Finite Group Databasebeta

← #26121652

Subgroups — #26121652

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689889C211yesmaximal, characteristic
2C2×#15689096C211yesmaximal, characteristic
3#20245878C211yesmaximal, characteristic
4#20243099C211yesmaximal, characteristic, complemented
5#20243098C211yesmaximal, characteristic, complemented
6C2×#15688883C211yesmaximal, characteristic, complemented
7C2×C47×#339C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9#15689889—22no—
10#15689889—22no—
11#15689889—22no—
12#15689889C414yescomplemented
13C2×#13094032—22no—
14C2×#13094030—22no—
15C22×#10625329C2211yescharacteristic
16C2×M16×D47—22no—
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19#15689096—22no—
20#15689096C414yescomplemented
21C2×#13093938—22no—
22#15689029C412yescomplemented
23#15689025C412yescomplemented
24#15689002—22no—
25C4×(C2×D47)⋊C4—22no—
26#15688884—22no—
27#15688883—22no—
28#15688883—22no—
29C22×M16×C47C2211yescharacteristic
30C2×#117×C47C2211yescharacteristic
31C22×C42×C47C2211yescharacteristic
32C47×#339—22no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350C2×C412yes—
42C2×#10625350C2×C412yes—
43C2×#10625350D412yes—
44C2×#10625350Q812yes—
45C22×Dic94C2×C411yescharacteristic
46C22×D188C2×C411yescharacteristic
47C22×C22⋊D47D412yes—
48C22×C4×D47D412yes—
49#13094032—28no—
50#13094032—28no—
51#13094030—28no—
52#13094030—28no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329C2×C412yes—
57C2×#10625329D412yes—
58C2×#10625329D412yes—
59C2×#10625324—24no—
60M16×D47—28no—
61M16×D47—28no—
62C22×C47⋊C8—21no—
63C2×C8×D47—24no—
64#13093983—21no—
65#13093983—21no—
66#13093983—21no—
67#13093983—21no—
68#13093983—21no—
69#13093983—21no—
70#13093983—21no—
71#13093983C2×C412yes—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2×C412yes—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×(C2×D47)⋊C4—24no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79#13093938—28no—
80C47⋊#95—24no—
81C23×C4×C47—21no—
82C23×C4×C47C2311yescharacteristic
83C2×M16×C47—21no—
84C2×M16×C47—21no—
85C2×M16×C47—21no—
86C2×M16×C47C2×C412yes—
87C2×M16×C47D412yes—
88C2×M16×C47D412yes—
89C22×C8×C47—21no—
90#117×C47—21no—
91#117×C47—21no—
92#117×C47—21no—
93#117×C47C2×C412yes—
94C2×C4⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47—21no—
98C2×C42×C47—21no—
99C2×C42×C47—21no—
100C2×C42×C47—21no—
101C2×C42×C47—21no—
102C2×C42×C47C2×C412yes—
103C24×C47C2×D411yescharacteristic
104C23×D47C22⋊C412yes—
105C22×C4×C47—22no—
106C22×C4×C47—44no—
107C22×C4×C47—41no—
108C22×C4×C47—41no—
109C22×C4×C47—41no—
110C22×C4×C47C22×C411yescharacteristic
111C22×C4×C47C22×C411yescharacteristic
112C22×C4×C47C22×C411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×Q811yescharacteristic
120C22×C4×C47#4111yescharacteristic
121C22×C4×C47#4111yescharacteristic
122C22×C4×C47#4111yescharacteristic
123C22×C4×C47#4111yescharacteristic
124C2×C22⋊D47—22no—
125C2×C22⋊D47—22no—
126C2×C22⋊D47—22no—
127C2×C22⋊D47—22no—
128C2×C22⋊D47C22⋊C414yes—
129C2×C22⋊D47C4⋊C414yes—
130C22×Dic47—22no—
131C22×Dic47—41no—
132C22×Dic47C22⋊C412yes—
133#10625350—22no—
134#10625350—22no—
135#10625350—22no—
136#10625350—22no—
137#10625350—22no—
138#10625350—22no—
139#10625350—22no—
140#10625350—22no—
141#10625350—22no—
142#10625350—22no—
143#10625350—22no—
144#10625350—22no—
145#10625350—22no—
146#10625350—22no—
147#10625350—22no—
148#10625350—22no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188C4212yes—
154C2×D188C4⋊C412yes—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47C22⋊C414yes—
