Finite Group Databasebeta

← #26121639

Subgroups — #26121639

Order 12032.

1562 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689597C211yesmaximal, characteristic
2C22×#13093940C211yesmaximal, characteristic
3C2×#15689024C211yesmaximal, characteristic
4#20243107C211yesmaximal, characteristic, complemented
5C2×#15688884C211yesmaximal, characteristic, complemented
6C2×#15688882C211yesmaximal, characteristic, complemented
7C47×#1736C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10#15689597—22no—
11#15689597—22no—
12#15689597—22no—
13#15689597C414yescomplemented
14C2×C4×D188—22no—
15C2×#13093986—22no—
16C2×#13093983C2211yescharacteristic
17C22×(C2×D47)⋊C4—22no—
18C2×C22⋊C4×D47—22no—
19C22×C4×Dic47C2211yescharacteristic
20C2×#13093940—22no—
21C2×#13093940—22no—
22C2×#13093940C414yescomplemented
23#15689024—22no—
24#15689024C414yescomplemented
25#15688885—22no—
26#15688884—22no—
27#15688882—22no—
28C22×M16×C47C2211yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C47×#340—22no—
32C24×D47D412yes—
33C22×D188—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C22×D188—21no—
41C22×D188C2×C412yes—
42C22×D188C2×C412yes—
43C22×D188C2×C412yes—
44C22×D188D412yes—
45C22×D188D412yes—
46C22×D188Q812yes—
47C23×Dic47—21no—
48C22×C4×D47—24no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329C2×C412yescomplemented
55C22×C47⋊C8—21no—
56C4×D188—28no—
57C4×D188—28no—
58#13093986—28no—
59#13093986—28no—
60#13093983—21no—
61#13093983—21no—
62#13093983C2×C412yes—
63#13093983D412yes—
64C2×C4⋊Dic47D412yes—
65C2×C22⋊Dic47—21no—
66C2×(C2×D47)⋊C4—24no—
67C2×(C2×D47)⋊C4—28no—
68C2×(C2×D47)⋊C4—48no—
69C22⋊C4×D47—28no—
70C22⋊C4×D47—28no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47—21no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47C2×C412yescomplemented
78#13093940—28no—
79C23×C4×C47—21no—
80C23×C4×C47C2311yescharacteristic
81C2×M16×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47—21no—
84C2×M16×C47—21no—
85C2×M16×C47—21no—
86C2×M16×C47C2×C412yescomplemented
87C22×C8×C47—21no—
88#117×C47—21no—
89#117×C47—21no—
90#117×C47C2×C412yes—
91#117×C47D412yes—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47C2×C412yescomplemented
98C2×C4⋊C4×C47D412yes—
99C2×C22⋊C4×C47—21no—
100C2×C42×C47—21no—
101C24×C47C2×D411yescharacteristic
102C23×D47—22no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—44no—
107C23×D47—44no—
108C23×D47C22⋊C414yes—
109C23×D47C22⋊C414yes—
110C23×D47C4⋊C414yes—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—44no—
114C22×C4×C47—41no—
115C22×C4×C47—41no—
116C22×C4×C47C22×C411yescharacteristic
117C22×C4×C47C22×C411yescharacteristic
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×Q811yescharacteristic
126C22×C4×C47#4111yescharacteristic
127C22×C4×C47#4111yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×C4×C47#4111yescharacteristic
130C22×Dic47—24no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—42no—
134C22×Dic47—41no—
135C22×Dic47—41no—
136C2×D188—22no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—22no—
148C2×D188—22no—
149C2×D188—22no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—22no—
156C2×D188—22no—
157C2×D188—22no—
158C2×D188—22no—
159C2×D188—22no—
160C2×D188C4214yes—
161C2×D188C22⋊C414yes—
162C2×D188C4⋊C414yes—
163C2×D188C4⋊C414yes—
164C2×C4×D47—216no—
165C2×C4×D47—84no—
166C2×C4×D47—84no—
167M16×C47—22no—
168M16×C47—22no—
169M16×C47—41no—
170M16×C47—41no—
171C2×C8×C47—22no—
172C2×C8×C47—41no—
173C2×C8×C47—41no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—41no—
178C4⋊C4×C47—41no—
179C4⋊C4×C47C22⋊C414yescomplemented
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C42×C47—22no—
