Finite Group Databasebeta

← #26217799

Subgroups — #26217799

Order 12032.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094055C211yesmaximal, characteristic
2#20256589C211yesmaximal, characteristic
3C2×#15689139C211yesmaximal, characteristic
4#20245974C211yesmaximal, characteristic, complemented
5#20243754C218yesmaximal, complemented
6#20243126C211yesmaximal, characteristic, complemented
7#20243097C211yesmaximal, characteristic, complemented
8C2×C47×#419C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C2×#13094055—24no—
11C2×#13094055—22no—
12C2×#13094055C418yescomplemented
13C2×#13094055C2214yes—
14C22×#10625337—21no—
15C22×#10625336—21no—
16C22×#10625329C2211yescharacteristic
17#15689604—22no—
18#15689603C2214yes—
19#15689600—21no—
20#15689597—21no—
21#15689596—22no—
22C4×#10625350—22no—
23#15689591—22no—
24C2×#13093983C2211yescharacteristic
25C2×#13093983C2211yescharacteristic
26#117×D47—22no—
27#15689139—24no—
28C2×#13093950C2214yes—
29#15689030—22no—
30#15689030C2214yescomplemented
31#15688986—22no—
32#15688891C2214yescomplemented
33#15688882C2214yescomplemented
34C23×Q8×C47C2211yescharacteristic
35C22×M16×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C47×#419—22no—
38C47×#419C2214yescomplemented
39C2×#103×C47—22no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D414yes—
46C2×#10625350D412yes—
47C2×#10625350C2312yes—
48C22×Dic94D411yescharacteristic
49C22×D188D411yescharacteristic
50C22×C22⋊D47—21no—
51C22×C4×D47—21no—
52#13094055—216no—
53#13094055—28no—
54#13094055C2×C4116yescomplemented
55C2×#10625337—28no—
56C2×#10625337—24no—
57C2×#10625336—28no—
58C2×#10625336—24no—
59C2×#10625329—22no—
60C2×#10625329—21no—
61C2×#10625329C2×C414yescomplemented
62C2×#10625329C2312yes—
63C22×C47⋊C8—21no—
64C4×Dic94—24no—
65#13093989—24no—
66C4×D188—24no—
67#13093986—24no—
68C4×C22⋊D47—28no—
69(C22⋊D47)⋊C4—28no—
70#13093983—21no—
71#13093983—21no—
72#13093983—21no—
73#13093983—21no—
74#13093983—22no—
75#13093983—22no—
76#13093982—24no—
77#13093981—24no—
78#13093980—24no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47C2312yes—
81C4⋊C4×D47—24no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47D412yes—
84C22⋊C4×D47—24no—
85C2×C4×Dic47D412yes—
86C2×C4×Dic47C2312yes—
87C42×D47—24no—
88#13093950—28no—
89C47⋊#95—28no—
90C22×Q8×C47—21no—
91C22×Q8×C47—22no—
92C22×Q8×C47—21no—
93C22×Q8×C47—21no—
94C22×Q8×C47—22no—
95C22×Q8×C47—21no—
96C22×Q8×C47—21no—
97C22×Q8×C47—22no—
98C22×Q8×C47C2×C414yescomplemented
99C22×Q8×C47C2312yes—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C2×M16×C47—21no—
103C2×M16×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47C2312yes—
106C22×C8×C47—21no—
107#117×C47—21no—
108#117×C47—21no—
109#117×C47D412yes—
110#117×C47C2312yes—
111C2×C4⋊C4×C47D412yes—
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114#103×C47—28no—
115C24×C47C2×D411yescharacteristic
116C23×D47—21no—
117C2×Q8×C47—24no—
118C2×Q8×C47—22no—
119C2×Q8×C47—24no—
120C2×Q8×C47—24no—
121C2×Q8×C47—24no—
122C2×Q8×C47—24no—
123C2×Q8×C47—22no—
124C2×Q8×C47—22no—
125C2×Q8×C47—22no—
126C2×Q8×C47—22no—
127C2×Q8×C47—41no—
128C2×Q8×C47—41no—
129C2×Q8×C47—42no—
