Finite Group Databasebeta

← #26215810

Subgroups — #26215810

Order 12032.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689604C211yesmaximal, characteristic
2C2×#15689149C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic
4C2×#15689120C211yesmaximal, characteristic
5#20246012C211yesmaximal, characteristic, complemented
6#20243402C212yesmaximal, complemented
7#20243397C212yesmaximal, complemented
8#20243392C212yesmaximal, complemented
9#20243387C212yesmaximal, complemented
10#20243107C211yesmaximal, characteristic, complemented
11C2×C47×#340C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689604—24no—
15#15689604C418yescomplemented
16C2×#13093989C2212yes—
17C2×#13093986C2212yes—
18C2×C4×C22⋊D47—22no—
19C2×#13093983C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21C2×#13093982—21no—
22C2×C4⋊C4×D47—21no—
23#15689149—24no—
24C2×Dic47⋊C8C2212yes—
25C2×Dic47⋊C8C2212yes—
26#15689122—24no—
27#15689120—24no—
28C2×#13093949C2211yescharacteristic
29C2×#13093949C2211yescharacteristic
30C2×#13093947C2212yes—
31C2×#13093946C2211yescharacteristic
32C2×#13093946C2211yescharacteristic
33C2×#13093944C2212yes—
34#15689019—21no—
35#15689019—21no—
36#15688996—22no—
37#15688992C2212yes—
38#15688988C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C23×C8×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350C2×C414yes—
50C2×#10625350D414yes—
51C2×#10625350Q814yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C22×C4×D47—21no—
56C22×C4×D47—22no—
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C22×C47⋊C8C2312yes—
61#13093989—24no—
62#13093989C2×C418yescomplemented
63#13093986—24no—
64#13093986C2×C418yescomplemented
65C4×C22⋊D47—216no—
66#13093983—22no—
67#13093983—22no—
68#13093983C2×C414yes—
69#13093983D414yes—
70#13093982—28no—
71C2×C4⋊Dic47D411yescharacteristic
72C2×C4⋊Dic47D411yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—22no—
76C2×Dic47⋊C4—22no—
77C2×Dic47⋊C4—22no—
78C4⋊C4×D47—28no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47D412yes—
81C2×(C2×D47)⋊C4—22no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2312yes—
85Dic47⋊C8—24no—
86Dic47⋊C8—24no—
87#13093949—22no—
88#13093949—22no—
89#13093947—24no—
90#13093946—22no—
91#13093946—22no—
92#13093944—24no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C23×C4×C47—21no—
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98C22×C8×C47—22no—
99C22×C8×C47—22no—
100C22×C8×C47—22no—
101C22×C8×C47C2311yescharacteristic
102C22×C8×C47C2311yescharacteristic
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2×C414yes—
109C2×C4⋊C4×C47C2312yes—
110C2×C4⋊C4×C47C2312yes—
111C24×C47C2×D411yescharacteristic
112C23×D47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—24no—
115C22×C4×C47—22no—
116C22×C4×C47—42no—
117C22×C4×C47—81no—
118C22×C4×C47—81no—
119C22×C4×C47—81no—
120C22×C4×C47—81no—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q812yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—24no—
130C2×C22⋊D47—24no—
131C2×C22⋊D47—24no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47C2×D412yes—
136C22×Dic47C2×D412yes—
137#10625350—28no—
138#10625350—28no—
139#10625350—28no—
140#10625350C4⋊C4116yescomplemented
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188C22×C412yes—
145C2×D188C2×D412yes—
146C2×D188C2×Q812yes—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—28no—
151C2×C4×D47—82no—
152C2×C4×D47—82no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94C22×C412yes—
157C2×Dic94C2×D412yes—
158C2×Dic94C2×Q812yes—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—44no—
166C4⋊C4×C47—24no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—42no—
171C4⋊C4×C47C22×C414yes—
172C22⋊Dic47—42no—
173C22⋊Dic47C22⋊C418yes—
174(C2×D47)⋊C4—28no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47C22⋊C414yes—
180C4⋊Dic47C22⋊C414yes—
181Dic47⋊C4—28no—
