Finite Group Databasebeta

← #26215748

Subgroups — #26215748

Order 12032.

1514 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic47C211yesmaximal, characteristic
2C22×Dic47⋊C8C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic
4#20245791C211yesmaximal, characteristic, complemented
5#20243401C212yesmaximal, complemented
6#20243396C212yesmaximal, complemented
7#20243391C212yesmaximal, complemented
8#20243386C212yesmaximal, complemented
9#20243105C211yesmaximal, characteristic, complemented
10#20243104C211yesmaximal, characteristic, complemented
11C2×C47×#416C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13#41×Dic47—24no—
14#41×Dic47C418yescomplemented
15C2×Q8×Dic47C2211yescharacteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×D4×Dic47—21no—
18C2×D4×Dic47—21no—
19C2×D4×Dic47C2211yescharacteristic
20C2×D4×Dic47C2211yescharacteristic
21C2×#13093983—21no—
22C2×#13093983C2211yescharacteristic
23C2×#13093983C2211yescharacteristic
24C22×C4×Dic47—21no—
25C22×C4×Dic47C2211yescharacteristic
26C2×Dic47⋊C8—24no—
27C2×Dic47⋊C8C2212yes—
28C2×Dic47⋊C8C2212yes—
29#15689119—24no—
30C2×#13093948C2212yes—
31C2×#13093945C2212yes—
32#15688992C2212yes—
33#15688988C2212yes—
34#15688981—21no—
35#15688981—21no—
36C4×Dic47⋊C4—21no—
37C4×Dic47⋊C4—21no—
38#15688886C2212yes—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C22×#41×C47C2211yescharacteristic
43C23×C8×C47C2211yescharacteristic
44C2×#117×C47C2211yescharacteristic
45C47×#416—24no—
46C2×#103×C47C2212yes—
47C2×#102×C47C2212yes—
48C23×Dic47—21no—
49C23×Dic47—21no—
50C23×Dic47D411yescharacteristic
51C23×Dic47D411yescharacteristic
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8C2312yes—
56Q8×Dic47—22no—
57Q8×Dic47—22no—
58Q8×Dic47C2×C414yes—
59Q8×Dic47C2×C414yes—
60D4×Dic47—28no—
61D4×Dic47—28no—
62D4×Dic47—22no—
63D4×Dic47—22no—
64D4×Dic47C2×C414yes—
65D4×Dic47C2×C414yes—
66#13093983—22no—
67#13093983—28no—
68#13093983—22no—
69#13093983—22no—
70#13093983C2×C414yes—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2312yes—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—22no—
77C2×Dic47⋊C4—22no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—28no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47C2×C414yes—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D414yes—
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93Dic47⋊C8—24no—
94Dic47⋊C8—24no—
95#13093948—24no—
96#13093945—24no—
97C47⋊#95—22no—
98C47⋊#95—22no—
99C2×#41×C47—22no—
100C2×#41×C47—22no—
101C2×#41×C47—22no—
102C2×#41×C47C2×C414yes—
103C2×#41×C47D414yes—
104C2×#41×C47Q814yes—
105C22×Q8×C47C2311yescharacteristic
106C22×D4×C47—21no—
107C22×D4×C47C2311yescharacteristic
108C23×C4×C47—21no—
109C23×C4×C47C2311yescharacteristic
110C22×C8×C47—22no—
111C22×C8×C47—22no—
112C22×C8×C47—22no—
113C22×C8×C47C2312yes—
114#117×C47—22no—
115#117×C47—22no—
116C2×C4⋊C4×C47C2312yes—
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—21no—
119#103×C47—24no—
120#102×C47—24no—
121C24×C47—21no—
122C24×C47C2×D411yescharacteristic
123#41×C47—28no—
124#41×C47—28no—
125#41×C47—28no—
126#41×C47C4⋊C4116yescomplemented
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C22×C412yes—
131C2×Q8×C47C2×D412yes—
132C2×Q8×C47C2×Q812yes—
133C2×D4×C47—24no—
134C2×D4×C47—24no—
135C2×D4×C47—24no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47C22×C412yes—
140C2×D4×C47C2×D412yes—
141C2×D4×C47C2×Q812yes—
142C22×C4×C47—21no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×Q812yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—24no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—22no—
161C22×Dic47—42no—
162C22×Dic47—44no—
163C22×Dic47—44no—
164C22×Dic47C22⋊C414yes—
165C22×Dic47C22⋊C414yes—
166C22×Dic47C2×D411yescharacteristic
167C22×Dic47C2×D411yescharacteristic
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D411yescharacteristic
170C2×C8×C47—22no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—44no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C22⋊C4×C47—42no—
177C42×C47—42no—
178C22⋊Dic47—24no—
179C22⋊Dic47—24no—
180C22⋊Dic47—24no—
181C22⋊Dic47—42no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—24no—
185C4⋊Dic47—24no—
186C4⋊Dic47—22no—
187C4⋊Dic47—22no—
