Finite Group Databasebeta

← #26217800

Subgroups — #26217800

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×#13094054C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic
3C2×#15689133C211yesmaximal, characteristic, complemented
4#20245972C211yesmaximal, characteristic, complemented
5#20243755C218yesmaximal, complemented
6#20243126C211yesmaximal, characteristic, complemented
7#20243098C211yesmaximal, characteristic, complemented
8C2×C47×#421C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C2×#13094054—24no—
11C2×#13094054—22no—
12C2×#13094054C418yescomplemented
13C2×#13094054C2214yes—
14C22×#10625337—21no—
15C22×#10625335—21no—
16C22×#10625329C2211yescharacteristic
17#15689606—22no—
18#15689606C2214yes—
19#15689601—21no—
20#15689598—22no—
21#15689598—21no—
22#41×Dic47—22no—
23#41×Dic47—22no—
24C2×#13093983—22no—
25C2×#13093983C2211yescharacteristic
26C2×#13093983C2211yescharacteristic
27#15689133—22no—
28#15689133C2214yescomplemented
29C2×#13093949—22no—
30#15689029C2214yescomplemented
31#15689028—22no—
32#15688984—22no—
33#15688891C2214yescomplemented
34#15688883C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#421—24no—
39C2×#104×C47C2214yes—
40C22×Dic94—21no—
41C22×Dic94—21no—
42C22×Dic94—21no—
43C22×Dic94—22no—
44C22×Dic94—21no—
45C22×Dic94—22no—
46C22×Dic94—21no—
47C22×Dic94—22no—
48C22×Dic94C2×C414yescomplemented
49C22×Dic94C2312yes—
50C23×Dic47—21no—
51#13094054—216no—
52#13094054—28no—
53#13094054C2×C4116yescomplemented
54C2×#10625337—28no—
55C2×#10625337—24no—
56C2×#10625335—28no—
57C2×#10625335—24no—
58C2×#10625329—22no—
59C2×#10625329—21no—
60C2×#10625329C2×C414yescomplemented
61C2×#10625329C2312yes—
62C22×C47⋊C8—21no—
63Q8×Dic47—24no—
64Q8×Dic47—24no—
65D4×Dic47—28no—
66D4×Dic47—24no—
67D4×Dic47—24no—
68D4×Dic47—28no—
69#13093983—24no—
70#13093983—24no—
71#13093983—21no—
72#13093983—21no—
73#13093983—21no—
74#13093983—21no—
75#13093983—21no—
76#13093983—21no—
77#13093983—24no—
78#13093983D412yes—
79#13093983C2312yes—
80C2×C4⋊Dic47—24no—
81C2×C4⋊Dic47D412yes—
82C2×C4⋊Dic47C2312yes—
83C2×C22⋊Dic47—24no—
84C2×C22⋊Dic47—21no—
85C2×C22⋊Dic47D412yes—
86C2×C4×Dic47—24no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D412yes—
89#13093949—28no—
90C47⋊#95—28no—
91C2×#41×C47—21no—
92C2×#41×C47—21no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47C2×C414yes—
96C2×#41×C47D414yes—
97C2×#41×C47D412yes—
98C2×#41×C47C2312yes—
99C22×Q8×C47D411yescharacteristic
100C22×D4×C47—21no—
101C22×D4×C47D411yescharacteristic
102C23×C4×C47—21no—
103C23×C4×C47C2311yescharacteristic
104C2×M16×C47—21no—
105C2×M16×C47—22no—
106C2×M16×C47—22no—
107C2×M16×C47C2312yes—
108C22×C8×C47—21no—
109#117×C47—22no—
110#117×C47—22no—
111C2×C4⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C42×C47C2312yes—
114#104×C47—28no—
115C24×C47—21no—
116C24×C47C2×D411yescharacteristic
117#41×C47—24no—
118#41×C47—24no—
119#41×C47—24no—
120#41×C47—24no—
121#41×C47—24no—
122#41×C47—22no—
123#41×C47—22no—
124#41×C47C22⋊C418yes—
125#41×C47C22×C414yes—
126#41×C47C2×D414yes—
127C2×Q8×C47—21no—
128C2×Q8×C47—22no—
129C2×Q8×C47C22⋊C414yes—
130C2×Q8×C47C2×D412yes—
131C2×D4×C47—24no—
132C2×D4×C47—22no—
133C2×D4×C47—24no—
134C2×D4×C47—22no—
135C2×D4×C47—21no—
136C2×D4×C47—22no—
137C2×D4×C47C22⋊C414yes—
138C2×D4×C47C2×D412yes—
139C22×C4×C47—21no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—28no—
158C22×Dic47—22no—
159C22×Dic47—41no—
160C22×Dic47—41no—
161C22×Dic47—42no—
162C22×Dic47C2×D414yes—
163C2×Dic94—24no—
164C2×Dic94—24no—
165C2×Dic94—24no—
166C2×Dic94—24no—
167C2×Dic94—22no—
168C2×Dic94—22no—
169C2×Dic94—22no—
170C2×Dic94—22no—
171C2×Dic94—24no—
172C2×Dic94—22no—
173C2×Dic94—41no—
174C2×Dic94—41no—
175C2×Dic94—42no—
176C2×Dic94—42no—
177C2×Dic94—42no—
178C2×Dic94—42no—
179C2×Dic94—42no—
180C2×Dic94C22×C414yes—
181M16×C47—22no—
182M16×C47—22no—