160C2×C4×D47C4⋊C414yes—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94C4212yes—
166C2×Dic94C4⋊C412yes—
167M16×C47—22no—
168M16×C47—22no—
169M16×C47—22no—
170M16×C47C22⋊C414yescomplemented
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—41no—
176C22⋊C4×C47—22no—
177C22⋊C4×C47—41no—
178C42×C47—22no—
179C42×C47—22no—
180C42×C47—22no—
181C42×C47—41no—
182C42×C47—41no—
183C42×C47—41no—
184C22⋊Dic47—22no—
185C22⋊Dic47—41no—
186C22⋊Dic47—41no—
187C22⋊Dic47—41no—
188(C2×D47)⋊C4—48no—
189C4⋊Dic47—22no—
190C4⋊Dic47—22no—
191C4⋊Dic47—22no—
192C4⋊Dic47—41no—
193C4×Dic47—22no—
194C4×Dic47—41no—
195C4×Dic47—41no—
196C4×Dic47—41no—
197#10625329—22no—
198#10625329—22no—
199#10625329—22no—
200#10625329C22⋊C414yescomplemented
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204#10625324—216no—
205C8×D47—216no—
206C23×C47—42no—
207C23×C47C2×C22⋊C411yescharacteristic
208C23×C47C2×C22⋊C411yescharacteristic
209C23×C47C2×C4⋊C411yescharacteristic
210C23×C47#12111yescharacteristic
211C23×C47#12111yescharacteristic
212C23×C47#12111yescharacteristic
213C23×C47#12111yescharacteristic
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—44no—
217C22×D47#9514yes—
218C2×C4×C47—24no—
219C2×C4×C47—42no—
220C2×C4×C47—44no—
221C2×C4×C47—44no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—82no—
225C2×C4×C47C2×C4211yescharacteristic
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47C2×C22⋊C411yescharacteristic
228C2×C4×C47C2×C22⋊C411yescharacteristic
229C2×C4×C47C2×C4⋊C411yescharacteristic
230C2×C4×C47C2×C4⋊C411yescharacteristic
231C2×C4×C47#11711yescharacteristic
232C2×C4×C47#11711yescharacteristic
233C2×C4×C47C4×D411yescharacteristic
234C2×C4×C47C4×D411yescharacteristic
235C2×C4×C47C4×D411yescharacteristic
236C2×C4×C47C4×D411yescharacteristic
237C2×C4×C47C4×D411yescharacteristic
238C2×C4×C47C4×D411yescharacteristic
239C2×C4×C47C4×D411yescharacteristic
240C2×C4×C47C4×D411yescharacteristic
241C2×C4×C47#12011yescharacteristic
242C2×C4×C47#12011yescharacteristic
243C2×C4×C47#12011yescharacteristic
244C2×C4×C47#12011yescharacteristic
245C2×C4×C47C22⋊Q811yescharacteristic
246C2×C4×C47C22⋊Q811yescharacteristic
247C2×C4×C47C22⋊Q811yescharacteristic
248C2×C4×C47C22⋊Q811yescharacteristic
249C2×C4×C47#12311yescharacteristic
250C2×C4×C47#12311yescharacteristic
251C2×C4×C47#12311yescharacteristic
252C2×C4×C47#12311yescharacteristic
253C22⋊D47—24no—
254C22⋊D47—24no—
255C22⋊D47—24no—
256C22⋊D47—24no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—44no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—82no—
263C2×Dic47#9514yes—
264D188—22no—
265D188—22no—
266D188—22no—
267D188—22no—
268C4×D47—24no—
269C4×D47—24no—
270C4×D47—24no—
271C4×D47—24no—
272Dic94—22no—
273Dic94—22no—
274Dic94—22no—
275Dic94—22no—
276C8×C47—24no—
277C47⋊C8—24no—
278#10751—471nomaximal, Sylow
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47C2×#9511yescharacteristic
287C22×C47#38011yescharacteristic
288C22×C47#38011yescharacteristic
289C22×C47#38611yescharacteristic
290C22×C47#38611yescharacteristic
291C22×C47#38611yescharacteristic
292C22×C47#38611yescharacteristic
293C2×D47—24no—
294C2×D47—44no—
295C2×D47—44no—
296C2×D47—82no—
297C4×C47—44no—
298C4×C47—82no—
299C4×C47C4×C22⋊C411yescharacteristic
300C4×C47C4×C22⋊C411yescharacteristic
301C4×C47#37911yescharacteristic
302C4×C47#38111yescharacteristic
303C4×C47#38511yescharacteristic
304C4×C47#38511yescharacteristic
305C4×C47#38511yescharacteristic
306C4×C47#38511yescharacteristic
307Dic47—24no—
308Dic47—82no—
