Finite Group Databasebeta

← #26217805

Subgroups — #26217805

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×#15689141C211yesmaximal, characteristic
4C2×#15689132C211yesmaximal, characteristic, complemented
5#20243760C218yesmaximal, complemented
6#20243127C211yesmaximal, characteristic, complemented
7#20243098C211yesmaximal, characteristic, complemented
8C47×#2310C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C2×#13094055—24no—
11C2×#13094055—22no—
12C2×#13094055C418yescomplemented
13C2×#13094055C2214yes—
14C22×#10625337—21no—
15C22×#10625336—21no—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18#15689604C2214yes—
19#15689603—22no—
20#15689600—21no—
21#15689597—21no—
22#15689596—22no—
23C4×#10625350—22no—
24#15689591—22no—
25C2×#13093983C2211yescharacteristic
26#117×D47—22no—
27#15689141—24no—
28C2×#13093952C2214yes—
29#15689132—22no—
30#15689132C2214yescomplemented
31C2×#13093948—22no—
32#15688892C2214yescomplemented
33#15688883C2214yescomplemented
34C23×Q8×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C47×#391—22no—
38C47×#391C2214yescomplemented
39C47×#381—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×#10625329—21no—
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329C2×C414yescomplemented
65C2×#10625329C2312yes—
66C2×#10625329C2312yes—
67C22×C47⋊C8—21no—
68C22×C47⋊C8—21no—
69C4×Dic94—24no—
70#13093989—24no—
71C4×D188—24no—
72#13093986—24no—
73C4×C22⋊D47—28no—
74(C22⋊D47)⋊C4—28no—
75#13093983—21no—
76#13093983—21no—
77#13093983D412yes—
78#13093983C2312yes—
79#13093982—24no—
80#13093981—24no—
81#13093980—24no—
82C2×C4⋊Dic47D412yes—
83C4⋊C4×D47—24no—
84C2×C22⋊Dic47—21no—
85C22⋊C4×D47—24no—
86C2×C4×Dic47—21no—
87C42×D47—24no—
88#13093952—28no—
89#13093948—28no—
90C22×Q8×C47—21no—
91C22×Q8×C47—22no—
92C22×Q8×C47—21no—
93C22×Q8×C47—21no—
94C22×Q8×C47—21no—
95C22×Q8×C47—21no—
96C22×Q8×C47—22no—
97C22×Q8×C47—22no—
98C22×Q8×C47C2×C414yescomplemented
99C22×Q8×C47C2312yes—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102#117×C47—21no—
103#117×C47—21no—
104#117×C47—21no—
105#117×C47—21no—
106#117×C47—22no—
107#117×C47—22no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47C2312yes—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47D412yes—
112C2×C42×C47D412yes—
113C2×C42×C47C2312yes—
114#95×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×D414yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47#4112yes—
150C22×C4×C47#4112yes—
151C22×C4×C47C2411yescharacteristic
152C2×C22⋊D47—24no—
153C2×C22⋊D47—22no—
154C2×C22⋊D47—24no—
155C2×C22⋊D47—22no—
156C22×Dic47—21no—
157C22×Dic47—22no—
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—
181C4⋊C4×C47—22no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—42no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—42no—
187C42×C47—22no—
188C42×C47—22no—
189C42×C47—22no—
190C42×C47—22no—
191C22⋊Dic47—22no—
192C22⋊Dic47—22no—
193#10625337—216no—
194#10625336—216no—
195(C2×D47)⋊C4—28no—
196C4⋊Dic47—22no—
197C4⋊Dic47C2×D414yes—
198Dic47⋊C4—28no—
199C4×Dic47—28no—
200C4×Dic47—22no—
201C4×Dic47—22no—
202#10625329—24no—
203#10625329—22no—
204#10625329—22no—
205#10625329—22no—
206#10625329—42no—
207#10625329C22×C414yes—
208C2×C47⋊C8—24no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—42no—
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—44no—
234C2×C4×C47—44no—
235C2×C4×C47—42no—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47C4×D412yes—
240C2×C4×C47C4×D412yes—
241C2×C4×C47#12011yescharacteristic
242C2×C4×C47#12011yescharacteristic
243C2×C4×C47#12112yes—
244C2×C4×C47#12112yes—
245C2×C4×C47#12112yes—
246C2×C4×C47#12312yes—
247C2×C4×C47#12312yes—
248C2×C4×C47C23×C411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C22⋊D47—24no—
255C22⋊D47—24no—
256C22⋊D47—28no—
257C2×Dic47—22no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—82no—
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—
271C47⋊C8—24no—
272C47⋊C8—42no—
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—82no—
290C4×C47#38612yes—
291C4×C47#38612yes—
292C4×C47C22×C22⋊C411yescharacteristic
293C4×C47C2×C4×D411yescharacteristic
294C4×C47C2×#12111yescharacteristic
295C4×C47C2×#12311yescharacteristic
296Dic47—24no—
297Dic47—44no—
298C22×#137—471no—
299#3428—471no—
300C2×#421—471no—
301C2×#419—471no—
302#2310D4711yescharacteristic, complemented
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×M16—471no—
323#520—474no—
324#520—942no—
