Finite Group Databasebeta

← #26224003

Subgroups — #26224003

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689423C211yesmaximal, characteristic
2#20246129C211yesmaximal, characteristic, complemented
3#20244345C211yesmaximal, characteristic, complemented
4#20244146C211yesmaximal, characteristic, complemented
5#20243980C211yesmaximal, characteristic, complemented
6#20243973C211yesmaximal, characteristic, complemented
7#20243635C211yesmaximal, characteristic
8#20243635C211yesmaximal, characteristic
9#20243388C211yesmaximal, characteristic
10#20243286C211yesmaximal, characteristic, complemented
11C4×#13093957C211yesmaximal, characteristic, complemented
12#20243245C211yesmaximal, characteristic, complemented
13#20243213C211yesmaximal, characteristic
14C4×#13093945C211yesmaximal, characteristic
15C47×#2275C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17#15689423C418yescomplemented
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689021C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689020—21no—
33#15689020—21no—
34#15689010—21no—
35#15689010—21no—
36#15688993C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C4×C4⋊C4×C47C2211yescharacteristic
53C4×C4⋊C4×C47C2211yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C22⋊C4×C47—21no—
56C4×C22⋊C4×C47—21no—
57C4×C22⋊C4×C47—21no—
58C47×#376C2211yescharacteristic
59C2×#95×C47—21no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094016—22no—
65#13094016C2×C414yescomplemented
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78#13093957—22no—
79#13093957—22no—
80#13093957C2×C414yescomplemented
81#13093955C2×C414yescomplemented
82C4⋊(C47⋊C8)—22no—
83C4⋊(C47⋊C8)C2×C414yescomplemented
84#13093945—22no—
85#13093945—22no—
86#13093945C2×C414yes—
87#13093945C2×C414yes—
88#13093935—22no—
89C4×C47⋊C8—22no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C22×D4×C47C2311yescharacteristic
95C23×C4×C47—21no—
96C23×C4×C47—21no—
97C23×C4×C47—21no—
98C4×D4×C47—22no—
99C4×D4×C47—22no—
100C4×D4×C47—22no—
101C4×D4×C47—22no—
102C4×D4×C47C2×C414yes—
103C4×D4×C47Q814yes—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47—21no—
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115C2×C42×C47C2311yescharacteristic
116#95×C47—21no—
117#95×C47—21no—
118#95×C47—21no—
119#95×C47D411yescharacteristic
120#95×C47D411yescharacteristic
121#95×C47D411yescharacteristic
122#95×C47D411yescharacteristic
123C24×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47C22×C412yes—
128C2×D4×C47C2×Q812yes—
129C2×D4×C47#4112yes—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—81no—
137C22×C4×C47—81no—
138C22×C4×C47—81no—
139C22×C4×C47—81no—
140C22×C4×C47—81no—
141C22×C4×C47—81no—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47C22×C412yes—
157C4⋊C4×C47C2×Q812yes—
158C4⋊C4×C47#4112yes—
159C22⋊C4×C47—24no—
160C22⋊C4×C47—24no—
161C22⋊C4×C47—24no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—81no—
165C42×C47—81no—
166C42×C47—81no—
167C42×C47C22×C412yes—
168C42×C47C2×Q812yes—
169C42×C47#4112yes—
170C42×C47#4112yes—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—81no—
178C4⋊Dic47—81no—
179C4⋊Dic47C22×C412yes—
180C4⋊Dic47C22×C412yes—
181Dic47⋊C4—24no—
182Dic47⋊C4—24no—
183Dic47⋊C4—81no—
184Dic47⋊C4—81no—
185C4×Dic47—81no—
186C4×Dic47—81no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8C22×C412yes—
198C2×C47⋊C8C22×C412yes—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C22×D411yescharacteristic
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—22no—
210D4×C47C4×Q814yes—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—84no—
235C2×C4×C47—84no—
236C2×C4×C47—84no—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47C22⋊Q812yes—
240C2×C4×C47C22⋊Q812yes—
241C2×C4×C47C23×C411yescharacteristic
242C2×C4×C47C22×Q811yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C47⋊C8—22no—
260C47⋊C8—22no—
261C47⋊C8—41no—
262C47⋊C8—41no—
263#21620—471nomaximal, Sylow
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47C2×C4×D411yescharacteristic
272C22×C47C2×C22⋊Q811yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47#57611yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57811yescharacteristic
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—22no—
282C4×C47—42no—
283C4×C47—41no—
284C4×C47—41no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47C2×C4×Q811yescharacteristic
288C4×C47C4×#4111yescharacteristic
289C4×C47#52911yescharacteristic
290C4×C47#53311yescharacteristic
291Dic47—42no—
292Dic47—42no—
293Dic47—81no—
294Dic47—81no—
295C2×#478—471no—
296#2330—471no—
297#2275D4711yescharacteristic
298#1989—471no—
299#1947—471no—
300#1921—471no—
301#1920—471no—
302#1862—471no—
303#1862—471no—
304#1802—471no—
305#1776—471no—
306C4×#107—471no—
307#1768—471no—
