Finite Group Databasebeta

← #26123696

Subgroups — #26123696

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093959C211yesmaximal, characteristic, complemented
2C2×#15689119C211yesmaximal, characteristic
3C2×#15689090C211yesmaximal, characteristic
4C2×#15688883C211yesmaximal, characteristic, complemented
5#20243086C211yesmaximal, characteristic, complemented
6C2×#15688878C211yesmaximal, characteristic, complemented
7C22×#104×C47C211yesmaximal, characteristic
8C22×#10625329C2211yescharacteristic
9C23×C47⋊C8C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12C2×#13093959—22no—
13C2×#13093959—22no—
14C2×#13093959—22no—
15#15689119—22no—
16#15689119—22no—
17C2×#13093948C411yescharacteristic, complemented
18C2×#13093948C411yescharacteristic, complemented
19C2×#13093945C411yescharacteristic, complemented
20C2×#13093945C411yescharacteristic, complemented
21#15689090—22no—
22#15689090—22no—
23C2×C22⋊(C47⋊C8)—22no—
24#15688883—22no—
25#15688878—22no—
26#15688877—22no—
27C22×#41×C47C2211yescharacteristic
28C22×M16×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C2×#104×C47—22no—
31C2×#104×C47—22no—
32C2×#104×C47—22no—
33C2×#10625329—21no—
34C2×#10625329—21no—
35C2×#10625329—21no—
36C2×#10625329—21no—
37C2×#10625329—21no—
38C2×#10625329D412yes—
39C22×C47⋊C8—21no—
40C22×C47⋊C8—21no—
41C22×C47⋊C8—21no—
42C22×C47⋊C8—21no—
43C22×C47⋊C8—21no—
44C22×C47⋊C8—21no—
45C22×C47⋊C8—21no—
46C22×C47⋊C8—21no—
47C22×C47⋊C8—21no—
48C22×C47⋊C8—21no—
49C22×C47⋊C8—21no—
50C22×C47⋊C8C2×C411yescharacteristic
51C22×C47⋊C8C2×C411yescharacteristic
52C22×C47⋊C8D412yes—
53C22×C47⋊C8D412yes—
54C22×C47⋊C8D412yes—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59C2×C4⋊Dic47C2×C411yescharacteristic
60C2×C4⋊Dic47C2×C411yescharacteristic
61C2×C22⋊Dic47—21no—
62C2×C4×Dic47—21no—
63#13093959—28no—
64#13093948—22no—
65#13093948—22no—
66#13093945—22no—
67#13093945—22no—
68C22⋊(C47⋊C8)—28no—
69C2×#41×C47—21no—
70C2×#41×C47—21no—
71C2×#41×C47—21no—
72C2×#41×C47—21no—
73C2×#41×C47—21no—
74C2×#41×C47—21no—
75C2×#41×C47—21no—
76C2×#41×C47—21no—
77C2×#41×C47—21no—
78C2×#41×C47—21no—
79C2×#41×C47D412yes—
80C2×#41×C47Q812yes—
81C22×Q8×C47C2×C411yescharacteristic
82C22×D4×C47—21no—
83C22×D4×C47C2×C411yescharacteristic
84C23×C4×C47—21no—
85C23×C4×C47—21no—
86C23×C4×C47C2311yescharacteristic
87C2×M16×C47—21no—
88C2×M16×C47—21no—
89C2×M16×C47—21no—
90C2×M16×C47—21no—
91C2×M16×C47—21no—
92C2×M16×C47D412yes—
93C22×C8×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47—21no—
98C2×C42×C47—21no—
99C2×C42×C47D412yes—
100#104×C47—28no—
101C24×C47—21no—
102C24×C47#4111yescharacteristic
103#41×C47—22no—
104#41×C47—22no—
105#41×C47—22no—
106#41×C47—22no—
107#41×C47—22no—
108#41×C47—22no—
109#41×C47—22no—
110#41×C47—22no—
111#41×C47—41no—
112#41×C47—41no—
113#41×C47—41no—
114#41×C47—41no—
115#41×C47—41no—
116#41×C47—41no—
117#41×C47—41no—
118#41×C47—41no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47C4⋊C412yes—
125C2×D4×C47—22no—
126C2×D4×C47—22no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—41no—
133C2×D4×C47—41no—
134C2×D4×C47—41no—
135C2×D4×C47—41no—
136C2×D4×C47C4⋊C412yes—
137C22×C4×C47—24no—
138C22×C4×C47—21no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—42no—
143C22×C4×C47—41no—
144C22×C4×C47—41no—
145C22×C4×C47—41no—
