Finite Group Databasebeta

← #26217363

Subgroups — #26217363

Order 12032.

1502 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256589C211yesmaximal, characteristic
2C2×#15689096C211yesmaximal, characteristic
3#20245878C211yesmaximal, characteristic
4#20245871C211yesmaximal, characteristic, complemented
5#20243667C218yesmaximal, complemented
6#20243121C211yesmaximal, characteristic, complemented
7#20243120C211yesmaximal, characteristic, complemented
8C47×#2296C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10#15689604—22no—
11#15689603—22no—
12#15689600—21no—
13#15689597—21no—
14#15689596C2214yes—
15C4×#10625350—22no—
16#15689591—22no—
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20#117×D47—22no—
21#15689096C418yescomplemented
22C2×#13093938C2214yes—
23#15689029—21no—
24#15689025—21no—
25#15689021—22no—
26#15689002C2214yes—
27#15688996C2214yes—
28C4×Dic47⋊C4—22no—
29C4×(C2×D47)⋊C4—22no—
30#15688890C2214yes—
31#15688889C2214yes—
32C2×#117×C47C2211yescharacteristic
33C2×#117×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C47×#389—22no—
36C47×#385C2214yes—
37C4×C4⋊C4×C47—22no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350Q814yes—
46C22×Dic94D411yescharacteristic
47C22×D188D411yescharacteristic
48C22×C22⋊D47C2312yes—
49C22×C4×D47D412yes—
50C4×Dic94—24no—
51#13093989—24no—
52C4×D188—24no—
53#13093986—24no—
54C4×C22⋊D47—28no—
55(C22⋊D47)⋊C4—28no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983—21no—
60#13093983—22no—
61#13093983—22no—
62#13093983—22no—
63#13093983C2×C414yescomplemented
64#13093982—24no—
65#13093981—24no—
66#13093980—24no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47D412yes—
70C2×Dic47⋊C4—24no—
71C4⋊C4×D47—24no—
72C2×C22⋊Dic47C2312yes—
73C2×C22⋊Dic47C2312yes—
74C2×C22⋊Dic47C2312yes—
75C2×(C2×D47)⋊C4—24no—
76C22⋊C4×D47—24no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47D412yes—
80C42×D47—24no—
81#13093938C2×C4116yescomplemented
82C47⋊#95—24no—
83C47⋊#95—24no—
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86#117×C47—21no—
87#117×C47—21no—
88#117×C47—21no—
89#117×C47—21no—
90#117×C47—22no—
91#117×C47C2×C414yescomplemented
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—24no—
94C2×C4⋊C4×C47D412yes—
95C2×C22⋊C4×C47C2312yes—
96C2×C22⋊C4×C47C2312yes—
97C2×C42×C47—21no—
98C2×C42×C47—21no—
99C2×C42×C47—21no—
100C2×C42×C47—21no—
101C2×C42×C47—21no—
102C2×C42×C47—21no—
103C2×C42×C47—21no—
104C2×C42×C47D412yes—
105#95×C47—24no—
106C24×C47C2411yescharacteristic
107C23×D47C2×D412yes—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—44no—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D414yes—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×Q812yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4112yes—
124C2×C22⋊D47—22no—
125C2×C22⋊D47—22no—
126C2×C22⋊D47—22no—
127C2×C22⋊D47—22no—
128C2×C22⋊D47C22×C414yes—
129C2×C22⋊D47C2×Q814yes—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47C2×D412yes—
133C22×Dic47C2×D414yes—
134#10625350—24no—
135#10625350—24no—
136#10625350—24no—
137#10625350—24no—
138#10625350—24no—
139#10625350—24no—
140#10625350—24no—
141#10625350—24no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—22no—
145C2×D188C4⋊C414yescomplemented
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—24no—
149C2×C4×D47—28no—
150C2×C4×D47C4⋊C418yescomplemented
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—22no—
154C2×Dic94C4⋊C414yescomplemented
155C4⋊C4×C47—24no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—48no—
159C22⋊C4×C47—22no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47C22×C414yes—
163C42×C47—24no—
164C42×C47—22no—
165C42×C47—22no—
166C42×C47—41no—
167C42×C47—41no—
168C42×C47—41no—
169C42×C47—41no—
170C22⋊Dic47—22no—
171C22⋊Dic47—22no—
172C22⋊Dic47—22no—
173C22⋊Dic47—22no—
174C22⋊Dic47—22no—
175C22⋊Dic47C22×C414yes—
176(C2×D47)⋊C4—28no—
177(C2×D47)⋊C4—48no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47—42no—