130C2×Q8×C47—42no—
131C2×Q8×C47—42no—
132C2×Q8×C47—42no—
133C2×Q8×C47—42no—
134C2×Q8×C47C22×C414yes—
135C22×C4×C47—24no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—41no—
139C22×C4×C47—41no—
140C22×C4×C47—42no—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47#4112yes—
149C22×C4×C47#4112yes—
150C22×C4×C47C2411yescharacteristic
151C2×C22⋊D47—24no—
152C2×C22⋊D47—22no—
153C2×C22⋊D47—24no—
154C2×C22⋊D47—22no—
155C22×Dic47—21no—
156C22×Dic47—22no—
157C22×Dic47C2×D414yes—
158#10625350—24no—
159#10625350—24no—
160#10625350—24no—
161#10625350—24no—
162#10625350—24no—
163#10625350—22no—
164#10625350—22no—
165#10625350C22⋊C418yes—
166#10625350C22×C414yes—
167#10625350C2×D414yes—
168C2×D188—21no—
169C2×D188—22no—
170C2×D188C22⋊C414yes—
171C2×D188C2×D412yes—
172C2×C4×D47—24no—
173C2×C4×D47—22no—
174C2×C4×D47—24no—
175C2×C4×D47—22no—
176C2×C4×D47—28no—
177C2×Dic94—21no—
178C2×Dic94—22no—
179C2×Dic94C22⋊C414yes—
180C2×Dic94C2×D412yes—
181M16×C47—22no—
182M16×C47—22no—
183M16×C47—42no—
184C2×C8×C47—22no—
185C2×C8×C47—42no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47C2×D414yes—
188C22⋊C4×C47—22no—
189C22⋊C4×C47—22no—
190C42×C47—22no—
191C42×C47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194C22⋊Dic47—42no—
195#10625337—216no—
196#10625336—216no—
197(C2×D47)⋊C4—28no—
198C4⋊Dic47—22no—
199C4⋊Dic47—22no—
200C4⋊Dic47—42no—
201Dic47⋊C4—28no—
202C4×Dic47—28no—
203C4×Dic47—22no—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207#10625329—24no—
208#10625329—22no—
209#10625329C22×C414yes—
210C2×C47⋊C8—24no—
211C2×C47⋊C8—22no—
212C23×C47—42no—
213C23×C47C4×D412yes—
214C23×C47#12011yescharacteristic
215C23×C47#12011yescharacteristic
216C23×C47#12112yes—
217C23×C47C22×D411yescharacteristic
218C22×D47—22no—
219C22×D47—24no—
220C22×D47—82no—
221Q8×C47—24no—
222Q8×C47—24no—
223Q8×C47—42no—
224Q8×C47—42no—
225Q8×C47—44no—
226Q8×C47—44no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—44no—
232C2×C4×C47—44no—
233C2×C4×C47—82no—
234C2×C4×C47C2×C22⋊C412yes—
235C2×C4×C47C2×C22⋊C412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47#12011yescharacteristic
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47#12112yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12112yes—
244C2×C4×C47#12312yes—
245C2×C4×C47#12312yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C22⋊D47—24no—
253C22⋊D47—24no—
254C22⋊D47—28no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—44no—
259C2×Dic47—44no—
260C2×Dic47—42no—
261D188—22no—
262D188—24no—
263D188C2×C22⋊C414yes—
264C4×D47—24no—
265C4×D47—24no—
266C4×D47—28no—
267C4×D47—88no—
268Dic94—22no—
269Dic94—24no—
270Dic94C2×C22⋊C414yes—
271C8×C47—42no—
272C47⋊C8—24no—
273#20944—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47#38612yes—
280C22×C47#38612yes—
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C2×#12011yescharacteristic
283C22×C47C2×#12111yescharacteristic
284C2×D47—24no—
285C2×D47—44no—
286C2×D47—84no—
287C4×C47—44no—
288C4×C47—44no—
289C4×C47#38612yes—
290C4×C47#38612yes—
291C4×C47C22×C22⋊C411yescharacteristic
292C4×C47C2×C4×D411yescharacteristic
293C4×C47C2×#12111yescharacteristic