183M16×C47—42no—
184C2×C8×C47—22no—
185C2×C8×C47—42no—
186C4⋊C4×C47—42no—
187C22⋊C4×C47—42no—
188C42×C47—22no—
189C42×C47—22no—
190C22⋊Dic47—28no—
191C22⋊Dic47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194C22⋊Dic47—22no—
195#10625337—216no—
196#10625335—216no—
197C4⋊Dic47—28no—
198C4⋊Dic47—22no—
199C4⋊Dic47—22no—
200C4⋊Dic47—22no—
201C4⋊Dic47C2×D414yes—
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—22no—
213C23×C47—24no—
214C23×C47—42no—
215C23×C47—82no—
216C23×C47C4×D412yes—
217C23×C47#12011yescharacteristic
218C23×C47#12011yescharacteristic
219C23×C47#12112yes—
220C23×C47C22×D411yescharacteristic
221Q8×C47—22no—
222Q8×C47—24no—
223Q8×C47C2×C22⋊C414yes—
224D4×C47—24no—
225D4×C47—24no—
226D4×C47—28no—
227D4×C47—22no—
228D4×C47—24no—
229D4×C47C2×C22⋊C414yes—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—28no—
235C2×C4×C47—44no—
236C2×C4×C47—42no—
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C4×D412yes—
240C2×C4×C47C4×D412yes—
241C2×C4×C47C4×D412yes—
242C2×C4×C47#12011yescharacteristic
243C2×C4×C47#12011yescharacteristic
244C2×C4×C47#12112yes—
245C2×C4×C47#12112yes—
246C2×C4×C47#12112yes—
247C2×C4×C47#12312yes—
248C2×C4×C47#12312yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—44no—
261C2×Dic47—44no—
262C2×Dic47—44no—
263C2×Dic47—82no—
264C2×Dic47—88no—
265Dic94—24no—
266Dic94—24no—
267Dic94—42no—
268Dic94—42no—
269Dic94—44no—
270Dic94—44no—
271C8×C47—42no—
272C47⋊C8—24no—
273#20944—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—24no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—44no—
281C22×C47—84no—
282C22×C47#38612yes—
283C22×C47#38612yes—
284C22×C47C2×C4×D411yescharacteristic
285C22×C47C2×#12011yescharacteristic
286C22×C47C2×#12111yescharacteristic
287C4×C47—24no—
288C4×C47—82no—
289C4×C47#38612yes—
290C4×C47#38612yes—
291C4×C47C22×C22⋊C411yescharacteristic
292C4×C47C2×C4×D411yescharacteristic
293C4×C47C2×#12111yescharacteristic
294C4×C47C2×#12311yescharacteristic
295Dic47—44no—
296Dic47—44no—
297Dic47—44no—
298C22×#137—471no—
299#3428—471no—
300C2×#421D4711yescharacteristic
301C2×#419—471no—
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×C47—44no—
311C2×C47C2×#38611yescharacteristic
312C2×C47#302212yes—
313C22×#41C2×D4711yescharacteristic
314C23×Q8—471no—
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—24no—
335C2×#104C2×D4714yes—
336#419—474no—
337#419—942no—
338C2×#103—942no—
339#391—474no—
340#391—942no—
341#381—942no—
342#343—474no—
343#339—474no—
344C47#2094411yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C2×#41—21no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41C4×D4714yes—
350C2×#41D18812yes—
351C2×#41C22⋊D4714yes—
352C2×#41C22×D4712yes—
353C22×Q8—474no—
354C22×Q8—472no—
355C22×Q8—941no—
356C22×Q8—942no—
357C22×Q8—941no—
358C22×Q8—941no—
359C22×Q8—941no—
360C22×Q8—941no—
361C22×Q8—942no—
362C22×Q8—942no—
363C22×Q8D18811yescharacteristic
364C22×D4—21no—
365C22×D4D18811yescharacteristic
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—941no—
382C2×M16—942no—
383C2×M16C22×D4712yes—
384C22×C8—21no—
385C22×C8—941no—
386C4×Q8—944no—
387C4×Q8—944no—
388C4×D4—948no—
389C4×D4—948no—
390C4×D4—944no—
391C4×D4—944no—
392#117—22no—
393#117—22no—
394#117—472no—
395#117—472no—
396#117—944no—
397#117—944no—
398#117—944no—
399#117—941no—
400#117—941no—
401#117—941no—
402#117—941no—
403#117—941no—
404#117—941no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—472no—
407C2×C4⋊C4—472no—
408C2×C4⋊C4—944no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—472no—
411C2×C22⋊C4—944no—
412C2×C22⋊C4—941no—
413C2×C42—472no—
414C2×C42—944no—
415C2×C42—941no—
416C2×C42C22×D4712yes—
417#104—28no—
418#103—948no—
419#95—948no—
420C24—21no—
421C24D4×D4711yescharacteristic
422#41—24no—
423#41—24no—
424#41—24no—
425#41—24no—
426#41—24no—
427#41—22no—
428#41—22no—
429#41(C2×D47)⋊C418yes—