309Dic47—82no—
310C2×#568—471no—
311C2×#410—471no—
312#2297—471no—
313#1733—471no—
314#1733—471no—
315C2×#339—471no—
316C2×#339D4711yescharacteristic, complemented
317C2×C47—21no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—21no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47C22⋊#9511yescharacteristic
324C2×C47#188412yes—
325D47—44no—
326C22×#41—471no—
327#568—474no—
328#568—942no—
329#568—942no—
330#568—942no—
331C2×#131—942no—
332C2×#131—942no—
333C22×M16—471no—
334C22×M16—942no—
335C22×M16C2×D4711yescharacteristic
336C2×#117—471no—
337C2×#117—471no—
338C2×#117C2×D4711yescharacteristic
339C22×C42C2×D4711yescharacteristic
340#410—474no—
341#410—942no—
342C2×#99—942no—
343#391—472no—
344#390—472no—
345#386—942no—
346C4×C22⋊C4—942no—
347#339—22no—
348#339—942no—
349#339—942no—
350#339—942no—
351C47#1075111yescharacteristic, complemented, Sylow
352C2×#41—472no—
353C2×#41—472no—
354C2×#41—472no—
355C2×#41—472no—
356C2×#41—941no—
357C2×#41—941no—
358C2×#41—941no—
359C2×#41—941no—
360C2×#41—941no—
361C2×#41—941no—
362C2×#41—941no—
363C2×#41—941no—
364C22×Q8—471no—
365C22×D4—472no—
366C22×D4—471no—
367C23×C4—21no—
368C23×C4—472no—
369C23×C4C22×D4711yescharacteristic
370#131—948no—
371#131—948no—
372#131—948no—
373#131—948no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—472no—
378C2×M16—472no—
379C2×M16—472no—
380C2×M16—944no—
381C2×M16—948no—
382C2×M16—948no—
383C2×M16—941no—
384C2×M16—941no—
385C2×M16—941no—
386C2×M16C4×D4712yes—
387C2×M16D18812yes—
388C2×M16C22⋊D4712yes—
389C22×C8—21no—
390C22×C8—944no—
391C22×C8—941no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—472no—
396#117—941no—
397#117—941no—
398#117—941no—
399#117—941no—
400#117—941no—
401#117—941no—
402#117—941no—
403#117C4×D4712yes—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—472no—
406C2×C4⋊C4—941no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—941no—
409C2×C22⋊C4—944no—
410C2×C22⋊C4—941no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—941no—
418C2×C42—941no—
419C2×C42C4×D4712yes—
420#99—948no—
421#95—944no—
422C24—472no—
423C24D4×D4711yescharacteristic
424#41—942no—
425#41—942no—
426#41—942no—
427#41—942no—
428#41—942no—
429#41—942no—
430#41—942no—
431#41—942no—
432#41—942no—
433#41—942no—
434#41—942no—
435#41—942no—
436#41—942no—
437#41—942no—
438#41—942no—
439#41—942no—
440C2×Q8—472no—
441C2×Q8—472no—
442C2×Q8—941no—
443C2×Q8—941no—
444C2×Q8—941no—
445C2×Q8—941no—
446C2×D4—474no—
447C2×D4—474no—
448C2×D4—472no—
449C2×D4—472no—
450C2×D4—942no—
451C2×D4—942no—
452C2×D4—942no—
453C2×D4—942no—
454C2×D4—941no—
455C2×D4—941no—
456C2×D4—941no—
457C2×D4—941no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—472no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—1881no—
472C22×C4C2×C4×D4711yescharacteristic
473C22×C4C2×C4×D4711yescharacteristic
474C22×C4C2×C4×D4711yescharacteristic
475C22×C4C2×D18811yescharacteristic
476C22×C4#1062535011yescharacteristic
477C22×C4#1062535011yescharacteristic
478C22×C4D4×D4711yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4#1062535211yescharacteristic
483C22×C4Q8×D4711yescharacteristic
484C22×C4#1062535411yescharacteristic
485C22×C4C2×C22⋊D4711yescharacteristic
486M16—22no—
487M16—22no—
488M16—22no—
489M16—474no—
490M16—9416no—
491M16—942no—
492M16—942no—
493M16—942no—
494M16(C2×D47)⋊C414yescomplemented
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—9416no—
499C2×C8—942no—
500C2×C8—942no—
501C2×C8—942no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—942no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—1881no—