294C4×C47C2×#12311yescharacteristic
295Dic47—24no—
296Dic47—44no—
297Dic47—82no—
298C22×#137—471no—
299#3428—471no—
300C2×#421—471no—
301C2×#419D4711yescharacteristic, complemented
302#2310—471no—
303#1882—478no—
304#1742—471no—
305#1733—471no—
306C2×C47—21no—
307C2×C47—22no—
308C2×C47—21no—
309C2×C47—22no—
310C2×C47C2×#38611yescharacteristic
311C2×C47#302212yes—
312D47—44no—
313C22×#41—471no—
314C23×Q8C2×D4711yescharacteristic
315C2×#137—478no—
316C2×#137—474no—
317C2×#137—942no—
318C2×#137—944no—
319C22×Q16—941no—
320C22×SD16—941no—
321C22×M16—471no—
322C22×M16C2×D4711yescharacteristic
323#520—474no—
324#520—942no—
325#519—941no—
326#518—942no—
327#518—941no—
328C4×#41—942no—
329C4×#41—942no—
330C2×#117—471no—
331C2×#117—471no—
332C2×#117—942no—
333C2×#117C2×D4711yescharacteristic
334#421—944no—
335C2×#104—474no—
336#419—22no—
337#419C2×D4714yescomplemented
338C2×#103—22no—
339#391—474no—
340#391—942no—
341#381—942no—
342#343—474no—
343#339—474no—
344C47#2094411yescharacteristic, complemented, Sylow
345C2×#41—474no—
346C2×#41—472no—
347C2×#41—474no—
348C2×#41—472no—
349C2×#41—942no—
350C2×#41—941no—
351C2×#41—942no—
352C2×#41—941no—
353C22×Q8—21no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8—21no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—21no—
360C22×Q8—21no—
361C22×Q8—471no—
362C22×Q8C4×D4714yescomplemented
363C22×Q8C22×D4712yes—
364C22×D4—471no—
365C22×D4—941no—
366C23×C4—21no—
367C23×C4—941no—
368C23×C4C22×D4711yescharacteristic
369#137—4716no—
370#137—9416no—
371#137—948no—
372C2×Q16—948no—
373C2×Q16—944no—
374C2×SD16—948no—
375C2×SD16—944no—
376C2×M16—21no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—474no—
380C2×M16—472no—
381C2×M16—942no—
382C2×M16—941no—
383C2×M16C22×D4712yes—
384C22×C8—21no—
385C22×C8—941no—
386C4×Q8—944no—
387C4×Q8—944no—
388C4×D4—944no—
389C4×D4—948no—
390C4×D4—944no—
391C4×D4—948no—
392#117—21no—
393#117—21no—
394#117—944no—
395#117—944no—
396#117—941no—
397#117—941no—
398#117—941no—
399#117—941no—
400#117—944no—
401#117—942no—
402#117—942no—
403#117C22⋊D4712yes—
404#117C22×D4712yes—
405C2×C4⋊C4—472no—
406C2×C4⋊C4—944no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4C22⋊D4712yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—472no—
411C2×C22⋊C4—944no—
412C2×C22⋊C4—941no—
413C2×C42—21no—
414C2×C42—472no—
415C2×C42—472no—
416C2×C42—944no—
417#104—948no—
418#103—28no—
419#95—948no—
420C24—941no—
421C24D4×D4711yescharacteristic
422#41—478no—
423#41—474no—
424#41—474no—
425#41—944no—
426#41—944no—
427#41—944no—
428#41—944no—
429#41—944no—
430#41—942no—
431#41—942no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—474no—
450C2×Q8—472no—
451C2×Q8—941no—
452C2×Q8—942no—
453C2×Q8C2×C4×D4714yes—
454C2×D4—474no—
455C2×D4—472no—
456C2×D4—944no—
457C2×D4—942no—
458C2×D4—944no—
459C2×D4—942no—
460C2×D4—941no—
461C2×D4—942no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—474no—
469C22×C4—948no—
470C22×C4—941no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4—944no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4C2×C4×D4712yes—
477C22×C4#1062535012yes—
