Finite Group Databasebeta

← #26217366

Subgroups — #26217366

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#20245876C211yesmaximal, characteristic
4#20245872C211yesmaximal, characteristic, complemented
5#20245870C211yesmaximal, characteristic, complemented
6#20243670C218yesmaximal, complemented
7#20243120C211yesmaximal, characteristic, complemented
8C47×#1740C211yesmaximal, 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
19#117×D47—22no—
20C22×C4×Dic47C2211yescharacteristic
21#15689096C418yescomplemented
22C2×#13093938C2214yes—
23#15689028—21no—
24#15689024—21no—
25#15689022—22no—
26#15689019—22no—
27#15689000C2214yes—
28#15688998C2214yes—
29#15688995C2214yes—
30C4×Dic47⋊C4—22no—
31C4×C4⋊Dic47—22no—
32C4×(C2×D47)⋊C4—22no—
33#15688889C2214yes—
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C47×#342C2214yes—
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—
49C23×Dic47C2312yes—
50C22×C4×D47D412yes—
51C4×Dic94—24no—
52#13093989—24no—
53C4×D188—24no—
54#13093986—24no—
55C4×C22⋊D47—28no—
56(C22⋊D47)⋊C4—28no—
57#13093983—21no—
58#13093983—21no—
59#13093983—21no—
60#13093983—21no—
61#13093983—22no—
62#13093983C2×C414yescomplemented
63#13093982—24no—
64#13093981—24no—
65#13093980—24no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—24no—
68C2×C4⋊Dic47D412yes—
69C2×Dic47⋊C4—24no—
70C4⋊C4×D47—24no—
71C2×C22⋊Dic47C2312yes—
72C2×C22⋊Dic47C2312yes—
73C2×(C2×D47)⋊C4—24no—
74C22⋊C4×D47—24no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D412yes—
83C42×D47—24no—
84#13093938C2×C4116yescomplemented
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95—24no—
88C23×C4×C47C2311yescharacteristic
89#117×C47—21no—
90#117×C47—21no—
91#117×C47—21no—
92#117×C47—21no—
93#117×C47—22no—
94#117×C47—22no—
95#117×C47—22no—
96#117×C47C2×C414yescomplemented
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47D412yes—
100C2×C22⋊C4×C47C2312yes—
101C2×C22⋊C4×C47C2312yes—
102C2×C22⋊C4×C47C2312yes—
103C2×C42×C47—21no—
104C2×C42×C47—21no—
105C2×C42×C47D412yes—
106C24×C47C2411yescharacteristic
107C23×D47C2×D412yes—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47C22×C412yes—
111C22×C4×C47C2×D414yes—
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×Q812yes—
119C22×C4×C47#4112yes—
120C22×C4×C47#4112yes—
121C2×C22⋊D47—22no—
122C2×C22⋊D47—22no—
123C2×C22⋊D47—22no—
124C2×C22⋊D47—22no—
125C2×C22⋊D47C22×C414yes—
126C2×C22⋊D47C2×Q814yes—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—44no—
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—42no—
159C22⋊C4×C47—22no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—22no—
164C22⋊C4×C47C22×C414yes—
165C42×C47—24no—
166C42×C47—22no—
167C42×C47—22no—
168C42×C47—42no—
169C22⋊Dic47—22no—
170C22⋊Dic47—22no—
171C22⋊Dic47—22no—
172C22⋊Dic47C22×C414yes—
173(C2×D47)⋊C4—28no—
174(C2×D47)⋊C4—48no—
175C4⋊Dic47—24no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178C4⋊Dic47—48no—
179Dic47⋊C4—28no—
180Dic47⋊C4—48no—
181C4×Dic47—28no—
182C4×Dic47—24no—
183C4×Dic47—22no—
184C4×Dic47—22no—
185C4×Dic47—41no—
186C4×Dic47—41no—
187C4×Dic47—41no—
188C4×Dic47—41no—
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—44no—
204C2×C4×C47—44no—
205C2×C4×C47—44no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C4⋊C412yes—
208C2×C4×C47C2×C4⋊C412yes—
209C2×C4×C47C4×D412yes—
210C2×C4×C47C4×D412yes—
211C2×C4×C47C4×D412yes—
212C2×C4×C47C4×D412yes—
213C2×C4×C47#12011yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47#12011yescharacteristic
216C2×C4×C47#12011yescharacteristic
217C2×C4×C47C22⋊Q812yes—
218C2×C4×C47C22⋊Q812yes—
219C2×C4×C47C22⋊Q812yes—
220C2×C4×C47C22⋊Q812yes—
221C2×C4×C47#12312yes—
222C2×C4×C47#12312yes—
223C22⋊D47—24no—
224C22⋊D47—24no—
225C22⋊D47—24no—
226C22⋊D47—24no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—44no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—44no—
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—82no—
270C4×C47#38512yes—
271C4×C47#38512yes—
272C4×C47#38512yes—
273C4×C47#38512yes—
274Dic47—24no—
275Dic47—44no—
276Dic47—44no—
277Dic47—44no—
278#3428—471no—
279C2×#410—471no—
280#2297—471no—
281#2296—471no—
282#2296—471no—
283#1867—478no—
284#1740—471no—
285#1740D4711yescharacteristic, complemented
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—942no—
306C2×#117C2×D4711yescharacteristic
307C2×#117C2×D4711yescharacteristic
308C2×#117C2×D4711yescharacteristic
309C22×C42—471no—
310#410—478no—
311C2×#99—474no—
312#391—941no—
313#390—941no—
314#389—942no—
315#389—942no—
316#386—474no—
317#385—474no—
318#385—474no—
319C4×C4⋊C4—942no—
320C4×C4⋊C4—942no—
321C4×C22⋊C4—942no—
322#342—474no—