146C22×C4×C47—41no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×Q811yescharacteristic
158C22×C4×C47#4111yescharacteristic
159C22×C4×C47#4111yescharacteristic
160C22×C4×C47#4111yescharacteristic
161C22×C4×C47#4111yescharacteristic
162C22×C4×C47#4111yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165M16×C47—22no—
166M16×C47—22no—
167M16×C47—41no—
168M16×C47—41no—
169C2×C8×C47—22no—
170C2×C8×C47—41no—
171C2×C8×C47—41no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—41no—
175C42×C47—41no—
176C22⋊Dic47—41no—
177C22⋊Dic47—41no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4×Dic47—41no—
183C4×Dic47—41no—
184#10625329—22no—
185#10625329—22no—
186#10625329—41no—
187#10625329—41no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—41no—
201C2×C47⋊C8—41no—
202C2×C47⋊C8—41no—
203C2×C47⋊C8—41no—
204C2×C47⋊C8—41no—
205C2×C47⋊C8—41no—
206C2×C47⋊C8C22⋊C412yes—
207C2×C47⋊C8C22⋊C412yes—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47C4×D411yescharacteristic
216C23×C47#12111yescharacteristic
217C23×C47C22⋊Q811yescharacteristic
218C23×C47#12311yescharacteristic
219C23×C47#12411yescharacteristic
220C23×C47#12411yescharacteristic
221C23×C47#12611yescharacteristic
222Q8×C47—22no—
223Q8×C47—22no—
224Q8×C47—41no—
225Q8×C47—41no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47—42no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—42no—
232D4×C47—41no—
233D4×C47—41no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—44no—
237C2×C4×C47—44no—
238C2×C4×C47—42no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C2×C4⋊C411yescharacteristic
248C2×C4×C47#11711yescharacteristic
249C2×C4×C47C4×D411yescharacteristic
250C2×C4×C47C4×D411yescharacteristic
251C2×C4×C47C4×D411yescharacteristic
252C2×C4×C47#12011yescharacteristic
253C2×C4×C47#12011yescharacteristic
254C2×C4×C47#12011yescharacteristic
255C2×C4×C47#12111yescharacteristic
256C2×C4×C47#12111yescharacteristic
257C2×C4×C47#12111yescharacteristic
258C2×C4×C47#12111yescharacteristic
259C2×C4×C47#12111yescharacteristic
260C2×C4×C47#12111yescharacteristic
261C2×C4×C47#12111yescharacteristic
262C2×C4×C47C22⋊Q811yescharacteristic
263C2×C4×C47C22⋊Q811yescharacteristic
264C2×C4×C47C22⋊Q811yescharacteristic
265C2×C4×C47C22⋊Q811yescharacteristic
266C2×C4×C47C22⋊Q811yescharacteristic
267C2×C4×C47#12311yescharacteristic
268C2×C4×C47#12311yescharacteristic
269C2×C4×C47#12311yescharacteristic
270C2×C4×C47#12311yescharacteristic
271C2×C4×C47#12411yescharacteristic
272C2×C4×C47#12611yescharacteristic
273C2×C4×C47#12711yescharacteristic
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C8×C47—42no—
279C47⋊C8—22no—
280C47⋊C8—22no—
281C47⋊C8—42no—
282C47⋊C8—42no—
283#11125—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×C47—44no—
293C22×C47—42no—
294C22×C47—82no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47#38511yescharacteristic
298C22×C47#38811yescharacteristic
299C22×C47#39011yescharacteristic
300C22×C47#39311yescharacteristic
301C22×C47#39411yescharacteristic
302C22×C47#39611yescharacteristic
303C22×C47#39711yescharacteristic
304C22×C47#44312yes—
305C22×C47#44412yes—
306C4×C47—44no—
307C4×C47—42no—
308C4×C47#38011yescharacteristic
309C4×C47#38511yescharacteristic
310C4×C47#38611yescharacteristic
311C4×C47#38811yescharacteristic
312C4×C47#39211yescharacteristic