182Dic47⋊C4—82no—
183Dic47⋊C4—82no—
184C4×Dic47—28no—
185C4×Dic47—42no—
186C4×Dic47C22×C414yes—
187C4×Dic47C2×D414yes—
188#10625329—44no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47C2×C22⋊C412yes—
197C23×C47#12112yes—
198C23×C47C22⋊Q812yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C2×C4×C47—24no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—84no—
214C2×C4×C47—84no—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47C2×C4⋊C414yes—
217C2×C4×C47#11714yes—
218C2×C4×C47#12114yes—
219C2×C4×C47#12112yes—
220C2×C4×C47C22⋊Q814yes—
221C2×C4×C47C22⋊Q812yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×Q811yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C22⋊D47—216no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47C2×C22⋊C414yes—
243C2×Dic47C2×C22⋊C414yes—
244D188—22no—
245D188—22no—
246D188—22no—
247D188C2×C4⋊C414yes—
248C4×D47—216no—
249C4×D47—88no—
250Dic94—22no—
251Dic94—22no—
252Dic94—22no—
253Dic94C2×C4⋊C414yes—
254C8×C47—41no—
255C8×C47—41no—
256C47⋊C8—42no—
257#20652—471nomaximal, Sylow
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47#38014yes—
265C22×C47C22×C22⋊C411yescharacteristic
266C22×C47C2×#12111yescharacteristic
267C22×C47C2×C22⋊Q811yescharacteristic
268C22×C47#57611yescharacteristic
269C22×C47#57711yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57811yescharacteristic
272C2×D47—24no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C4×C47—44no—
280C4×C47—82no—
281C4×C47#38014yes—
282C4×C47C22×C4⋊C411yescharacteristic
283C4×C47C2×#11711yescharacteristic
284C4×C47C2×#12111yescharacteristic
285C4×C47C2×C22⋊Q811yescharacteristic
286Dic47—24no—
287Dic47—24no—
288Dic47—24no—
289Dic47—81no—
290Dic47—81no—
291C2×#520—471no—
292C2×#424—471no—
293C2×#417—471no—
294C2×#416—471no—
295#2317—471no—
296#1805—472no—
297#1804—472no—
298#1803—472no—
299#1802—472no—
300#1736—471no—
301C2×#340D4711yescharacteristic, complemented
302C2×C47—24no—
303C2×C47C2×#38011yescharacteristic
304C2×C47#289711yescharacteristic
305C2×C47#289811yescharacteristic
306C2×C47#304511yescharacteristic
307C2×C47#304611yescharacteristic
308C2×C47#308311yescharacteristic
309C2×C47#308411yescharacteristic
310D47—44no—
311C22×#41—471no—
312C22×M16—471no—
313C23×C8C2×D4711yescharacteristic
314#520—478no—
315#520—944no—
316C2×C4×Q8—472no—
317C2×C4×D4—472no—
318C2×C4×D4—942no—
319C2×#117—471no—
320C2×#117—471no—
321C2×#117—941no—
322C22×C4⋊C4—941no—
323C22×C4⋊C4C2×D4711yescharacteristic
324C22×C4⋊C4C2×D4711yescharacteristic
325#424—944no—
326C2×C4⋊C8—472no—
327C2×C4⋊C8—472no—
328#417—944no—
329#416—944no—
330C2×#103—471no—
331C2×#103—471no—
332C2×#103—472no—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102—472no—
336#389—941no—
337#389—941no—
338#385—942no—
339#384—472no—
340#382—472no—
341#340—472no—
342#340—472no—
343#340C2×D4712yes—
344#340C2×D4712yes—
345C47#2065211yescharacteristic, complemented, Sylow
346C2×#41—474no—
347C2×#41—474no—
348C2×#41—474no—
349C2×#41—942no—
350C2×#41—942no—
351C2×#41—942no—
352C22×Q8—471no—
353C22×D4—471no—
354C22×D4—941no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—942no—
358C23×C4—941no—
359C23×C4C22×D4711yescharacteristic
360C2×M16—942no—
361C2×M16—942no—
362C2×M16—942no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—472no—
367C22×C8C22×D4711yescharacteristic
368C22×C8C22×D4711yescharacteristic
369C4×Q8—478no—
370C4×Q8—944no—
371C4×D4—478no—
372C4×D4—944no—
373C4×D4—9416no—
374#117—474no—
375#117—474no—
376#117—948no—
377#117—942no—
378#117—942no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4C4×D4714yes—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—472no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—941no—
398C2×C42—472no—
399C2×C42—942no—
400C2×C42—941no—
401C4⋊C8—944no—
402C4⋊C8—944no—
403#103—944no—
404#103—942no—
405#103—942no—
406#102—944no—
407#102—942no—