323#342C2×D4714yes—
324C47#2088811yescharacteristic, complemented, Sylow
325C2×#41—474no—
326C2×#41—474no—
327C2×#41—941no—
328C2×#41—941no—
329C2×#41—942no—
330C2×#41—941no—
331C2×#41—941no—
332C2×#41—942no—
333C22×Q8—471no—
334C22×D4—472no—
335C22×D4—471no—
336C23×C4—472no—
337C23×C4—472no—
338C23×C4C22×D4711yescharacteristic
339C4×Q8—944no—
340C4×Q8—944no—
341C4×D4—944no—
342C4×D4—948no—
343C4×D4—944no—
344C4×D4—948no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—474no—
353#117—944no—
354#117—944no—
355#117—944no—
356#117—941no—
357#117—941no—
358#117—942no—
359#117—941no—
360#117—941no—
361#117C4×D4714yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4D18812yes—
370C2×C22⋊C4—472no—
371C2×C22⋊C4—472no—
372C2×C22⋊C4—944no—
373C2×C22⋊C4—944no—
374C2×C22⋊C4C22×D4712yes—
375C2×C22⋊C4C22×D4712yes—
376C2×C22⋊C4C22×D4712yes—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—472no—
380C2×C42—944no—
381C2×C42—941no—
382C2×C42—941no—
383C2×C42—941no—
384C2×C42—941no—
385C2×C42—941no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42D18812yes—
389#99—4716no—
390#95—944no—
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—472no—
420C22×C4—478no—
421C22×C4—474no—
422C22×C4—948no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—944no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—1884no—
431C22×C4C2×C4×D4712yes—
432C22×C4C2×D18814yes—
433C22×C4C2×D18811yescharacteristic
434C22×C4C2×D18811yescharacteristic
435C22×C4#1062535012yes—
436C22×C4D4×D4711yescharacteristic
437C22×C4D4×D4711yescharacteristic
438C22×C4D4×D4711yescharacteristic
439C22×C4D4×D4711yescharacteristic
440C22×C4#1062535212yes—
441C22×C4Q8×D4712yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—948no—
447C4⋊C4—944no—
448C4⋊C4—942no—
449C4⋊C4—942no—
450C4⋊C4—1888no—
451C4⋊C4—1888no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—474no—
458C22⋊C4—948no—
459C22⋊C4—942no—
460C22⋊C4—942no—
461C22⋊C4—942no—
462C22⋊C4—1888no—
463C22⋊C4C2×C4×D4714yes—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—948no—
469C42—944no—
470C42—942no—
471C42—942no—
472C42—1881no—
473C42—1881no—
474C42—1881no—
475C42—1881no—
476C23—42no—
477C23—474no—
478C23—942no—
479C23—942no—
480C23—1884no—
481C23C22×C4×D4711yescharacteristic
482C23C2×D4×D4711yescharacteristic
483C23C2×D4×D4711yescharacteristic
484C23C22×D18811yescharacteristic
485C23C2×Q8×D4711yescharacteristic
486C23C2×#1062535211yescharacteristic
487C23C2×#1062535011yescharacteristic
488Q8—944no—
489Q8—944no—
490D4—944no—
491D4—944no—
492D4—944no—
493D4—944no—
494D4—944no—
495D4—944no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—474no—
503C2×C4—942no—
504C2×C4—942no—
505C2×C4—948no—
506C2×C4—948no—
507C2×C4—944no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—3768no—
513C2×C4—1884no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1884no—
519C2×C4C4⋊C4×D4712yes—
520C2×C4C4⋊C4×D4712yes—
521C2×C4(C22⋊D47)⋊C412yes—
522C2×C4(C22⋊D47)⋊C412yes—
523C2×C4C4×D18812yes—
524C2×C4C4×D18812yes—
525C2×C4#1309399411yescharacteristic
526C2×C4#1309399411yescharacteristic
527C2×C4#1309399411yescharacteristic
528C2×C4#1309399411yescharacteristic
529C2×C4#1309400212yes—
530C2×C4#1309400212yes—
531C2×C4#1309400412yes—
532C2×C4#1309400412yes—
533C2×C4#1309400612yes—
534C2×C4#1309400612yes—
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×(C22⋊D47)⋊C411yescharacteristic
548C22C2×C4×D18811yescharacteristic
549C22C2×#1309399411yescharacteristic
550C22C2×#1309400211yescharacteristic
551C22C2×#1309400411yescharacteristic
552C22C2×#1309400611yescharacteristic
553C4—44no—
554C4—44no—
555C4—82no—
556C4—944no—
557C4—1884no—
558C4—1884no—
559C4—1884no—
560C4#1568899712yes—
561C4#1568899712yes—
562C4#1568899712yes—
563C4#1568899712yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—1884no—
571C2C2×#1568899711yescharacteristic
572C2#2025278212yes—
subgroupG/H
centerC2×C4#13093994
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1740C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC24×C47C24
hypercenter#1740D47
hypocenterC47#20888
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. #20245872C2semidirect
5. #20245870C2semidirect
6. #20243670C2semidirect
7. #20243120C2semidirect
8. C47×#1740C2semidirect
21. #15689096C4semidirect
62. #13093983C2×C4semidirect
84. #13093938C2×C4semidirect
96. #117×C47C2×C4semidirect
145. C2×D188C4⋊C4semidirect
150. C2×C4×D47C4⋊C4semidirect
154. C2×Dic94C4⋊C4semidirect
285. #1740D47semidirect
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
#202458762
#202458722
#202458702
#202436702
#202431202
C47×#17402
#2088847