508C22⋊C4—22no—
509C22⋊C4—41no—
510C22⋊C4—942no—
511C22⋊C4—1888no—
512C22⋊C4—1881no—
513C22⋊C4—1881no—
514C22⋊C4—1881no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—942no—
522C42—1881no—
523C42—1881no—
524C42—1881no—
525C23—42no—
526C23—474no—
527C23—942no—
528C23—942no—
529C23—1884no—
530C23C22⋊C4×D4711yescharacteristic
531C23C22⋊C4×D4711yescharacteristic
532C23C4⋊C4×D4711yescharacteristic
533C23#1309399511yescharacteristic
534C23#1309399711yescharacteristic
535C23#1309399711yescharacteristic
536C23#1309399911yescharacteristic
537Q8—942no—
538Q8—942no—
539Q8—942no—
540Q8—942no—
541D4—944no—
542D4—944no—
543D4—944no—
544D4—944no—
545D4—942no—
546D4—942no—
547D4—942no—
548D4—942no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—474no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—3762no—
564C2×C4—1884no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4C42×D4711yescharacteristic
568C2×C4C22⋊C4×D4711yescharacteristic
569C2×C4C22⋊C4×D4711yescharacteristic
570C2×C4C2×(C2×D47)⋊C411yescharacteristic
571C2×C4C4⋊C4×D4711yescharacteristic
572C2×C4C4⋊C4×D4711yescharacteristic
573C2×C4#1309398111yescharacteristic
574C2×C4#1309398211yescharacteristic
575C2×C4(C22⋊D47)⋊C411yescharacteristic
576C2×C4(C22⋊D47)⋊C411yescharacteristic
577C2×C4C4×C22⋊D4711yescharacteristic
578C2×C4C4×C22⋊D4711yescharacteristic
579C2×C4#1309398611yescharacteristic
580C2×C4#1309398611yescharacteristic
581C2×C4C4×D18811yescharacteristic
582C2×C4C4×D18811yescharacteristic
583C2×C4#1309399211yescharacteristic
584C2×C4#1309399211yescharacteristic
585C2×C4#1309399411yescharacteristic
586C2×C4#1309399411yescharacteristic
587C2×C4#1309400011yescharacteristic
588C2×C4#1309400211yescharacteristic
589C2×C4#1309400211yescharacteristic
590C2×C4#1309400411yescharacteristic
591C2×C4#1309400611yescharacteristic
592C2×C4#1309400611yescharacteristic
593C2×C4#1309400911yescharacteristic
594C2×C4#1309400911yescharacteristic
595C8—24no—
596C8—944no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—944no—
605C22—3762no—
606C22—1884no—
607C22—1884no—
608C22#95×D4711yescharacteristic
609C22#1568898011yescharacteristic
610C22#1568898211yescharacteristic
611C22#1568899911yescharacteristic
612C22#1568899911yescharacteristic
613C22#1568900111yescharacteristic
614C22#1568900111yescharacteristic
615C4—44no—
616C4—82no—
617C4—944no—
618C4—3762no—
619C4—3762no—
620C4#1568897011yescharacteristic
621C4C4×(C2×D47)⋊C411yescharacteristic
622C4#1568897811yescharacteristic
623C4#1568898511yescharacteristic
624C4#1568899411yescharacteristic
625C4#1568899411yescharacteristic
626C4#1568899711yescharacteristic
627C4#1568899711yescharacteristic
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—1884no—
635C2(C2×D47)⋊#9511yescharacteristic
636C2#2024377012yes—
subgroupG/H
centerC2×C4#13093994
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×#339C2
socleC22×C47#386
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×#339D47
hypocenterC47#10751
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. #20243099C2semidirect
5. #20243098C2semidirect
6. C2×#15688883C2semidirect
7. C2×C47×#339C2semidirect
12. #15689889C4semidirect
20. #15689096C4semidirect
22. #15689029C4semidirect
23. #15689025C4semidirect
170. M16×C47C22⋊C4semidirect
200. #10625329C22⋊C4semidirect
316. C2×#339D47semidirect
351. C47#10751semidirect
494. M16(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156898892
C2×#156890962
#202458782
#202430992
#202430982
C2×#156888832
C2×C47×#3392
#1075147