Finite Group Databasebeta

← #26224765

Subgroups — #26224765

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689505C211yesmaximal, characteristic
2C2×#15689233C211yesmaximal, characteristic
3C2×#15689228C211yesmaximal, characteristic, complemented
4C2×C4×#10625336C211yesmaximal, characteristic, complemented
5C2×C4×#10625335C211yesmaximal, characteristic, complemented
6#20246099C211yesmaximal, characteristic, complemented
7#20244263C211yesmaximal, characteristic
8#20244258C211yesmaximal, characteristic, complemented
9#20244245C211yesmaximal, characteristic, complemented
10#20244241C211yesmaximal, characteristic, complemented
11#20244205C211yesmaximal, characteristic, complemented
12#20244202C211yesmaximal, characteristic, complemented
13#20243544C211yesmaximal, characteristic, complemented
14#20243075C211yesmaximal, characteristic, complemented
15C47×#2326C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625335C2211yescharacteristic
18C22×#10625336C2211yescharacteristic
19C22×#10625334C2211yescharacteristic
20C2×#13094012C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23#15689505C418yescomplemented
24#15689233—24no—
25#15689228—24no—
26C4×#10625336—24no—
27C4×#10625335—24no—
28C2×#13093955C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31C2×#13093955C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093948C2211yescharacteristic
34C2×#13093947C2211yescharacteristic
35C2×#13093947C2211yescharacteristic
36C2×#13093945C2211yescharacteristic
37C2×#13093945C2211yescharacteristic
38C2×#13093944C2211yescharacteristic
39C2×#13093944C2211yescharacteristic
40C2×#13093935C2211yescharacteristic
41C2×#13093935C2211yescharacteristic
42C2×#13093935C2211yescharacteristic
43C2×C4×C47⋊C8C2211yescharacteristic
44C2×C4×C47⋊C8C2211yescharacteristic
45C2×C4×C47⋊C8C2211yescharacteristic
46#15689029C2211yescharacteristic
47#15689025C2211yescharacteristic
48#15689015—22no—
49#15688997—22no—
50#15688993C2211yescharacteristic
51C2×#124×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#390C2211yescharacteristic
56C47×#388—22no—
57C47×#385—22no—
58C47×#384C2211yescharacteristic
59C47×#376C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yescomplemented
66C2×#10625335—22no—
67C2×#10625335—22no—
68C2×#10625335C2×C414yescomplemented
69C2×#10625336—22no—
70C2×#10625336—22no—
71C2×#10625336C2×C414yescomplemented
72C2×#10625334—22no—
73C2×#10625334—22no—
74C2×#10625334C2×C414yes—
75C22×C47⋊C8C2311yescharacteristic
76C22×C47⋊C8C2311yescharacteristic
77C22×C47⋊C8C2311yescharacteristic
78C22×C47⋊C8C2311yescharacteristic
79#13094012—22no—
80#13094012C2×C414yescomplemented
81C4×Dic94—22no—
82C4×Dic94—22no—
83C4×D188—22no—
84C4×D188—22no—
85C2×C4⋊Dic47—22no—
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×Dic47⋊C4—21no—
89C2×(C2×D47)⋊C4—22no—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—22no—
92C2×C4×Dic47—21no—
93#13093955—22no—
94#13093955—22no—
95#13093955—22no—
96#13093955—22no—
97#13093948—22no—
98#13093948—22no—
99#13093947—22no—
100#13093947—22no—
101#13093945—22no—
102#13093945—22no—
103#13093944—22no—
104#13093944—22no—
105#13093935—22no—
106#13093935—22no—
107#13093935D414yes—
108C4×C47⋊C8—22no—
109C4×C47⋊C8C2×C414yescomplemented
110C4×C47⋊C8D414yes—
111C47⋊#95—22no—
112C22×Q8×C47C2311yescharacteristic
113C22×D4×C47C2311yescharacteristic
114C23×C4×C47—21no—
115#124×C47—22no—
116#124×C47C2×C414yescomplemented
117C4×Q8×C47—22no—
118C4×Q8×C47—22no—
119C4×D4×C47—22no—
120C4×D4×C47—22no—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—22no—
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C22⋊C4×C47—22no—
126C2×C22⋊C4×C47—21no—
127C2×C22⋊C4×C47—22no—
128C2×C42×C47—21no—
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#95×C47—22no—
133#95×C47D412yes—
134#95×C47D412yes—
135C24×C47—21no—
136C23×D47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47C22×C412yes—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47C22×C412yes—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—42no—
153C22×C4×C47—82no—
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2411yescharacteristic
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C2×D188—21no—
163C2×D188—21no—
164C2×D188—21no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188C22×C412yes—
168C2×C4×D47—42no—
