Finite Group Databasebeta

← #26216098

Subgroups — #26216098

Order 12032.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689090C211yesmaximal, characteristic
2#20245980C211yesmaximal, characteristic
3#20243483C212yesmaximal, complemented
4#20243483C212yesmaximal, complemented
5#20243478C212yesmaximal, complemented
6#20243474C212yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243107C211yesmaximal, characteristic, complemented
9#20243104C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C2×C47×#520C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14C2×#13093983C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16#15689090—24no—
17C2×C22⋊(C47⋊C8)C2212yes—
18C2×C22⋊(C47⋊C8)C2212yes—
19#15689030—21no—
20#15689026—21no—
21#15689003—22no—
22#15688983C2212yes—
23#15688983C2212yes—
24#15688885C2211yescharacteristic
25#15688885C2212yes—
26#15688885C2212yes—
27#15688885C2212yes—
28#15688885C2211yescharacteristic
29#15688885C2212yes—
30#15688885C2212yes—
31#15688885C2211yescharacteristic
32#15688885C2211yescharacteristic
33#15688885C2212yes—
34C22×#41×C47C2211yescharacteristic
35C47×#520—24no—
36C47×#520C418yescomplemented
37C2×C4×Q8×C47—21no—
38C2×C4×D4×C47—22no—
39C2×C4×D4×C47—21no—
40C2×#117×C47C2211yescharacteristic
41C2×#117×C47C2212yes—
42C22×C4⋊C4×C47C2211yescharacteristic
43C22×C4⋊C4×C47C2212yes—
44C23×Dic47—21no—
45C2×#10625329—22no—
46C2×#10625329—22no—
47C2×#10625329—22no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2311yescharacteristic
53C22×C47⋊C8C2312yes—
54#13093983—22no—
55#13093983—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47C2312yes—
60C2×C4⋊Dic47C2311yescharacteristic
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C22⋊Dic47—21no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C4×Dic47—21no—
66C22⋊(C47⋊C8)—24no—
67C22⋊(C47⋊C8)—24no—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C2×#41×C47—22no—
71C2×#41×C47—22no—
72C2×#41×C47—22no—
73C2×#41×C47—22no—
74C2×#41×C47C2×C414yes—
75C2×#41×C47D414yes—
76C22×Q8×C47D411yescharacteristic
77C22×D4×C47D412yes—
78C22×D4×C47D411yescharacteristic
79C23×C4×C47—21no—
80C23×C4×C47—22no—
81C23×C4×C47C2312yes—
82C23×C4×C47C2311yescharacteristic
83C4×Q8×C47—28no—
84C4×D4×C47—28no—
85C4×D4×C47—216no—
86#117×C47—22no—
87#117×C47—24no—
88#117×C47C2×C414yes—
89#117×C47C2×C418yescomplemented
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47—24no—
94C2×C4⋊C4×C47C2×C414yes—
95C2×C4⋊C4×C47C2×C418yescomplemented
96C2×C4⋊C4×C47C2312yes—
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—22no—
101C2×C42×C47—21no—
102C2×C42×C47—22no—
103C24×C47C2×D412yes—
104C24×C47C2×D411yescharacteristic
105#41×C47—28no—
106#41×C47—28no—
107#41×C47—28no—
108#41×C47—28no—
109C2×Q8×C47—22no—
110C2×Q8×C47—22no—
111C2×Q8×C47C22⋊C414yes—
112C2×D4×C47—24no—
113C2×D4×C47—24no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47C22⋊C418yes—
117C2×D4×C47C22⋊C414yes—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—28no—
127C22×C4×C47—81no—
128C22×C4×C47—82no—
129C22×C4×C47—81no—
130C22×C4×C47—82no—
131C22×C4×C47C22×C414yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D412yes—
137C22×C4×C47#4112yes—
138C22×C4×C47#4112yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×C4×C47C2411yescharacteristic
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—42no—
146C22×Dic47—81no—
147C22×Dic47—81no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—28no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47C22×C414yes—
154C4⋊C4×C47C22×C412yes—
155C4⋊C4×C47C22×C412yes—
156C22⋊C4×C47—24no—
157C22⋊C4×C47—28no—
158C42×C47—24no—
159C42×C47—28no—
160C22⋊Dic47—42no—
161C22⋊Dic47—82no—
