C4#1309405312yes—
529C2×C4#1309405612yes—
530C2×C4C22×C4×D4711yescharacteristic
531C2×C4C2×D4×D4711yescharacteristic
532C2×C4#1309406911yescharacteristic
533C2×C4#1309407011yescharacteristic
534C2×C4#1309407111yescharacteristic
535C2×C4#1309407211yescharacteristic
536C8—42no—
537C8—1882no—
538C22—942no—
539C22—942no—
540C22—942no—
541C22—1881no—
542C22—1881no—
543C22—1881no—
544C22—1881no—
545C22C2×C22⋊C4×D4711yescharacteristic
546C22#1568968911yescharacteristic
547C22#1568969611yescharacteristic
548C22C2×#1309405311yescharacteristic
549C22C2×#1309405611yescharacteristic
550C22#1568991911yescharacteristic
551C22#1568992211yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—942no—
557C4—942no—
558C4—942no—
559C4—3762no—
560C4—1882no—
561C4—1882no—
562C4#1568913014yes—
563C4#1568960311yescharacteristic
564C4#1568960411yescharacteristic
565C4#1568970211yescharacteristic
566C4#1568970311yescharacteristic
567C2—1882no—
568C2#2024597711yescharacteristic
569C2C2×#1568913011yescharacteristic
570C2#2025190411yescharacteristic
571C2#2025439711yescharacteristic
572C2#2025439811yescharacteristic
573C2#2025460011yescharacteristic
574C2#2025460111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1803C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1803D47
hypocenterC47#21870
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #20244359C2semidirect
5. #20244206C2semidirect
8. #20243394C2semidirect
9. #20243393C2semidirect
10. #20243390C2semidirect
11. C47×#1803C2semidirect
13. #15689677C4semidirect
290. #1803D47semidirect
333. C47#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×#156891792
#202457932
#202443592
#202442062
#202436992
#202436972
#202433942
#202433932
#202433902
C47×#18032
#2187047