183C42×C47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186(C2×D47)⋊C4—216no—
187(C2×D47)⋊C4—416no—
188C4⋊Dic47—22no—
189C4⋊Dic47C22⋊C414yescomplemented
190C4×Dic47—22no—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47—22no—
194C4×Dic47—41no—
195C4×Dic47—41no—
196#10625329—22no—
197#10625329—22no—
198#10625329—41no—
199#10625329—41no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—41no—
202C2×C47⋊C8—41no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47C2×C22⋊C411yescharacteristic
206C23×C47C2×C22⋊C411yescharacteristic
207C23×C47C2×C4⋊C411yescharacteristic
208C23×C47#12011yescharacteristic
209C23×C47#12011yescharacteristic
210C23×C47#12311yescharacteristic
211C23×C47#12311yescharacteristic
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—44no—
219C22×D47—44no—
220C22×D47—44no—
221C22×D47—48no—
222C22×D47—44no—
223C22×D47—48no—
224C22×D47—84no—
225C22×D47—84no—
226C22×D47—82no—
227C22×D47—82no—
228C22×D47#9518yes—
229C2×C4×C47—24no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—44no—
233C2×C4×C47—42no—
234C2×C4×C47—82no—
235C2×C4×C47—82no—
236C2×C4×C47C2×C4211yescharacteristic
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C2×C22⋊C411yescharacteristic
239C2×C4×C47C2×C22⋊C411yescharacteristic
240C2×C4×C47C2×C4⋊C411yescharacteristic
241C2×C4×C47C2×C4⋊C411yescharacteristic
242C2×C4×C47#11711yescharacteristic
243C2×C4×C47#11711yescharacteristic
244C2×C4×C47C4×D411yescharacteristic
245C2×C4×C47C4×D411yescharacteristic
246C2×C4×C47C4×D411yescharacteristic
247C2×C4×C47C4×D411yescharacteristic
248C2×C4×C47C4×D411yescharacteristic
249C2×C4×C47C4×D411yescharacteristic
250C2×C4×C47C4×D411yescharacteristic
251C2×C4×C47C4×D411yescharacteristic
252C2×C4×C47#12011yescharacteristic
253C2×C4×C47#12011yescharacteristic
254C2×C4×C47#12111yescharacteristic
255C2×C4×C47#12111yescharacteristic
256C2×C4×C47#12111yescharacteristic
257C2×C4×C47#12111yescharacteristic
258C2×C4×C47C22⋊Q811yescharacteristic
259C2×C4×C47C22⋊Q811yescharacteristic
260C2×C4×C47C22⋊Q811yescharacteristic
261C2×C4×C47C22⋊Q811yescharacteristic
262C2×C4×C47#12311yescharacteristic
263C2×C4×C47#12311yescharacteristic
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—44no—
267C2×Dic47—44no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272D188—24no—
273D188—24no—
274D188—24no—
275D188—24no—
276D188—24no—
277D188—24no—
278D188—24no—
279D188—24no—
280C4×D47—816no—
281C8×C47—42no—
282C47⋊C8—42no—
283#10749—471nomaximal, Sylow
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—21no—
289C22×C47—21no—
290C22×C47—21no—
291C22×C47—21no—
292C22×C47C2×#9511yescharacteristic
293C22×C47#37911yescharacteristic
294C22×C47#38111yescharacteristic
295C22×C47#38511yescharacteristic
296C22×C47#38511yescharacteristic
297C22×C47#38611yescharacteristic
298C22×C47#38611yescharacteristic
299C22×C47#45312yes—
300C22×C47#45512yes—
301C2×D47—28no—
302C2×D47—48no—
303C2×D47—48no—
304C2×D47—44no—
305C2×D47—44no—
306C2×D47—48no—
307C2×D47—84no—
308C2×D47—84no—
309C2×D47—84no—
310C4×C47—44no—
311C4×C47—82no—
312C4×C47C4×C22⋊C411yescharacteristic
313C4×C47C4×C22⋊C411yescharacteristic
314C4×C47#38011yescharacteristic
315C4×C47#38011yescharacteristic
316C4×C47#38511yescharacteristic
317C4×C47#38511yescharacteristic
318C4×C47#38611yescharacteristic
319C4×C47#38611yescharacteristic
320Dic47—44no—
321Dic47—44no—
322C2×#518—471no—
323C22×#100—471no—
324C2×#390—471no—
325#1736—471no—
326#1736D4711yescharacteristic, complemented
327C2×#339—471no—
328C2×#339—471no—
329C2×C47—21no—
330C2×C47—21no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47C22⋊#9511yescharacteristic
334C2×C47#179212yes—
335C2×C47#185612yes—
336C2×C47#188511yescharacteristic