408#102—942no—
409#95—942no—
410#95—942no—
411C24—941no—
412C24D4×D4711yescharacteristic
413#41—4716no—
414#41—948no—
415#41—948no—
416#41—948no—
417C2×Q8—472no—
418C2×Q8—472no—
419C2×Q8—472no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×D4—472no—
424C2×D4—472no—
425C2×D4—472no—
426C2×D4—944no—
427C2×D4—944no—
428C2×D4—944no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—948no—
443C22×C4—941no—
444C22×C4—944no—
445C22×C4—944no—
446C22×C4—944no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—3762no—
450C22×C4—3762no—
451C22×C4C2×C4×D4712yes—
452C22×C4#1062535012yes—
453C22×C4#1062535012yes—
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4712yes—
456C22×C4D4×D4712yes—
457C22×C4Q8×D4712yes—
458C22×C4C23×D4711yescharacteristic
459M16—1884no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—942no—
468C2×C8—942no—
469C2×C8—942no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—474no—
476C4⋊C4—474no—
477C4⋊C4—948no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—3762no—
483C4⋊C4—3762no—
484C4⋊C4C2×C4×D4714yes—
485C22⋊C4—478no—
486C22⋊C4—948no—
487C22⋊C4—1882no—
488C42—474no—
489C42—474no—
490C42—948no—
491C42—1882no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—942no—
497C23—942no—
498C23—942no—
499C23—1881no—
500C23—1881no—
501C23—1881no—
502C23—1881no—
503C23C22⋊C4×D4712yes—
504C23#1309399712yes—
505C23#1309400212yes—
506C23C2×D4×D4711yescharacteristic
507Q8—474no—
508Q8—942no—
509Q8—942no—
510Q8—942no—
511D4—474no—
512D4—9416no—
513D4—942no—
514D4—942no—
515D4—942no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—474no—
525C2×C4—474no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—942no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—9416no—
534C2×C4—3768no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4C22⋊C4×D4712yes—
542C2×C4C4⋊C4×D4714yes—
543C2×C4#1309398114yes—
544C2×C4#1309399712yes—
545C2×C4#1309399714yes—
546C2×C4#1309400212yes—
547C2×C4#1309400214yes—
548C2×C4C22×C4×D4711yescharacteristic
549C2×C4C2×D4×D4711yescharacteristic
550C2×C4C2×D4×D4711yescharacteristic
551C2×C4C2×Q8×D4711yescharacteristic
552C2×C4C2×#1062535011yescharacteristic
553C2×C4C2×#1062535011yescharacteristic
554C8—41no—
555C8—41no—
556C8—1882no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—944no—
564C22—1882no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22—1882no—
569C22—1882no—
570C22#1568898214yes—
571C22C2×C22⋊C4×D4711yescharacteristic
572C22C2×#1309399711yescharacteristic
573C22C2×#1309400211yescharacteristic
574C22#1568992411yescharacteristic
575C22#1568992911yescharacteristic
576C22#1568992911yescharacteristic
577C22#1568993611yescharacteristic
578C4—44no—
579C4—82no—
580C4—944no—
581C4—944no—
582C4—944no—
583C4—3761no—
584C4—3761no—
585C4#1568898214yes—
586C4C2×C4⋊C4×D4711yescharacteristic
587C4C2×#1309398111yescharacteristic
588C4C2×#1309399711yescharacteristic
589C4C2×#1309400211yescharacteristic
590C2—24no—
591C2—1884no—
592C2C2×#1568898211yescharacteristic
593C2#2025190911yescharacteristic
594C2#2025191611yescharacteristic
595C2#2025293211yescharacteristic
596C2#2025294311yescharacteristic
597C2#2025324411yescharacteristic
598C2#2025325511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47#20652
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#15689149C2semidirect
5. #20246012C2semidirect
6. #20243402C2semidirect
7. #20243397C2semidirect
8. #20243392C2semidirect
9. #20243387C2semidirect
10. #20243107C2semidirect
11. C2×C47×#340C2semidirect
15. #15689604C4semidirect
62. #13093989C2×C4semidirect
64. #13093986C2×C4semidirect
140. #10625350C4⋊C4semidirect
301. C2×#340D47semidirect
345. C47#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#156896042
C2×#156891492
C2×#156891222
C2×#156891202
#202460122
#202434022
#202433972
#202433922
#202433872
#202431072
C2×C47×#3402
#2065247