188C4⋊Dic47C22×C412yes—
189C4⋊Dic47C22×C412yes—
190Dic47⋊C4—44no—
191Dic47⋊C4—44no—
192C4×Dic47—24no—
193C4×Dic47—24no—
194C4×Dic47—24no—
195C4×Dic47—42no—
196C4×Dic47C22⋊C414yes—
197C4×Dic47C22⋊C414yes—
198C4×Dic47C22×C412yes—
199C4×Dic47C22×C412yes—
200C4×Dic47C2×D412yes—
201C4×Dic47C2×D412yes—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C2×C47⋊C8—44no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—42no—
212C23×C47—42no—
213C23×C47C2×C22⋊C412yes—
214C23×C47#12112yes—
215C23×C47C22⋊Q812yes—
216C23×C47C22×D411yescharacteristic
217Q8×C47—22no—
218Q8×C47—22no—
219Q8×C47—22no—
220Q8×C47C2×C4⋊C414yes—
221D4×C47—216no—
222D4×C47—22no—
223D4×C47—22no—
224D4×C47—22no—
225D4×C47C2×C4⋊C414yes—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—216no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47C2×C22⋊C412yes—
234C2×C4×C47C2×C4⋊C414yes—
235C2×C4×C47#11714yes—
236C2×C4×C47#12114yes—
237C2×C4×C47#12112yes—
238C2×C4×C47C22⋊Q814yes—
239C2×C4×C47C22⋊Q812yes—
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×Q811yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—28no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—44no—
253C2×Dic47—48no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—48no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47C2×C22⋊C412yes—
261C2×Dic47C2×C22⋊C412yes—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C2×Dic47#13514yes—
265C8×C47—42no—
266C47⋊C8—42no—
267#20645—471nomaximal, Sylow
268C22×C47—24no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—24no—
272C22×C47—24no—
273C22×C47—44no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—44no—
277C22×C47#38014yes—
278C22×C47C22×C22⋊C411yescharacteristic
279C22×C47C2×#12111yescharacteristic
280C22×C47C2×C22⋊Q811yescharacteristic
281C22×C47C2×#13512yes—
282C22×C47#57512yes—
283C4×C47—24no—
284C4×C47—82no—
285C4×C47#38014yes—
286C4×C47C22×C4⋊C411yescharacteristic
287C4×C47C2×#11711yescharacteristic
288C4×C47C2×#12111yescharacteristic
289C4×C47C2×C22⋊Q811yescharacteristic
290Dic47—22no—
291Dic47—22no—
292Dic47—22no—
293Dic47—42no—
294Dic47—42no—
295Dic47—82no—
296Dic47#41614yescomplemented
297C2×C4×#41—471no—
298C22×C4⋊C8—471no—
299C2×#416—471no—
300C2×#416D4711yescharacteristic
301#2285—471no—
302#1805—472no—
303#1804—472no—
304#1803—472no—
305#1802—472no—
306#1735—471no—
307#1735—471no—
308C2×C47—24no—
309C2×C47—44no—
310C2×C47C2×#38011yescharacteristic
311C2×C47C2×#41611yescharacteristic
312C2×C47#289611yescharacteristic
313C2×C47#303512yes—
314C2×C47#307612yes—
315C22×#41C2×D4711yescharacteristic
316C23×C8—471no—
317C23×C8C2×D4711yescharacteristic
318C4×#41—478no—
319C4×#41—944no—
320C2×C4×Q8—471no—
321C2×C4×Q8—471no—
322C2×C4×D4—471no—
323C2×C4×D4—471no—
324C2×C4×D4—941no—
325C2×C4×D4—941no—
326C2×#117—471no—
327C2×#117—471no—
328C2×#117—941no—
329C2×#117C2×D4711yescharacteristic
330C22×C42—471no—
331C22×C42—941no—
332C2×C4⋊C8—472no—
333C2×C4⋊C8—472no—
334C2×C4⋊C8—944no—
335#416—24no—
336#416—944no—
337C2×#103—472no—
338C2×#103C2×D4712yes—
339C2×#102—472no—
340C2×#102C2×D4712yes—
341#384—472no—
342#382—472no—
343#380—941no—
344#380—941no—
345C4×C4⋊C4—941no—
346C4×C4⋊C4—941no—
347#340—472no—
348#340—472no—
349#340—472no—
350#340—472no—
351C47#2064511yescharacteristic, complemented, Sylow
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41Dic9414yes—
356C2×#41C4×D4714yes—
357C2×#41C22⋊D4714yes—
358C22×Q8C22×D4711yescharacteristic
359C22×D4—21no—
360C22×D4C22×D4711yescharacteristic
361C23×C4—21no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—941no—
365C23×C4—941no—
366C23×C4C22×D4711yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—22no—
370C22×C8—472no—
371C22×C8—942no—
372C22×C8—942no—
373C22×C8—942no—
374C22×C8C22×D4712yes—
375C4×Q8—474no—
376C4×Q8—474no—
377C4×Q8—942no—
378C4×Q8—942no—
379C4×D4—474no—
380C4×D4—474no—
381C4×D4—942no—
382C4×D4—942no—
383C4×D4—948no—
384C4×D4—948no—
385#117—22no—
386#117—22no—
387#117—474no—
388#117—948no—
389#117—942no—
390#117—942no—
391#117—942no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—472no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—942no—
399C2×C4⋊C4C22×D4712yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C42—21no—