308#1762—471no—
309C4×#102—471no—
310C2×C47—44no—
311C2×C47C4×C22⋊Q811yescharacteristic
312C2×C47#294611yescharacteristic
313C2×C47#295111yescharacteristic
314C2×C47#308111yescharacteristic
315C2×C47#308311yescharacteristic
316C2×C47#312011yescharacteristic
317C2×C47#312211yescharacteristic
318C2×#125—471no—
319C2×C4×D4C2×D4711yescharacteristic
320C2×C4×D4C2×D4711yescharacteristic
321C2×C4×D4C2×D4711yescharacteristic
322#478—478no—
323C2×#107—471no—
324C2×#107—471no—
325C2×#107—471no—
326C2×#106—471no—
327C2×C4⋊C8—471no—
328C2×C4⋊C8—471no—
329C2×#102—471no—
330C2×#102—471no—
331C2×#102—471no—
332C2×#102—471no—
333C2×#97—471no—
334C2×C4×C8—471no—
335#389—471no—
336#389—471no—
337#389—941no—
338#389—941no—
339#387—941no—
340#387—941no—
341#384—471no—
342C4×C4⋊C4—471no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4C2×D4711yescharacteristic
345C4×C4⋊C4C2×D4711yescharacteristic
346C4×C4⋊C4C2×D4711yescharacteristic
347C4×C22⋊C4—21no—
348C4×C22⋊C4—21no—
349C4×C22⋊C4—21no—
350#376C2×D4711yescharacteristic
351C2×#95—21no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#336—471no—
361#336—471no—
362C47#2162011yescharacteristic, complemented, Sylow
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—21no—
367C22×C8—471no—
368C22×C8—471no—
369C22×C8—471no—
370C22×C8—471no—
371#125—474no—
372#125—942no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4Dic9414yes—
378C4×D4C4×D4714yes—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42C22×D4711yescharacteristic
401C2×C42C22×D4711yescharacteristic
402C2×C42C22×D4711yescharacteristic
403#107—474no—
404#107—942no—
405#107—942no—
406#106—474no—
407C4⋊C8—474no—
408C4⋊C8—942no—
409#102—474no—
410#102—474no—
411#102—942no—
412#102—942no—
413#97—942no—
414C4×C8—942no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95C22⋊D4711yescharacteristic
423#95C22⋊D4711yescharacteristic
424#95C22⋊D4711yescharacteristic
425#95C22⋊D4711yescharacteristic
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×Dic9412yes—
431C2×D4C2×C4×D4712yes—
432C2×D4#1062535012yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456C2×C8—472no—
457C2×C8—472no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—472no—
472C4⋊C4—472no—
473C4⋊C4—944no—
474C4⋊C4—944no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—3761no—
482C4⋊C4—3761no—
483C4⋊C4—3761no—
484C4⋊C4—3761no—
485C4⋊C4C2×Dic9412yes—
486C4⋊C4C2×C4×D4712yes—
487C4⋊C4#1062535012yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—81no—
496C42—3761no—
497C42—3761no—
498C42C2×Dic9412yes—
499C42C2×C4×D4712yes—
500C42#1062535012yes—
501C42#1062535012yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D4711yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4C4×Dic9414yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4C4×C22⋊D4712yes—
553C2×C4C4×C22⋊D4712yes—
554C2×C4#1309400312yes—
555C2×C4#1309400312yes—
556C2×C4C22×C4×D4711yescharacteristic
557C2×C4C22×Dic9411yescharacteristic
558C2×C4C2×#1062535011yescharacteristic
559C2×C4C2×#1062535011yescharacteristic
560C2×C4C2×#1062535011yescharacteristic
561C2×C4C2×#1062535011yescharacteristic
562C8—942no—
563C8—942no—
564C8—1881no—
565C8—1881no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×C4×C22⋊D4711yescharacteristic
574C22C2×#1309400311yescharacteristic
575C22#1568963411yescharacteristic
576C22#1568992611yescharacteristic
577C22#1568992611yescharacteristic
578C22#1568993411yescharacteristic
579C22#1568993811yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—42no—
589C4—3761no—
590C4—3761no—
591C4—1882no—
592C4—1882no—
593C4C2×C4×Dic9411yescharacteristic
594C4C4×#1062535011yescharacteristic
595C4#1568965711yescharacteristic
596C4#1568968011yescharacteristic
597C2—44no—
598C2C4×#1309400311yescharacteristic
599C2#2025222011yescharacteristic
600C2#2025226111yescharacteristic
601C2#2025321911yescharacteristic
602C2#2025323911yescharacteristic
603C2#2025360211yescharacteristic
604C2#2025363111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2275C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#2275D47
hypocenterC47#21620
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #20246129C2semidirect
3. #20244345C2semidirect
4. #20244146C2semidirect
5. #20243980C2semidirect
6. #20243973C2semidirect
10. #20243286C2semidirect
11. C4×#13093957C2semidirect
12. #20243245C2semidirect
17. #15689423C4semidirect
65. #13094016C2×C4semidirect
80. #13093957C2×C4semidirect
81. #13093955C2×C4semidirect
83. C4⋊(C47⋊C8)C2×C4semidirect
362. C47#21620semidirect

Maximal subgroups

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

subgroupindex
C2×#156894232
#202461292
#202443452
#202441462
#202439802
#202439732
#202436352
#202433882
#202432862
C4×#130939572
#202432452
#202432132
C4×#130939452
C47×#22752
#2162047