162C22⋊Dic47—82no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—42no—
170C4×Dic47—42no—
171#10625329—44no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—44no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47C2×C22⋊C414yes—
193C23×C47C2×C22⋊C412yes—
194C23×C47#12312yes—
195C23×C47#12312yes—
196C23×C47C22×D411yescharacteristic
197Q8×C47—28no—
198D4×C47—216no—
199D4×C47—28no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—42no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—84no—
216C2×C4×C47—88no—
217C2×C4×C47C2×C22⋊C414yes—
218C2×C4×C47C2×C22⋊C412yes—
219C2×C4×C47#11714yes—
220C2×C4×C47#11714yes—
221C2×C4×C47#12314yes—
222C2×C4×C47#12314yes—
223C2×C4×C47#12312yes—
224C2×C4×C47#12312yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—42no—
234C2×Dic47—41no—
235C2×Dic47—42no—
236C2×Dic47—41no—
237C2×Dic47—42no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—84no—
241C47⋊C8—41no—
242C47⋊C8—41no—
243C47⋊C8—42no—
244#20703—471nomaximal, Sylow
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47#38114yes—
259C22×C47C22×C22⋊C411yescharacteristic
260C22×C47C2×#12311yescharacteristic
261C22×C47C2×#12311yescharacteristic
262C22×C47#57611yescharacteristic
263C22×C47#57711yescharacteristic
264C22×C47#57711yescharacteristic
265C22×C47#57811yescharacteristic
266C4×C47—24no—
267C4×C47—42no—
268C4×C47—44no—
269C4×C47—42no—
270C4×C47#38114yes—
271C4×C47C2×#11711yescharacteristic
272C4×C47C2×#11711yescharacteristic
273C4×C47C2×#12311yescharacteristic
274C4×C47C2×#12311yescharacteristic
275Dic47—82no—
276Dic47—81no—
277Dic47—81no—
278C2×#520D4711yescharacteristic
279C2×#408—471no—
280#2311—471no—
281#1825—472no—
282#1825—472no—
283#1824—472no—
284#1823—472no—
285#1736—471no—
286#1736—471no—
287#1735—471no—
288C2×#340—471no—
289C2×C47—24no—
290C2×C47—44no—
291C2×C47C2×#38111yescharacteristic
292C2×C47#289711yescharacteristic
293C2×C47#289811yescharacteristic
294C2×C47#309611yescharacteristic
295C2×C47#309711yescharacteristic
296C2×C47#309811yescharacteristic
297C2×C47#309911yescharacteristic
298C22×#41C2×D4711yescharacteristic
299C22×M16—471no—
300C23×C8—471no—
301#520—24no—
302#520Dic4718yescomplemented
303C2×C4×Q8—21no—
304C2×C4×D4—22no—
305C2×C4×D4—21no—
306C2×#117—471no—
307C2×#117C2×D4711yescharacteristic
308C2×#117C2×D4712yes—
309C22×C4⋊C4—471no—
310C22×C4⋊C4C2×D4711yescharacteristic
311C22×C4⋊C4C2×D4712yes—
312#408—944no—
313C2×C22⋊C8—472no—
314C2×C22⋊C8—472no—
315#391—941no—
316#390—941no—
317#386—942no—
318#380—472no—
319#380—472no—
320#340—471no—
321#340—472no—
322#340—472no—
323#340—472no—
324#340—471no—
325#340—472no—
326#340—472no—
327#340—471no—
328#340—471no—
329#340—472no—
330C47#2070311yescharacteristic, complemented, Sylow
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41C2×Dic4714yes—
336C2×#41C22⋊D4714yes—
337C22×Q8C22⋊D4711yescharacteristic
338C22×D4C22⋊D4712yes—
339C22×D4C22⋊D4711yescharacteristic
340C23×C4—21no—
341C23×C4—22no—
342C23×C4—941no—
343C23×C4C22×D4712yes—
344C23×C4C22×D4711yescharacteristic
345C2×M16—942no—
346C2×M16—942no—
347C2×M16—942no—
348C22×C8—471no—
349C22×C8—471no—
350C22×C8—472no—
351C22×C8—942no—
352C22×C8—942no—
353C22×C8—942no—
354C4×Q8—28no—
355C4×D4—28no—
356C4×D4—216no—
357#117—22no—
358#117—24no—
359#117—942no—
360#117—942no—
361#117C2×Dic4714yes—
362#117C2×Dic4718yescomplemented
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—472no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C2×Dic4714yes—
374C2×C4⋊C4C2×Dic4718yescomplemented
375C2×C4⋊C4C22×D4712yes—
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—941no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—941no—
