Finite Group Databasebeta

← #26243628

Subgroups — #26243628

Order 12032.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic, complemented
2C2×#15689124C211yesmaximal, characteristic
3C2×#15689119C211yesmaximal, characteristic
4#20244539C212yesmaximal, complemented
5#20244534C212yesmaximal, complemented
6#20244529C212yesmaximal, complemented
7#20244524C212yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C2×#15688885C211yesmaximal, characteristic, complemented
10#20243085C211yesmaximal, characteristic, complemented
11C47×#2311C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14#15689751—24no—
15#15689751—24no—
16C2×#13094024C2212yes—
17C2×#13094023—21no—
18C2×#13094016C2212yes—
19C2×#13094015—21no—
20C2×#13094001—22no—
21C2×#13093983C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689124—24no—
24#15689119—24no—
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2212yes—
27C2×#13093948C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×#13093945C2212yes—
30C2×#13093945C2211yescharacteristic
31#15688885C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35#15688878C2212yes—
36#15688878C2212yes—
37C22×#41×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C47×#391C2212yes—
41C47×#390C2212yes—
42C47×#386—22no—
43C47×#380—21no—
44C47×#380—21no—
45C22×Dic94—22no—
46C23×Dic47—21no—
47C2×#10625329—22no—
48C2×#10625329—22no—
49C2×#10625329—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8C2311yescharacteristic
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2312yes—
56#13094024—24no—
57#13094024—24no—
58#13094023—28no—
59#13094016—24no—
60#13094016—24no—
61#13094015—28no—
62#13094001—216no—
63#13093983—22no—
64#13093983D414yes—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47D414yes—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—22no—
74C2×C22⋊Dic47—21no—
75C2×C4×Dic47—21no—
76#13093948—22no—
77#13093948—22no—
78#13093948—24no—
79#13093945—22no—
80#13093945—22no—
81#13093945—24no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47Q814yescomplemented
88C22×Q8×C47C2311yescharacteristic
89C22×D4×C47—21no—
90C22×D4×C47C2311yescharacteristic
91C23×C4×C47—21no—
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94#117×C47—22no—
95#117×C47—22no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47—22no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47C2312yes—
106#95×C47—22no—
107#95×C47—22no—
108C24×C47—21no—
109C24×C47C2×D411yescharacteristic
110#41×C47—28no—
111#41×C47—28no—
112#41×C47—44no—
113#41×C47—44no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47C2×Q812yes—
120C2×D4×C47—24no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—42no—
127C2×D4×C47—42no—
128C2×D4×C47C2×Q812yes—
129C22×C4×C47—21no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—81no—
136C22×C4×C47—81no—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×Q812yes—
143C22×C4×C47#4112yes—
144C22×C4×C47#4112yes—
145C22×C4×C47#4112yes—
146C22×C4×C47C2411yescharacteristic
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—81no—
152C22×Dic47—81no—
153C2×Dic94—28no—
154C2×Dic94—82no—
155C2×Dic94—82no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—42no—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—82no—
163C22⋊C4×C47—82no—
164C42×C47—22no—
165C42×C47—22no—
166C22⋊Dic47—24no—
167C4⋊Dic47—28no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47C2×D414yes—
173C4⋊Dic47C2×D412yes—
174C4⋊Dic47C2×D412yes—
175Dic47⋊C4—28no—
176Dic47⋊C4—28no—
177C4×Dic47—24no—
178#10625329—44no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—44no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—22no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47#12012yes—
201C23×C47C22⋊Q812yes—
202C23×C47#12412yes—
203C23×C47C22×D411yescharacteristic
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—41no—
207Q8×C47—41no—
208D4×C47—22no—
209D4×C47—22no—
210D4×C47—48no—
211D4×C47—41no—
212D4×C47—41no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—48no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—84no—
227C2×C4×C47#12012yes—
228C2×C4×C47C22⋊Q814yes—
229C2×C4×C47C22⋊Q814yes—
230C2×C4×C47C22⋊Q812yes—
231C2×C4×C47#12414yes—
232C2×C4×C47#12414yes—
233C2×C4×C47#12412yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×Q811yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—41no—
244C2×Dic47—42no—
245C2×Dic47—41no—
246C2×Dic47—42no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—84no—
250Dic94—88no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8—42no—
254#23458—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—81no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—81no—
268C22×C47#39314yes—
269C22×C47C2×#12011yescharacteristic
270C22×C47C2×C22⋊Q811yescharacteristic
271C22×C47C2×#12411yescharacteristic
272C22×C47#57611yescharacteristic
273C22×C47#57711yescharacteristic
274C22×C47#57711yescharacteristic
275C22×C47#57811yescharacteristic
276C4×C47—42no—
277C4×C47—42no—
278C4×C47—42no—
279C4×C47—82no—
280C4×C47#39314yes—
281C4×C47C2×C22⋊Q811yescharacteristic
282C4×C47C2×C22⋊Q811yescharacteristic
283C4×C47C2×#12411yescharacteristic
284C4×C47C2×#12411yescharacteristic
285Dic47—42no—
286Dic47—44no—
287Dic47—42no—
288C2×#544—471no—
289C2×#417—471no—
290C2×#416—471no—
291#2311D4711yescharacteristic
292#2026—472no—
293#2025—472no—
294#2024—472no—
295#2023—472no—
296#1736—471no—
297C2×#340—471no—
298#1729—471no—
299C2×C47—24no—
300C2×C47—82no—
301C2×C47C2×#39311yescharacteristic