337C2×C47#188711yescharacteristic
338D47—48no—
339C23×D4—471no—
340C22×M16—471no—
341C22×M16C2×D4711yescharacteristic
342#518—474no—
343#518—942no—
344#518—942no—
345#518—942no—
346C2×C4×D4—942no—
347C2×C4×D4—942no—
348C2×#117—471no—
349C2×#117C2×D4711yescharacteristic
350C22×C4⋊C4C2×D4711yescharacteristic
351C22×C22⋊C4—942no—
352C22×C22⋊C4—942no—
353C22×C42—471no—
354C2×#100—474no—
355C2×#100—942no—
356C2×#100—942no—
357#390—474no—
358#390—942no—
359#340—22no—
360#340—942no—
361#339—942no—
362#339—942no—
363C47#1074911yescharacteristic, complemented, Sylow
364C25—472no—
365C22×D4—472no—
366C22×D4—472no—
367C22×D4—472no—
368C22×D4—472no—
369C22×D4—472no—
370C22×D4—472no—
371C22×D4—941no—
372C22×D4—941no—
373C22×D4—941no—
374C22×D4—941no—
375C22×D4—941no—
376C22×D4—941no—
377C22×D4—941no—
378C22×D4—941no—
379C23×C4—21no—
380C23×C4—944no—
381C23×C4—941no—
382C23×C4C22×D4711yescharacteristic
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—472no—
389C2×M16—941no—
390C2×M16—941no—
391C2×M16—941no—
392C2×M16—941no—
393C2×M16—941no—
394C2×M16C4×D4712yescomplemented
395C22×C8—21no—
396C22×C8—941no—
397C4×D4—948no—
398C4×D4—948no—
399C4×D4—948no—
400C4×D4—948no—
401#117—21no—
402#117—21no—
403#117—472no—
404#117—472no—
405#117—941no—
406#117—941no—
407#117C4×D4712yes—
408#117D18812yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—472no—
415C2×C4⋊C4C4×D4712yescomplemented
416C2×C4⋊C4C22⋊D4712yes—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—948no—
419C2×C22⋊C4—948no—
420C2×C22⋊C4—944no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—948no—
423C2×C22⋊C4—1888no—
424C2×C42—21no—
425C2×C42—472no—
426C2×C42—941no—
427C2×C42—941no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432#100—948no—
433C24—474no—
434C24—474no—
435C24—474no—
436C24—942no—
437C24—942no—
438C24—942no—
439C24—942no—
440C24—1884no—
441C24—1884no—
442C24D4×D4711yescharacteristic
443C2×D4—474no—
444C2×D4—474no—
445C2×D4—474no—
446C2×D4—474no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—942no—
451C2×D4—942no—
452C2×D4—942no—
453C2×D4—942no—
454C2×D4—942no—
455C2×D4—942no—
456C2×D4—942no—
457C2×D4—942no—
458C2×D4—942no—
459C2×D4—942no—
460C2×D4—942no—
461C2×D4—942no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—942no—
465C2×D4—942no—
466C2×D4—942no—
467C2×D4—942no—
468C2×D4—942no—
469C2×D4—942no—
470C2×D4—942no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—9416no—
477C22×C4—944no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—3764no—
481C22×C4—3764no—
482C22×C4—1882no—
483C22×C4—1881no—
484C22×C4—1881no—
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4C2×C4×D4711yescharacteristic
487C22×C4C2×C4×D4711yescharacteristic
488C22×C4C2×D18811yescharacteristic
489C22×C4#1062535011yescharacteristic
490C22×C4#1062535011yescharacteristic
491C22×C4D4×D4711yescharacteristic
492C22×C4D4×D4711yescharacteristic
493C22×C4D4×D4711yescharacteristic
494C22×C4D4×D4711yescharacteristic
495C22×C4#1062535211yescharacteristic
496C22×C4Q8×D4711yescharacteristic
497C22×C4#1062535411yescharacteristic
498C22×C4C2×C22⋊D4711yescharacteristic
499M16—22no—
500M16—22no—
501M16—41no—
502M16—41no—
503M16—942no—
504M16—942no—
505M16—1881no—
506M16—1881no—
507C2×C8—22no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—942no—
511C2×C8—1881no—
512C2×C8—1881no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—474no—
519C4⋊C4—942no—
520C4⋊C4(C2×D47)⋊C414yescomplemented
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—9416no—
524C22⋊C4—942no—
525C22⋊C4—942no—
526C22⋊C4—18816no—
527C42—22no—
528C42—22no—
529C42—942no—
530C42—942no—
531C42—942no—
532C42—942no—
533C42—1881no—
534C42—1881no—
535C23—22no—
536C23—22no—