169C2×C4×D47—82no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94C22×C412yes—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47—82no—
182C22⋊C4×C47—28no—
183C22⋊C4×C47—42no—
184C22⋊C4×C47—82no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—42no—
188C42×C47C22×C412yes—
189C42×C47C2×D412yes—
190C42×C47C2×D412yes—
191C42×C47#4112yes—
192#10625337—28no—
193#10625336—28no—
194#10625335—28no—
195#10625334—28no—
196(C2×D47)⋊C4—28no—
197(C2×D47)⋊C4—42no—
198(C2×D47)⋊C4—82no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—82no—
204Dic47⋊C4—42no—
205C4×Dic47—42no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8C22×C412yes—
211C2×C47⋊C8C22×C412yes—
212C2×C47⋊C8C2×D412yes—
213C2×C47⋊C8C2×D412yes—
214C2×C47⋊C8C2×D412yes—
215C2×C47⋊C8C2×D412yes—
216C2×C47⋊C8#4112yes—
217C2×C47⋊C8#4112yes—
218C23×C47—22no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—42no—
222C23×C47—42no—
223C23×C47C22×D411yescharacteristic
224C22×D47—22no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—42no—
228C22×D47—42no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—41no—
236D4×C47—41no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—84no—
252C2×C4×C47C4×D414yes—
253C2×C4×C47#12712yes—
254C2×C4×C47#12712yes—
255C2×C4×C47C23×C411yescharacteristic
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×Dic47—22no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268D188—22no—
269D188—22no—
270D188—41no—
271D188—41no—
272C4×D47—84no—
273Dic94—22no—
274Dic94—22no—
275Dic94—41no—
276Dic94—41no—
277C47⋊C8—22no—
278C47⋊C8—22no—
279C47⋊C8C4×D414yes—
280C47⋊C8C4×D414yes—
281#21694—471nomaximal, Sylow
282C22×C47—42no—
283C22×C47—44no—
284C22×C47—81no—
285C22×C47—81no—
286C22×C47—82no—
287C22×C47C2×C4×D411yescharacteristic
288C22×C47C2×#12711yescharacteristic
289C22×C47#53211yescharacteristic
290C22×C47#57712yes—
291C22×C47#57712yes—
292C2×D47—42no—
293C2×D47—44no—
294C2×D47—81no—
295C2×D47—81no—
296C2×D47—82no—
297C4×C47—22no—
298C4×C47—22no—
299C4×C47—24no—
300C4×C47—42no—
301C4×C47—42no—
302C4×C47—81no—
303C4×C47—81no—
304C4×C47C2×C4×D411yescharacteristic
305C4×C47C2×C4×D411yescharacteristic
306C4×C47#53211yescharacteristic
307C4×C47#53211yescharacteristic
308Dic47—42no—
309Dic47—81no—
310Dic47—81no—
311C2×#495—471no—
312C2×#442—471no—
313C2×#441—471no—
314C2×C4×SD16—471no—
315C2×C4×SD16—471no—
316#2326—471no—
317#2326D4711yescharacteristic, complemented
318#1972—471no—
319#1971—471no—
320#1968—471no—
321#1968—471no—
322#1959—471no—
323#1959—471no—
324#1844—471no—
325#1726—471no—
326C2×C47—82no—
327C2×C47C4×#12711yescharacteristic
328C2×C47#295212yes—
329C2×C47#315411yescharacteristic
330C2×C47#315411yescharacteristic
331C2×C47#317312yes—
332D47—82no—
333C22×Q16—471no—
334C22×SD16—471no—
335C22×SD16—471no—
336C22×D8—471no—
337C2×#124—471no—
338C2×#124C2×D4711yescharacteristic
339C2×C4×Q8—471no—
340C2×C4×Q8C2×D4711yescharacteristic
341C2×C4×D4—471no—
342C2×C4×D4C2×D4711yescharacteristic
343#495—478no—
344#442—944no—
345#441—944no—
346C4×SD16—944no—
347C4×SD16—944no—
348C2×#106—471no—
349C2×#106—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#103—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×#103—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#102—471no—
360C2×#97—471no—
361C2×#97—471no—
362C2×#97—471no—
363C2×C4×C8—471no—
364C2×C4×C8—471no—
365C2×C4×C8—471no—
366#391—471no—
367#391C2×D4711yescharacteristic
368#390—471no—
369#390C2×D4711yescharacteristic
370#388—22no—
371#388—942no—
372#385—22no—
373#385—942no—
374#384—471no—
375#384C2×D4711yescharacteristic
376#376C2×D4711yescharacteristic
377C47#2169411yescharacteristic, complemented, Sylow
378C22×Q8—471no—
379C22×Q8C22×D4711yescharacteristic
380C22×D4—471no—
381C22×D4C22×D4711yescharacteristic
382C23×C4—21no—
383C23×C4—941no—
384C2×Q16—474no—
385C2×Q16—942no—
386C2×Q16—942no—
387C2×SD16—474no—
388C2×SD16—474no—
389C2×SD16—942no—
390C2×SD16—942no—
391C2×SD16—942no—
392C2×SD16—942no—
393C2×D8—474no—
394C2×D8—942no—
395C2×D8—942no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—471no—
400#124—22no—
401#124—474no—
402#124—942no—
403#124C4×D4714yescomplemented
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—942no—
407C4×Q8—942no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—942no—