313C4×C47#39411yescharacteristic
314C4×C47#39611yescharacteristic
315C4×C47#39911yescharacteristic
316Dic47—81no—
317Dic47—81no—
318C22×#108—471no—
319C22×#104D4711yescharacteristic
320C2×#416—471no—
321C2×#408—471no—
322C2×#339—471no—
323#1729—471no—
324C2×#336—471no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47—82no—
330C2×C47#143911yescharacteristic
331C2×C47#186612yes—
332C2×C47#192912yes—
333C2×C47#197411yescharacteristic
334C2×C47#197511yescharacteristic
335C22×#41C2×D4711yescharacteristic
336C22×M16—471no—
337C22×M16C2×D4711yescharacteristic
338C23×C8—471no—
339C23×C8—471no—
340C2×#117—471no—
341C22×C42C2×D4711yescharacteristic
342C2×#108—942no—
343C2×#108—942no—
344C2×#108—942no—
345C2×#104—22no—
346C2×#104—22no—
347C2×#104—22no—
348#416—942no—
349#416—942no—
350C2×#103—471no—
351C2×#103—471no—
352C2×#102—471no—
353C2×#102—471no—
354#408—942no—
355#408—942no—
356C2×C22⋊C8—942no—
357#339—942no—
358#336—942no—
359#336—942no—
360C47#1112511yescharacteristic, complemented, Sylow
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41Dic9412yes—
372C2×#41C22⋊D4712yes—
373C22×Q8C4×D4711yescharacteristic
374C22×D4—21no—
375C22×D4C4×D4711yescharacteristic
376C23×C4—21no—
377C23×C4—21no—
378C23×C4C22×D4711yescharacteristic
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—472no—
385C2×M16—941no—
386C2×M16—941no—
387C2×M16—941no—
388C2×M16—941no—
389C2×M16—941no—
390C2×M16C22⋊D4712yes—
391C22×C8—21no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—472no—
395C22×C8—472no—
396C22×C8—472no—
397C22×C8—941no—
398C22×C8—941no—
399C22×C8—941no—
400C22×C8—941no—
401C22×C8—941no—
402C22×C8—941no—
403C22×C8—941no—
404C22×C8—941no—
405C22×C8—941no—
406C22×C8—941no—
407C22×C8—941no—
408#117—941no—
409#117—941no—
410#117—941no—
411#117—941no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C22⋊C4—941no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—941no—
421C2×C42C22⋊D4712yes—
422#108—948no—
423#104—28no—
424#103—942no—
425#103—942no—
426#102—942no—
427#102—942no—
428C22⋊C8—948no—
429C24—21no—
430C24#1062535211yescharacteristic
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—22no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446#41—41no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8Dic47⋊C412yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4Dic47⋊C412yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4C2×Dic9411yescharacteristic
480C22×C4C2×C4×D4711yescharacteristic
481C22×C4#1062535011yescharacteristic
482C22×C4#1062535011yescharacteristic
483C22×C4D4×D4711yescharacteristic
484C22×C4D4×D4711yescharacteristic
485C22×C4D4×D4711yescharacteristic
486C22×C4D4×D4711yescharacteristic
487C22×C4#1062535211yescharacteristic
488C22×C4#1062535211yescharacteristic
489C22×C4#1062535211yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493M16—22no—
494M16—22no—
495M16—41no—
496M16—41no—
497M16—942no—
498M16—942no—
499M16—1881no—
500M16—1881no—
501C2×C8—22no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—472no—
505C2×C8—472no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—942no—
511C2×C8—942no—
512C2×C8—942no—
513C2×C8—942no—
514C2×C8—942no—
515C2×C8—942no—
516C2×C8—942no—
517C2×C8—942no—
518C2×C8—1881no—
519C2×C8—1881no—
520C2×C8—1881no—
521C2×C8—1881no—
522C2×C8—1881no—