182Dic47⋊C4—28no—
183Dic47⋊C4—48no—
184C4×Dic47—28no—
185C4×Dic47—24no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C4×Dic47—42no—
189C23×C47—42no—
190C23×C47C23×C411yescharacteristic
191C23×C47C22×D411yescharacteristic
192C23×C47C22×D411yescharacteristic
193C23×C47C22×D411yescharacteristic
194C23×C47C22×Q811yescharacteristic
195C23×C47C2×#4111yescharacteristic
196C23×C47C2×#4111yescharacteristic
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—44no—
200C22×D47C2×C4⋊C414yes—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—44no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—44no—
212C2×C4×C47C2×C4⋊C412yes—
213C2×C4×C47C2×C4⋊C412yes—
214C2×C4×C47C4×D412yes—
215C2×C4×C47C4×D412yes—
216C2×C4×C47C4×D412yes—
217C2×C4×C47C4×D412yes—
218C2×C4×C47#12011yescharacteristic
219C2×C4×C47#12011yescharacteristic
220C2×C4×C47#12011yescharacteristic
221C2×C4×C47#12011yescharacteristic
222C2×C4×C47C22⋊Q812yes—
223C2×C4×C47C22⋊Q812yes—
224C2×C4×C47C22⋊Q812yes—
225C2×C4×C47C22⋊Q812yes—
226C2×C4×C47#12312yes—
227C2×C4×C47#12312yes—
228C22⋊D47—24no—
229C22⋊D47—24no—
230C22⋊D47—24no—
231C22⋊D47—24no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—44no—
237C2×Dic47—44no—
238C2×Dic47—44no—
239C2×Dic47—42no—
240C2×Dic47C2×C4⋊C414yes—
241D188—24no—
242D188—24no—
243C4×D47—28no—
244C4×D47—28no—
245C4×D47—88no—
246Dic94—24no—
247Dic94—24no—
248#20888—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—41no—
253C22×C47—41no—
254C22×C47—41no—
255C22×C47—41no—
256C22×C47C22×C4⋊C411yescharacteristic
257C22×C47C2×C4×D411yescharacteristic
258C22×C47C2×C4×D411yescharacteristic
259C22×C47C2×#12011yescharacteristic
260C22×C47C2×C22⋊Q811yescharacteristic
261C22×C47C2×C22⋊Q811yescharacteristic
262C22×C47C2×#12311yescharacteristic
263C2×D47—24no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47—82no—
267C4×C47—44no—
268C4×C47—44no—
269C4×C47—44no—
270C4×C47#38512yes—
271C4×C47#38512yes—
272C4×C47#38512yes—
273C4×C47#38512yes—
274Dic47—24no—
275Dic47—44no—
276Dic47—44no—
277Dic47—82no—
278#3428—471no—
279C2×#410—471no—
280#2297—471no—
281#2296—471no—
282#2296D4711yescharacteristic, complemented
283#1867—478no—
284#1740—471no—
285#1740—471no—
286C2×C47—21no—
287C2×C47—21no—
288C2×C47—21no—
289C2×C47—21no—
290C2×C47—21no—
291C2×C47—21no—
292C2×C47C2×#38511yescharacteristic
293C2×C47#302512yes—
294D47—44no—
295C22×#41—471no—
296#520—942no—
297#520—942no—
298#519—941no—
299#518—474no—
300#518—941no—
301C4×#41—942no—
302C4×#41—942no—
303C2×#117—471no—
304C2×#117—471no—
305C2×#117—471no—
306C2×#117—942no—
307C2×#117C2×D4711yescharacteristic
308C2×#117C2×D4711yescharacteristic
309C22×C42C2×D4711yescharacteristic
310#410—478no—
311C2×#99—474no—
312#391—941no—
313#390—941no—
314#389—22no—
315#389—942no—
316#386—474no—
317#385—474no—
318#385C2×D4714yes—
319C4×C4⋊C4—22no—
320C4×C4⋊C4—942no—
321C4×C22⋊C4—942no—
322#342—474no—
323#342—474no—
324C47#2088811yescharacteristic, complemented, Sylow
325C2×#41—474no—
326C2×#41—474no—
327C2×#41—942no—
328C2×#41—941no—
329C2×#41—941no—
330C2×#41—942no—
331C2×#41—941no—
332C2×#41—941no—
333C22×Q8—471no—
334C22×D4—472no—
335C22×D4—471no—
336C23×C4—472no—
337C23×C4C22×D4712yes—
338C23×C4C22×D4711yescharacteristic
339C4×Q8—944no—
340C4×Q8—944no—
341C4×D4—944no—
342C4×D4—948no—
343C4×D4—944no—
344C4×D4—948no—
345#117—22no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—474no—
351#117—944no—
352#117—944no—
353#117—944no—
354#117—941no—
355#117—941no—
356#117—942no—
357#117—941no—
358#117—941no—
359#117—942no—
360#117—942no—
361#117C4×D4714yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4C22⋊D4712yes—
370C2×C22⋊C4—472no—
371C2×C22⋊C4—472no—
372C2×C22⋊C4—472no—
373C2×C22⋊C4—944no—
374C2×C22⋊C4—944no—
375C2×C22⋊C4C22×D4712yes—
376C2×C22⋊C4C22×D4712yes—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—472no—
385C2×C42—944no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42C22⋊D4712yes—
389#99—4716no—
390#95—24no—