386C22⋊C8—944no—
387C22⋊C8—944no—
388#95—942no—
389#95—942no—
390C24C2×C22⋊D4712yes—
391C24C2×C22⋊D4711yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8C22⋊Dic4714yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4C22⋊Dic4718yes—
404C2×D4C22⋊Dic4714yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—941no—
419C22×C4—941no—
420C22×C4—942no—
421C22×C4—3761no—
422C22×C4—3761no—
423C22×C4—1882no—
424C22×C4#1062535012yes—
425C22×C4#1062535012yes—
426C22×C4#1062535012yes—
427C22×C4#1062535012yes—
428C22×C4C22×Dic4714yes—
429C22×C4C22×Dic4712yes—
430C22×C4C2×C22⋊D4712yes—
431C22×C4C2×C22⋊D4714yes—
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C2×C22⋊D4712yes—
434C22×C4C23×D4711yescharacteristic
435M16—1884no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—942no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—1884no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—942no—
452C4⋊C4—942no—
453C4⋊C4—941no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—941no—
457C4⋊C4—1882no—
458C4⋊C4C22×Dic4714yes—
459C4⋊C4C22×Dic4712yes—
460C4⋊C4C22×Dic4712yes—
461C22⋊C4—24no—
462C22⋊C4—28no—
463C22⋊C4—3762no—
464C22⋊C4—3762no—
465C22⋊C4—1882no—
466C42—24no—
467C42—28no—
468C42—1882no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23C2×C22⋊Dic4714yes—
480C23C2×C22⋊Dic4712yes—
481C23#1309400712yes—
482C23#1309400712yes—
483C23C22×C22⋊D4711yescharacteristic
484Q8—28no—
485D4—216no—
486D4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—3764no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1882no—
508C2×C4—1881no—
509C2×C4—1882no—
510C2×C4—1881no—
511C2×C4—1882no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4C2×C22⋊Dic4714yes—
515C2×C4C2×C22⋊Dic4712yes—
516C2×C4#1309398314yes—
517C2×C4#1309398314yes—
518C2×C4#1309400712yes—
519C2×C4#1309400712yes—
520C2×C4#1309400714yes—
521C2×C4#1309400714yes—
522C2×C4C23×Dic4711yescharacteristic
523C2×C4C22×C22⋊D4711yescharacteristic
524C2×C4C2×#1062535011yescharacteristic
525C2×C4C2×#1062535011yescharacteristic
526C2×C4C2×#1062535011yescharacteristic
527C2×C4C2×#1062535011yescharacteristic
528C8—1881no—
529C8—1881no—
530C8—1882no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22#1568898614yes—
545C22C22×C22⋊Dic4711yescharacteristic
546C22C2×#1309400711yescharacteristic
547C22C2×#1309400711yescharacteristic
548C22#1568992611yescharacteristic
549C22#1568993111yescharacteristic
550C22#1568993411yescharacteristic
551C22#1568993811yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—3762no—
557C4—3761no—
558C4—3761no—
559C4#1568898614yes—
560C4C2×#1309398311yescharacteristic
561C4C2×#1309398311yescharacteristic
562C4C2×#1309400711yescharacteristic
563C4C2×#1309400711yescharacteristic
564C2—24no—
565C2—44no—
566C2C2×#1568898611yescharacteristic
567C2#2025191411yescharacteristic
568C2#2025192111yescharacteristic
569C2#2025334611yescharacteristic
570C2#2025335711yescharacteristic
571C2#2025336611yescharacteristic
572C2#2025337711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47#20703
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #20243483C2semidirect
4. #20243483C2semidirect
5. #20243478C2semidirect
6. #20243474C2semidirect
7. #20243109C2semidirect
8. #20243107C2semidirect
9. #20243104C2semidirect
10. C2×#15688885C2semidirect
36. C47×#520C4semidirect
89. #117×C47C2×C4semidirect
95. C2×C4⋊C4×C47C2×C4semidirect
302. #520Dic47semidirect
330. C47#20703semidirect
362. #117C2×Dic47semidirect
374. C2×C4⋊C4C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890902
#202459802
#202434832
#202434782
#202434742
#202431092
#202431072
#202431042
C2×#156888852
C2×C47×#5202
#2070347