302C2×C47#301211yescharacteristic
303C2×C47#301311yescharacteristic
304C2×C47#308311yescharacteristic
305C2×C47#308411yescharacteristic
306C2×C47#313611yescharacteristic
307C2×C47#313711yescharacteristic
308C22×#41C2×D4711yescharacteristic
309C22×M16—471no—
310C23×C8—471no—
311#544—944no—
312#544—944no—
313C2×C4⋊Q8—472no—
314C2×C4⋊Q8—941no—
315C2×#125—472no—
316C2×#125—941no—
317C2×C22⋊Q8—942no—
318C2×#117—471no—
319C2×#117C2×D4711yescharacteristic
320C22×C4⋊C4—471no—
321C22×C4⋊C4C2×D4711yescharacteristic
322#417—944no—
323#416—944no—
324C2×#103—471no—
325C2×#103—471no—
326C2×#103—472no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#102—472no—
330#391C2×D4712yes—
331#390C2×D4712yes—
332#386—22no—
333#380—21no—
334#380—21no—
335#340—472no—
336#340—472no—
337#340—472no—
338#340—472no—
339#336—472no—
340#336—472no—
341C47#2345811yescharacteristic, complemented, Sylow
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41Dic9414yescomplemented
348C22×Q8—942no—
349C22×Q8C22×D4711yescharacteristic
350C22×D4—21no—
351C22×D4C22×D4711yescharacteristic
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—942no—
357C2×M16—942no—
358C2×M16—942no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—472no—
362C22×C8—942no—
363C22×C8—942no—
364C22×C8—942no—
365C4⋊Q8—944no—
366C4⋊Q8—944no—
367C4⋊Q8—948no—
368#125—944no—
369#125—944no—
370#125—948no—
371C22⋊Q8—9416no—
372#117—22no—
373#117—22no—
374#117—474no—
375#117—942no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—474no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4C22⋊D4711yescharacteristic
389C2×C4⋊C4C22⋊D4711yescharacteristic
390C2×C4⋊C4C22×D4712yes—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4C22⋊D4712yes—
395C2×C42—941no—
396C2×C42C22×D4712yes—
397#103—942no—
398#103—942no—
399#103—944no—
400#102—942no—
401#102—942no—
402#102—944no—
403#95—22no—
404#95—22no—
405C24—21no—
406C24C2×C22⋊D4711yescharacteristic
407#41—28no—
408#41—28no—
409#41—44no—
410#41—44no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—948no—
417C2×Q8—3762no—
418C2×Q8—3762no—
419C2×Q8C2×Dic9412yes—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×Dic9412yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—944no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—942no—
441C22×C4—3761no—
442C22×C4—3761no—
443C22×C4C2×Dic9412yes—
444C22×C4#1062535012yes—
445C22×C4D4×D4712yes—
446C22×C4D4×D4712yes—
447C22×C4#1062535212yes—
448C22×C4#1062535212yes—
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4712yes—
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C23×D4711yescharacteristic
453M16—1884no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—942no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—942no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—942no—
463C2×C8—1884no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—474no—
470C4⋊C4—472no—
471C4⋊C4—472no—
472C4⋊C4—948no—
473C4⋊C4—948no—
474C4⋊C4—948no—
475C4⋊C4—944no—
476C4⋊C4—942no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C22⋊C4—24no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—944no—
483C42—22no—
484C42—22no—
485C42—944no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1309399212yes—
498C23#1309400312yes—
499C23#1309401112yes—
500C23C22×C22⋊D4711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—3768no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—48no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—944no—
528C2×C4—3764no—
529C2×C4—1882no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4#1309399212yes—
536C2×C4#1309400114yes—
537C2×C4#1309400114yes—
538C2×C4#1309400312yes—
539C2×C4#1309401014yes—
540C2×C4#1309401014yes—
541C2×C4#1309401112yes—
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C2×D4×D4711yescharacteristic
544C2×C4C22×Dic9411yescharacteristic
545C2×C4C2×#1062535211yescharacteristic
546C2×C4C2×#1062535211yescharacteristic
547C2×C4C2×#1062535011yescharacteristic
548C8—1881no—
549C8—1881no—
550C8—1882no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#1568903514yes—
565C22C2×#1309399211yescharacteristic
566C22C2×#1309400311yescharacteristic
567C22C2×#1309401111yescharacteristic
568C22#1568992611yescharacteristic
569C22#1568993111yescharacteristic
570C22#1568993411yescharacteristic
571C22#1568993811yescharacteristic
572C4—42no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—1884no—
577C4—1882no—
578C4—1882no—
579C4#1568903514yes—
580C4C2×#1309400111yescharacteristic
581C4C2×#1309400111yescharacteristic
582C4C2×#1309401011yescharacteristic
583C4C2×#1309401011yescharacteristic
584C2—24no—
585C2—82no—
586C2C2×#1568903511yescharacteristic
587C2#2025268911yescharacteristic
588C2#2025270011yescharacteristic
589C2#2025323911yescharacteristic
590C2#2025325011yescharacteristic
591C2#2025374211yescharacteristic
592C2#2025375311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
hypercenter#2311D47
hypocenterC47#23458
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C2×#15689751C2semidirect
4. #20244539C2semidirect
5. #20244534C2semidirect
6. #20244529C2semidirect
7. #20244524C2semidirect
8. #20243107C2semidirect
9. C2×#15688885C2semidirect
10. #20243085C2semidirect
87. C2×#41×C47Q8semidirect
341. C47#23458semidirect
347. C2×#41Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
C2×#156891242
C2×#156891192
#202445392
#202445342
#202445292
#202445242
#202431072
C2×#156888852
#202430852
C47×#23112
#2345847