325#519—941no—
326#518—941no—
327#518—942no—
328C4×#41—942no—
329C4×#41—942no—
330C2×#117—471no—
331C2×#117—942no—
332C2×#117C2×D4711yescharacteristic
333C2×#117C2×D4711yescharacteristic
334#421—944no—
335C2×#104—474no—
336#419—474no—
337#419—942no—
338C2×#103—942no—
339#391—22no—
340#391C2×D4714yescomplemented
341#381—22no—
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—22no—
354C22×Q8—21no—
355C22×Q8—21no—
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—21no—
359C22×Q8—21no—
360C22×Q8—22no—
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—474no—
377C2×M16—472no—
378C2×M16—472no—
379C2×M16—942no—
380C2×M16—941no—
381C2×M16—941no—
382C2×M16—942no—
383C2×M16—942no—
384C22×C8—941no—
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—21no—
395#117—21no—
396#117—22no—
397#117—22no—
398#117—472no—
399#117—472no—
400#117—944no—
401#117—944no—
402#117—944no—
403#117—941no—
404#117—941no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—472no—
407C2×C4⋊C4—944no—
408C2×C4⋊C4C22×D4712yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4C22⋊D4712yes—
413C2×C42—944no—
414C2×C42—941no—
415C2×C42C22⋊D4712yes—
416C2×C42C22×D4712yes—
417#104—948no—
418#103—948no—
419#95—28no—
420C24—941no—
421C24C2×C22⋊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—948no—
469C22×C4—941no—
470C22×C4—944no—
471C22×C4—942no—
472C22×C4—944no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4C2×C4×D4712yes—
476C22×C4#1062535012yes—
477C22×C4D4×D4711yescharacteristic
478C22×C4D4×D4711yescharacteristic
479C22×C4D4×D4712yes—
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4#1062535212yes—
483C22×C4C2×C22⋊D4714yes—
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486Q16—9416no—
487SD16—9416no—
488M16—474no—
489M16—944no—
490M16—942no—
491M16—942no—
492M16—942no—
493M16—1882no—
494C2×C8—944no—
495C2×C8—942no—
496C2×C8—942no—
497C2×C8—1882no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—474no—
502C4⋊C4—948no—
503C4⋊C4—942no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—948no—
508C22⋊C4—942no—
509C22⋊C4—942no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—948no—
515C42—942no—
516C42—942no—
517C23—42no—
518C23—944no—
519C23—942no—
520C23—3762no—
521C23C4×C22⋊D4712yes—
522C23#1309399211yescharacteristic
523C23#1309399211yescharacteristic
524C23#1309399512yes—
525C23C22×C22⋊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—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—944no—
551C2×C4—942no—
552C2×C4—948no—
553C2×C4—944no—
554C2×C4—944no—
555C2×C4—944no—
556C2×C4—3768no—
557C2×C4—3762no—
558C2×C4C22⋊C4×D4712yes—
559C2×C4C22⋊C4×D4712yes—
560C2×C4(C22⋊D47)⋊C412yes—
561C2×C4(C22⋊D47)⋊C412yes—
562C2×C4C4×C22⋊D4712yes—
563C2×C4#1309399211yescharacteristic
564C2×C4#1309399211yescharacteristic
565C2×C4#1309399512yes—
566C2×C4#1309399712yes—
567C2×C4#1309399712yes—
568C2×C4#1309400612yes—
569C2×C4#1309400612yes—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4C2×D4×D4711yescharacteristic
574C2×C4C2×#1062535211yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C8—944no—
577C8—1882no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—944no—
584C22—3764no—
585C22—1884no—
586C22#1568899912yes—
587C22#1568899912yes—
588C22C2×C4×C22⋊D4711yescharacteristic
589C22C2×#1309399211yescharacteristic
590C22C2×#1309399511yescharacteristic
591C4—44no—
592C4—44no—
593C4—82no—
594C4—944no—
595C4—1884no—
596C4#1568899912yes—
597C4#1568899912yes—
598C4C2×C22⋊C4×D4711yescharacteristic
599C4C2×(C22⋊D47)⋊C411yescharacteristic
600C4C2×#1309399711yescharacteristic
601C4C2×#1309400611yescharacteristic
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—1884no—
607C2C2×#1568899911yescharacteristic
608C2#2025276612yes—
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2310C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#2310D47
hypocenterC47#20944
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C2×#15689132C2semidirect
5. #20243760C2semidirect
6. #20243127C2semidirect
7. #20243098C2semidirect
8. C47×#2310C2semidirect
12. C2×#13094055C4semidirect
30. #15689132C22semidirect
32. #15688892C22semidirect
33. #15688883C22semidirect
38. C47×#391C22semidirect
54. #13094055C2×C4semidirect
64. C2×#10625329C2×C4semidirect
98. C22×Q8×C47C2×C4semidirect
302. #2310D47semidirect
340. #391C2×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×#156891412
C2×#156891322
#202437602
#202431272
#202430982
C47×#23102
#2094447