391#95—944no—
392#95—944no—
393C24—472no—
394C24C23×D4711yescharacteristic
395#41—944no—
396#41—944no—
397#41—944no—
398#41—944no—
399#41—944no—
400#41—944no—
401#41—944no—
402#41—944no—
403C2×Q8—474no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×Q8—942no—
407C2×D4—474no—
408C2×D4—474no—
409C2×D4—474no—
410C2×D4—942no—
411C2×D4—942no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—941no—
415C2×D4—941no—
416C2×D4—942no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—472no—
423C22×C4—478no—
424C22×C4—474no—
425C22×C4—948no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—944no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4C2×C4×D4712yes—
432C22×C4#1062535012yes—
433C22×C4D4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4711yescharacteristic
436C22×C4D4×D4711yescharacteristic
437C22×C4Q8×D4712yes—
438C22×C4#1062535412yes—
439C22×C4C2×C22⋊D4714yes—
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—948no—
447C4⋊C4—944no—
448C4⋊C4—942no—
449C4⋊C4—942no—
450C4⋊C4—1888no—
451C4⋊C4—1882no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—474no—
456C22⋊C4—948no—
457C22⋊C4—942no—
458C22⋊C4—942no—
459C22⋊C4—942no—
460C22⋊C4—942no—
461C22⋊C4—942no—
462C22⋊C4—1888no—
463C22⋊C4C2×C4×D4714yes—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—948no—
472C42—944no—
473C42—942no—
474C42—942no—
475C42—1882no—
476C23—42no—
477C23—474no—
478C23—942no—
479C23—942no—
480C23—1884no—
481C23C22×C4×D4711yescharacteristic
482C23C2×D4×D4711yescharacteristic
483C23C2×D4×D4711yescharacteristic
484C23C22×C22⋊D4711yescharacteristic
485C23C2×Q8×D4711yescharacteristic
486C23C2×#1062535011yescharacteristic
487C23C2×#1062535411yescharacteristic
488Q8—944no—
489Q8—944no—
490D4—944no—
491D4—944no—
492D4—944no—
493D4—944no—
494D4—944no—
495D4—944no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—474no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—948no—
511C2×C4—948no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—3768no—
515C2×C4—1884no—
516C2×C4—1882no—
517C2×C4—1884no—
518C2×C4—1884no—
519C2×C4C4⋊C4×D4712yes—
520C2×C4C4⋊C4×D4712yes—
521C2×C4C4×C22⋊D4712yes—
522C2×C4C4×C22⋊D4712yes—
523C2×C4#1309398612yes—
524C2×C4#1309398612yes—
525C2×C4#1309399211yescharacteristic
526C2×C4#1309399211yescharacteristic
527C2×C4#1309399211yescharacteristic
528C2×C4#1309399211yescharacteristic
529C2×C4#1309400012yes—
530C2×C4#1309400012yes—
531C2×C4#1309400212yes—
532C2×C4#1309400212yes—
533C2×C4#1309400912yes—
534C2×C4#1309400912yes—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—944no—
543C22—3762no—
544C22—1884no—
545C22—1884no—
546C22C2×C4⋊C4×D4711yescharacteristic
547C22C2×C4×C22⋊D4711yescharacteristic
548C22C2×#1309398611yescharacteristic
549C22C2×#1309399211yescharacteristic
550C22C2×#1309400011yescharacteristic
551C22C2×#1309400211yescharacteristic
552C22C2×#1309400911yescharacteristic
553C4—44no—
554C4—44no—
555C4—44no—
556C4—944no—
557C4—3762no—
558C4—1884no—
559C4—1884no—
560C4#1568899412yes—
561C4#1568899412yes—
562C4#1568899412yes—
563C4#1568899412yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—1884no—
571C2C2×#1568899411yescharacteristic
572C2#2025277912yes—
subgroupG/H
centerC2×C4#13093992
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#2296C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC24×C47C24
hypercenter#2296D47
hypocenterC47#20888
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
4. #20245871C2semidirect
5. #20243667C2semidirect
6. #20243121C2semidirect
7. #20243120C2semidirect
8. C47×#2296C2semidirect
21. #15689096C4semidirect
63. #13093983C2×C4semidirect
81. #13093938C2×C4semidirect
91. #117×C47C2×C4semidirect
145. C2×D188C4⋊C4semidirect
150. C2×C4×D47C4⋊C4semidirect
154. C2×Dic94C4⋊C4semidirect
282. #2296D47semidirect
324. C47#20888semidirect
361. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565892
C2×#156890962
#202458782
#202458712
#202436672
#202431212
#202431202
C47×#22962
#2088847