523C2×C8—1881no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C22⋊C4—1881no—
529C22⋊C4—1881no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—41no—
534C42—1881no—
535C42—1881no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23(C22⋊D47)⋊C411yescharacteristic
544C23#1309399611yescharacteristic
545C23#1309400111yescharacteristic
546C23#1309400511yescharacteristic
547C23#1309401011yescharacteristic
548C23#1309401111yescharacteristic
549C23#1309401711yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—41no—
553Q8—41no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—41no—
561D4—41no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4C22⋊C4×D4711yescharacteristic
579C2×C4C2×Dic47⋊C411yescharacteristic
580C2×C4#1309398011yescharacteristic
581C2×C4(C22⋊D47)⋊C411yescharacteristic
582C2×C4C4×C22⋊D4711yescharacteristic
583C2×C4C4×C22⋊D4711yescharacteristic
584C2×C4#1309399211yescharacteristic
585C2×C4#1309399211yescharacteristic
586C2×C4#1309399311yescharacteristic
587C2×C4#1309399511yescharacteristic
588C2×C4#1309399511yescharacteristic
589C2×C4#1309399611yescharacteristic
590C2×C4#1309399611yescharacteristic
591C2×C4#1309399611yescharacteristic
592C2×C4#1309399711yescharacteristic
593C2×C4#1309399711yescharacteristic
594C2×C4#1309400111yescharacteristic
595C2×C4#1309400111yescharacteristic
596C2×C4#1309400111yescharacteristic
597C2×C4#1309400311yescharacteristic
598C2×C4#1309400311yescharacteristic
599C2×C4#1309400511yescharacteristic
600C2×C4#1309400611yescharacteristic
601C2×C4#1309400611yescharacteristic
602C2×C4#1309400711yescharacteristic
603C2×C4#1309401011yescharacteristic
604C2×C4#1309401711yescharacteristic
605C2×C4#1309402011yescharacteristic
606C8—42no—
607C8—942no—
608C8—942no—
609C8—1882no—
610C8—1882no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—82no—
624C22#1568899511yescharacteristic
625C22#1568901711yescharacteristic
626C22#1568902311yescharacteristic
627C22#1568903511yescharacteristic
628C22#1568903711yescharacteristic
629C22#1568904611yescharacteristic
630C22#1568905111yescharacteristic
631C22#1568923712yes—
632C22#1568924212yes—
633C4—44no—
634C4—42no—
635C4—3761no—
636C4—3761no—
637C4#1568898111yescharacteristic
638C4#1568899611yescharacteristic
639C4#1568899911yescharacteristic
640C4#1568901111yescharacteristic
641C4#1568903111yescharacteristic
642C4#1568903811yescharacteristic
643C4#1568904611yescharacteristic
644C4#1568905911yescharacteristic
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—82no—
650C2#2024163111yescharacteristic
651C2#2024366112yes—
652C2#2024403912yes—
653C2#2024427311yescharacteristic
654C2#2024427811yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC22×#104×C47C2
socleC23×C47C4×D4
prime radicalC22×#41×C47C22
cosocleC23×C4×C47C23
hypercenterC22×#104D47
hypocenterC47#11125
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
1. C22×#13093959C2semidirect
4. C2×#15688883C2semidirect
5. #20243086C2semidirect
6. C2×#15688878C2semidirect
17. C2×#13093948C4semidirect
18. C2×#13093948C4semidirect
19. C2×#13093945C4semidirect
20. C2×#13093945C4semidirect
360. C47#11125semidirect

Maximal subgroups

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

subgroupindex
C22×#130939592
C2×#156891192
C2×#156890902
C2×#156888832
#202430862
C2×#156888782
C22×#104×C472
#1112547