537C23—478no—
538C23—944no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—944no—
544C23—3762no—
545C23—3762no—
546C23—3764no—
547C23—3764no—
548C23—1884no—
549C23—1884no—
550C23—1888no—
551C23—1888no—
552C23—1884no—
553C23—1884no—
554C23C22⋊C4×D4711yescharacteristic
555C23C22⋊C4×D4711yescharacteristic
556C23C4⋊C4×D4711yescharacteristic
557C23#1309399211yescharacteristic
558C23#1309399411yescharacteristic
559C23#1309400611yescharacteristic
560C23#1309400911yescharacteristic
561D4—944no—
562D4—944no—
563D4—944no—
564D4—944no—
565D4—944no—
566D4—944no—
567D4—944no—
568D4—944no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—37616no—
579C2×C4—1882no—
580C2×C4—1884no—
581C2×C4—1884no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4C42×D4711yescharacteristic
586C2×C4C22⋊C4×D4711yescharacteristic
587C2×C4C22⋊C4×D4711yescharacteristic
588C2×C4C2×(C2×D47)⋊C411yescharacteristic
589C2×C4C4⋊C4×D4711yescharacteristic
590C2×C4C4⋊C4×D4711yescharacteristic
591C2×C4#1309398111yescharacteristic
592C2×C4#1309398211yescharacteristic
593C2×C4(C22⋊D47)⋊C411yescharacteristic
594C2×C4(C22⋊D47)⋊C411yescharacteristic
595C2×C4C4×C22⋊D4711yescharacteristic
596C2×C4C4×C22⋊D4711yescharacteristic
597C2×C4#1309398611yescharacteristic
598C2×C4#1309398611yescharacteristic
599C2×C4C4×D18811yescharacteristic
600C2×C4C4×D18811yescharacteristic
601C2×C4#1309399211yescharacteristic
602C2×C4#1309399411yescharacteristic
603C2×C4#1309399511yescharacteristic
604C2×C4#1309399711yescharacteristic
605C2×C4#1309399711yescharacteristic
606C2×C4#1309399911yescharacteristic
607C2×C4#1309400011yescharacteristic
608C2×C4#1309400211yescharacteristic
609C2×C4#1309400211yescharacteristic
610C2×C4#1309400411yescharacteristic
611C2×C4#1309400611yescharacteristic
612C2×C4#1309400911yescharacteristic
613C8—42no—
614C8—1882no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—948no—
624C22—3764no—
625C22—3764no—
626C22—3764no—
627C22—1888no—
628C22—1884no—
629C22—1884no—
630C22—1888no—
631C22—1888no—
632C22#95×D4711yescharacteristic
633C22#1568897811yescharacteristic
634C22#1568898511yescharacteristic
635C22#1568899411yescharacteristic
636C22#1568899711yescharacteristic
637C22#1568899911yescharacteristic
638C22#1568900111yescharacteristic
639C22#1568928912yes—
640C22#1568929912yes—
641C4—44no—
642C4—82no—
643C4—1884no—
644C4—1884no—
645C4#1568897011yescharacteristic
646C4C4×(C2×D47)⋊C411yescharacteristic
647C4#1568898011yescharacteristic
648C4#1568898211yescharacteristic
649C4#1568899411yescharacteristic
650C4#1568899711yescharacteristic
651C4#1568899911yescharacteristic
652C4#1568900111yescharacteristic
653C2—21no—
654C2—21no—
655C2—21no—
656C2—21no—
657C2—1888no—
658C2(C2×D47)⋊#9511yescharacteristic
659C2#2024334012yes—
660C2#2024359612yes—
661C2#2024377311yescharacteristic
662C2#2024378311yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1736C2
socleC23×C47#120
prime radicalC23×D188C22
cosocleC23×C4×C47C23
hypercenter#1736D47
hypocenterC47#10749
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20243107C2semidirect
5. C2×#15688884C2semidirect
6. C2×#15688882C2semidirect
7. C47×#1736C2semidirect
13. #15689597C4semidirect
22. C2×#13093940C4semidirect
24. #15689024C4semidirect
54. C2×#10625329C2×C4semidirect
77. C2×C4×Dic47C2×C4semidirect
86. C2×M16×C47C2×C4semidirect
97. C2×C4⋊C4×C47C2×C4semidirect
179. C4⋊C4×C47C22⋊C4semidirect
189. C4⋊Dic47C22⋊C4semidirect
326. #1736D47semidirect
363. C47#10749semidirect
394. C2×M16C4×D47semidirect
415. C2×C4⋊C4C4×D47semidirect
520. C4⋊C4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156895972
C22×#130939402
C2×#156890242
#202431072
C2×#156888842
C2×#156888822
C47×#17362
#1074947