411C4×D4—942no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C22⋊C4—22no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—941no—
428C2×C42C22×D4711yescharacteristic
429C2×C42C22×D4711yescharacteristic
430C2×C42C22×D4711yescharacteristic
431#106—942no—
432#106—942no—
433#106—942no—
434#106—942no—
435#103—942no—
436#103—942no—
437#103—942no—
438#103—942no—
439#102—942no—
440#102—942no—
441#102—942no—
442#102—942no—
443#97—474no—
444#97—942no—
445#97—942no—
446C4×C8—474no—
447C4×C8—474no—
448C4×C8—942no—
449#95—22no—
450#95—942no—
451#95C22⋊D4712yes—
452#95C22⋊D4712yes—
453C24—21no—
454C24—941no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—472no—
461C2×Q8—941no—
462C2×Q8—941no—
463C2×Q8—941no—
464C2×Q8—941no—
465C2×Q8—941no—
466C2×Q8C2×C4×D4712yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—472no—
473C2×D4—941no—
474C2×D4—941no—
475C2×D4—941no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4C2×C4×D4712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—3762no—
488C22×C4—1882no—
489C22×C4C2×C22⋊D4712yes—
490C22×C4C2×C22⋊D4712yes—
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493C22×C4C23×D4711yescharacteristic
494Q16—948no—
495SD16—948no—
496SD16—948no—
497D8—948no—
498C2×C8—472no—
499C2×C8—472no—
500C2×C8—472no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—472no—
504C2×C8—472no—
505C2×C8—472no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—82no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—3762no—
521C4⋊C4—1882no—
522C22⋊C4—28no—
523C22⋊C4—42no—
524C22⋊C4—82no—
525C22⋊C4—948no—
526C22⋊C4—3762no—
527C22⋊C4—1882no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—1882no—
532C42C2×C4×D4712yes—
533C42#1062535012yes—
534C42D4×D4712yes—
535C42D4×D4712yes—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—942no—
542C23—1881no—
543C23—1881no—
544C23—1882no—
545C23—1882no—
546C23C22×C22⋊D4711yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—942no—
552Q8—942no—
553Q8—1881no—
554Q8—1881no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—942no—
560D4—942no—
561D4—1881no—
562D4—1881no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—942no—
579C2×C4—3764no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4C4×C22⋊D4714yes—
587C2×C4#1309402012yes—
588C2×C4#1309402012yes—
589C2×C4C22×C4×D4711yescharacteristic
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4C2×D4×D4711yescharacteristic
592C2×C4#41×D4711yescharacteristic
593C2×C4#41×D4711yescharacteristic
594C2×C4C2×#1062535011yescharacteristic
595C8—474no—
596C8—474no—
597C8—942no—
598C8—942no—
599C22—44no—
600C22—42no—
601C22—82no—
602C22—81no—
603C22—81no—
604C22—3762no—
605C22—3761no—
606C22—3761no—
607C22—1884no—
608C22—1882no—
609C22C2×C4×C22⋊D4711yescharacteristic
610C22C2×#1309402011yescharacteristic
611C22#1568966911yescharacteristic
612C22#1568993112yes—
613C22#1568993412yes—
614C4—24no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—3761no—
622C4—3761no—
623C4—1882no—
624C4C4×D4×D4711yescharacteristic
625C4C4×D4×D4711yescharacteristic
626C4#1568967011yescharacteristic
627C4#1568967011yescharacteristic
628C2—82no—
629C2—3762no—
630C2C4×#1309402011yescharacteristic
631C2#2025226912yes—
632C2#2025386011yescharacteristic
633C2#2025386611yescharacteristic
634C2#2025399912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2326C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
cosocleC22×C4×C47C24
hypercenter#2326D47
hypocenterC47#21694
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. C2×#15689228C2semidirect
4. C2×C4×#10625336C2semidirect
5. C2×C4×#10625335C2semidirect
6. #20246099C2semidirect
8. #20244258C2semidirect
9. #20244245C2semidirect
10. #20244241C2semidirect
11. #20244205C2semidirect
12. #20244202C2semidirect
13. #20243544C2semidirect
14. #20243075C2semidirect
15. C47×#2326C2semidirect
23. #15689505C4semidirect
65. C2×#10625337C2×C4semidirect
68. C2×#10625335C2×C4semidirect
71. C2×#10625336C2×C4semidirect
80. #13094012C2×C4semidirect
109. C4×C47⋊C8C2×C4semidirect
116. #124×C47C2×C4semidirect
317. #2326D47semidirect
377. C47#21694semidirect
403. #124C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895052
C2×#156892332
C2×#156892282
C2×C4×#106253362
C2×C4×#106253352
#202460992
#202442632
#202442582
#202442452
#202442412
#202442052
#202442022
#202435442
#202430752
C47×#23262
#2169447