478C22×C4D4×D4711yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4712yes—
481C22×C4D4×D4711yescharacteristic
482C22×C4#1062535212yes—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486Q16—9416no—
487SD16—9416no—
488M16—22no—
489M16—22no—
490M16—42no—
491M16—474no—
492M16—944no—
493M16—942no—
494C2×C8—22no—
495C2×C8—42no—
496C2×C8—944no—
497C2×C8—942no—
498C4⋊C4—22no—
499C4⋊C4—948no—
500C4⋊C4—942no—
501C4⋊C4—942no—
502C4⋊C4—1882no—
503C4⋊C4C2×C22⋊D4714yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—948no—
507C22⋊C4—942no—
508C22⋊C4—942no—
509C22⋊C4—1882no—
510C42—22no—
511C42—22no—
512C42—948no—
513C42—942no—
514C42—942no—
515C42—942no—
516C42—942no—
517C23—42no—
518C23—944no—
519C23—942no—
520C23—3762no—
521C23(C22⋊D47)⋊C412yes—
522C23#1309399211yescharacteristic
523C23#1309399211yescharacteristic
524C23#1309399712yes—
525C23C2×D4×D4711yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—44no—
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—474no—
533Q8—944no—
534Q8—942no—
535D4—474no—
536D4—948no—
537D4—944no—
538D4—944no—
539D4—944no—
540D4—942no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—82no—
548C2×C4—944no—
549C2×C4—942no—
550C2×C4—948no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—3768no—
555C2×C4—1884no—
556C2×C4—1882no—
557C2×C4—1884no—
558C2×C4C22⋊C4×D4712yes—
559C2×C4C22⋊C4×D4712yes—
560C2×C4(C22⋊D47)⋊C412yes—
561C2×C4C4×C22⋊D4712yes—
562C2×C4C4×C22⋊D4712yes—
563C2×C4#1309399211yescharacteristic
564C2×C4#1309399211yescharacteristic
565C2×C4#1309399512yes—
566C2×C4#1309399512yes—
567C2×C4#1309399712yes—
568C2×C4#1309400612yes—
569C2×C4#1309400612yes—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4C22×C22⋊D4711yescharacteristic
574C2×C4C2×#1062535211yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C8—42no—
577C8—944no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—944no—
584C22—3764no—
585C22—1884no—
586C22#1568899912yes—
587C22#1568899912yes—
588C22C2×(C22⋊D47)⋊C411yescharacteristic
589C22C2×#1309399211yescharacteristic
590C22C2×#1309399711yescharacteristic
591C4—44no—
592C4—44no—
593C4—944no—
594C4—3762no—
595C4—1884no—
596C4#1568899912yes—
597C4#1568899912yes—
598C4C2×C22⋊C4×D4711yescharacteristic
599C4C2×C4×C22⋊D4711yescharacteristic
600C4C2×#1309399511yescharacteristic
601C4C2×#1309400611yescharacteristic
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—1884no—
607C2C2×#1568899911yescharacteristic
608C2#2025276412yes—
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#419D47
hypocenterC47#20944
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. #20245974C2semidirect
5. #20243754C2semidirect
6. #20243126C2semidirect
7. #20243097C2semidirect
8. C2×C47×#419C2semidirect
12. C2×#13094055C4semidirect
30. #15689030C22semidirect
32. #15688891C22semidirect
33. #15688882C22semidirect
38. C47×#419C22semidirect
54. #13094055C2×C4semidirect
61. C2×#10625329C2×C4semidirect
98. C22×Q8×C47C2×C4semidirect
301. C2×#419D47semidirect
337. #419C2×D47semidirect
344. C47#20944semidirect
362. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940552
#202565892
C2×#156891392
#202459742
#202437542
#202431262
#202430972
C2×C47×#4192
#2094447