430#41C2×C4×D4714yes—
431#41C2×C22⋊D4714yes—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—474no—
435C2×Q8—944no—
436C2×Q8—942no—
437C2×Q8—944no—
438C2×Q8—944no—
439C2×Q8—944no—
440C2×Q8—944no—
441C2×Q8—942no—
442C2×Q8—942no—
443C2×Q8—942no—
444C2×Q8—942no—
445C2×Q8—1881no—
446C2×Q8—1881no—
447C2×Q8—1882no—
448C2×Q8—1882no—
449C2×Q8—1882no—
450C2×Q8—1882no—
451C2×Q8—1882no—
452C2×Q8(C2×D47)⋊C414yes—
453C2×Q8C2×D18812yes—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4(C2×D47)⋊C414yes—
461C2×D4C2×D18812yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—474no—
469C22×C4—948no—
470C22×C4—942no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4—1881no—
474C22×C4—1881no—
475C22×C4—1882no—
476C22×C4C2×C4×D4712yes—
477C22×C4C2×D18811yescharacteristic
478C22×C4C2×D18811yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4#1062535212yes—
483C22×C4#1062535212yes—
484C22×C4C2×C22⋊D4712yes—
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—42no—
499C4⋊C4—474no—
500C4⋊C4—948no—
501C4⋊C4—942no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C22⋊C4—42no—
505C22⋊C4—948no—
506C22⋊C4—942no—
507C22⋊C4—942no—
508C22⋊C4—942no—
509C22⋊C4—942no—
510C42—22no—
511C42—22no—
512C42—948no—
513C42—942no—
514C42—942no—
515C42—942no—
516C42—942no—
517C23—22no—
518C23—24no—
519C23—42no—
520C23—82no—
521C23(C22⋊D47)⋊C412yes—
522C23#1309399411yescharacteristic
523C23#1309399411yescharacteristic
524C23#1309399612yes—
525C23C2×D4×D4711yescharacteristic
526Q8—24no—
527Q8—22no—
528Q8—944no—
529Q8—944no—
530Q8—1884no—
531Q8—1884no—
532Q8—1882no—
533Q8—1882no—
534Q8C2×(C2×D47)⋊C414yes—
535D4—28no—
536D4—24no—
537D4—24no—
538D4—24no—
539D4—22no—
540D4C2×(C2×D47)⋊C414yes—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—3762no—
552C2×C4—3768no—
553C2×C4—1884no—
554C2×C4—1884no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1884no—
558C2×C4C2×(C2×D47)⋊C412yes—
559C2×C4C2×(C2×D47)⋊C412yes—
560C2×C4(C22⋊D47)⋊C412yes—
561C2×C4(C22⋊D47)⋊C412yes—
562C2×C4(C22⋊D47)⋊C412yes—
563C2×C4#1309399411yescharacteristic
564C2×C4#1309399411yescharacteristic
565C2×C4#1309399612yes—
566C2×C4#1309399612yes—
567C2×C4#1309399612yes—
568C2×C4#1309400812yes—
569C2×C4#1309400812yes—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4C22×C22⋊D4711yescharacteristic
573C2×C4C22×D18811yescharacteristic
574C2×C4C2×#1062535211yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C8—42no—
577C8—944no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—84no—
586C22#1568900012yes—
587C22#1568900012yes—
588C22C2×(C22⋊D47)⋊C411yescharacteristic
589C22C2×#1309399411yescharacteristic
590C22C2×#1309399611yescharacteristic
591C4—24no—
592C4—82no—
593C4—1884no—
594C4—1884no—
595C4—1884no—
596C4#1568900012yes—
597C4#1568900012yes—
598C4C22×(C2×D47)⋊C411yescharacteristic
599C4C2×(C22⋊D47)⋊C411yescharacteristic
600C4C2×#1309399611yescharacteristic
601C4C2×#1309400811yescharacteristic
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—44no—
607C2C2×#1568900011yescharacteristic
608C2#2025276812yes—
subgroupG/H
centerC22C2×#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#421C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#421D47
hypocenterC47#20944
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
3. C2×#15689133C2semidirect
4. #20245972C2semidirect
5. #20243755C2semidirect
6. #20243126C2semidirect
7. #20243098C2semidirect
12. C2×#13094054C4semidirect
28. #15689133C22semidirect
30. #15689029C22semidirect
33. #15688891C22semidirect
34. #15688883C22semidirect
48. C22×Dic94C2×C4semidirect
53. #13094054C2×C4semidirect
60. C2×#10625329C2×C4semidirect
344. C47#20944semidirect

Maximal subgroups

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

subgroupindex
C22×#130940542
#202565912
C2×#156891332
#202459722
#202437552
#202431262
#202430982
C2×C47×#4212
#2094447