406C2×C42—474no—
407C2×C42—474no—
408C2×C42—471no—
409C2×C42—471no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—948no—
413C2×C42—941no—
414C2×C42—941no—
415C2×C42—941no—
416C2×C42—941no—
417C4⋊C8—944no—
418C4⋊C8—944no—
419#103—24no—
420#103—944no—
421#102—24no—
422#102—944no—
423#95—942no—
424#95—942no—
425C24—21no—
426C24D4×D4711yescharacteristic
427#41—28no—
428#41—28no—
429#41—28no—
430#41Dic47⋊C4116yescomplemented
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic9412yes—
435C2×Q8C2×C4×D4712yes—
436C2×Q8C2×C22⋊D4712yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4C2×Dic9412yes—
444C2×D4C2×C4×D4712yes—
445C2×D4C2×C22⋊D4712yes—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—471no—
452C22×C4—474no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—474no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—944no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—942no—
463C22×C4—1884no—
464C22×C4—1882no—
465C22×C4—1884no—
466C22×C4C2×Dic9412yes—
467C22×C4C2×C4×D4712yes—
468C22×C4D4×D4712yes—
469C22×C4D4×D4711yescharacteristic
470C22×C4#1062535212yes—
471C22×C4#1062535212yes—
472C22×C4C2×C22⋊D4712yes—
473C22×C4C23×D4711yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—942no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8—1884no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—472no—
485C4⋊C4—472no—
486C4⋊C4—944no—
487C4⋊C4—944no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—942no—
491C4⋊C4—942no—
492C4⋊C4—1884no—
493C4⋊C4—1884no—
494C22⋊C4—42no—
495C22⋊C4—944no—
496C22⋊C4—944no—
497C22⋊C4—944no—
498C22⋊C4—1882no—
499C42—42no—
500C42—474no—
501C42—474no—
502C42—472no—
503C42—472no—
504C42—472no—
505C42—472no—
506C42—944no—
507C42—944no—
508C42—944no—
509C42—1882no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23C22⋊C4×D4712yes—
518C23#1309399612yes—
519C23#1309400112yes—
520C23C2×D4×D4711yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8C2×Dic47⋊C414yes—
525D4—216no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4C2×Dic47⋊C414yes—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—216no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—472no—
538C2×C4—474no—
539C2×C4—472no—
540C2×C4—472no—
541C2×C4—472no—
542C2×C4—948no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—1888no—
549C2×C4—1884no—
550C2×C4—1888no—
551C2×C4—1882no—
552C2×C4—1882no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4C22⋊C4×D4712yes—
557C2×C4C2×Dic47⋊C414yes—
558C2×C4#1309398014yes—
559C2×C4#1309399612yes—
560C2×C4#1309399614yes—
561C2×C4#1309400112yes—
562C2×C4#1309400114yes—
563C2×C4C22×C4×D4711yescharacteristic
564C2×C4C2×D4×D4711yescharacteristic
565C2×C4C22×C22⋊D4711yescharacteristic
566C2×C4C22×Dic9411yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4C2×#1062535211yescharacteristic
569C8—42no—
570C8—1882no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22#1568898114yes—
581C22C2×C22⋊C4×D4711yescharacteristic
582C22C2×#1309399611yescharacteristic
583C22C2×#1309400111yescharacteristic
584C22#135×D4712yes—
585C22#1568992012yes—
586C4—24no—
587C4—82no—
588C4—474no—
589C4—942no—
590C4—942no—
591C4—942no—
592C4—3762no—
593C4—1882no—
594C4—1882no—
595C4#1568898114yes—
596C4C22×Dic47⋊C411yescharacteristic
597C4C2×#1309398011yescharacteristic
598C4C2×#1309399611yescharacteristic
599C4C2×#1309400111yescharacteristic
600C2—24no—
601C2—44no—
602C2C2×#1568898111yescharacteristic
603C2#416×D4711yescharacteristic
604C2#2025190311yescharacteristic
605C2#2025283912yes—
606C2#2025317212yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#416C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#416D47
hypocenterC47#20645
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×Dic47⋊C8C2semidirect
4. #20245791C2semidirect
5. #20243401C2semidirect
6. #20243396C2semidirect
7. #20243391C2semidirect
8. #20243386C2semidirect
9. #20243105C2semidirect
10. #20243104C2semidirect
14. #41×Dic47C4semidirect
126. #41×C47C4⋊C4semidirect
296. Dic47#416semidirect
351. C47#20645semidirect
430. #41Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic472
C22×Dic47⋊C82
C2×#156891192
#202457912
#202434012
#202433962
#202433912
#202433862
#202431052
#202431042
C2×C47×#4162
#2064547