Finite Group Databasebeta

← #26289019

Subgroups — #26289019

Order 12032.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247821C211yesmaximal, characteristic
2#20244796C211yesmaximal, characteristic, complemented
3#20244783C211yesmaximal, characteristic, complemented
4#20244588C211yesmaximal, characteristic
5#20244588C211yesmaximal, characteristic
6#20244146C211yesmaximal, characteristic, complemented
7#20244118C211yesmaximal, characteristic, complemented
8#20243996C211yesmaximal, characteristic
9#20243980C211yesmaximal, characteristic, complemented
10#20243977C211yesmaximal, characteristic, complemented
11#20243642C211yesmaximal, characteristic, complemented
12#20243576C211yesmaximal, characteristic
13#20243478C211yesmaximal, characteristic
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#2360C211yesmaximal, characteristic
16C2×#13093957C2211yescharacteristic
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689065—21no—
29#15689046—21no—
30#15689021C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689016—21no—
33#15689010—21no—
34#15688993C2211yescharacteristic
35#15688983C2211yescharacteristic
36#15688983C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C22⋊Q8×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C22×C4⋊C4×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#389—21no—
53C47×#389C2211yescharacteristic
54C47×#387—21no—
55C47×#385—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47—21no—
58C4×C22⋊C4×C47C2211yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×C4×Dic47—21no—
74#13093957—22no—
75#13093957—22no—
76#13093957—22no—
77#13093957—41no—
78C4⋊(C47⋊C8)—22no—
79#13093948—22no—
80#13093948—22no—
81#13093948—22no—
82#13093948—41no—
83C22⋊(C47⋊C8)—22no—
84C22⋊(C47⋊C8)—41no—
85C4×C47⋊C8—41no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C22×Q8×C47C2311yescharacteristic
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94C22⋊Q8×C47—22no—
95C22⋊Q8×C47—22no—
96C4×Q8×C47—22no—
97C4×Q8×C47—41no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—41no—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47D411yescharacteristic
109C2×C22⋊C4×C47D411yescharacteristic
110C2×C42×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114#95×C47—21no—
115#95×C47—21no—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47C2×D411yescharacteristic
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—42no—
138C22×C4×C47—42no—
139C22×C4×C47—81no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C4⋊C4×C47—24no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—42no—
156C4⋊C4×C47—42no—
157C4⋊C4×C47—81no—
158C4⋊C4×C47#4112yes—
159C4⋊C4×C47#4112yes—
160C22⋊C4×C47—24no—
161C22⋊C4×C47—41no—
162C22⋊C4×C47Q1614yes—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—42no—
168C42×C47—81no—
169C22⋊Dic47—81no—
170C22⋊Dic47—81no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—81no—
180C4⋊Dic47—81no—
181Dic47⋊C4—42no—
182Dic47⋊C4—81no—
183C4×Dic47—81no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
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#4112yes—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C2×Q1612yes—
203C23×C47C22×D411yescharacteristic
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—41no—
207Q8×C47—41no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—22no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
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—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—84no—
238C2×C4×C47C2×Q1612yes—
239C2×C4×C47C2×Q1612yes—
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47H(5,𝔽2)11yescharacteristic
245C2×C4×C472−1+411yescharacteristic
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
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—
258C47⋊C8—22no—
259C47⋊C8—41no—
260C47⋊C8—41no—
261C47⋊C8—41no—
262#27036—471nomaximal, Sylow
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47#52511yescharacteristic
271C22×C47#54611yescharacteristic
272C22×C47#54711yescharacteristic
273C22×C47C22×Q1611yescharacteristic
274C22×C47#57511yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—41no—
280C4×C47—42no—
281C4×C47—41no—
282C4×C47—42no—
283C4×C47—41no—
284C4×C47—42no—
285C4×C47—81no—
286C4×C47#52911yescharacteristic
287C4×C47#53911yescharacteristic
288C4×C47#55311yescharacteristic
289C4×C47#55611yescharacteristic
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293Dic47—81no—
294#2497—471no—
295#2360D4711yescharacteristic
296#2072—471no—
297#2068—471no—
298#2036—471no—
299#2036—471no—
300#1947—471no—
301#1942—471no—
302#1923—471no—
303#1921—471no—
304#1921—471no—
305#1863—471no—
306#1852—471no—
307#1824—471no—
308C4×#103—471no—
309C2×C47—42no—
310C2×C47#246611yescharacteristic
311C2×C47#310411yescharacteristic
312C2×C47#311911yescharacteristic
313C2×C47#329911yescharacteristic
314C2×C47#331911yescharacteristic
315C2×C47#333811yescharacteristic
316C2×C47#335111yescharacteristic
317C2×C22⋊Q8C2×D4711yescharacteristic
318C2×C4×Q8C2×D4711yescharacteristic
319C22×C4⋊C4C2×D4711yescharacteristic
320C2×#107—471no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#107—471no—
324C2×C4⋊C8—471no—
325C2×#103—471no—
326C2×#103—471no—
327C2×#103—471no—
328C2×#103—471no—
329C2×C22⋊C8—471no—
330C2×C22⋊C8—471no—
331C2×C4×C8—471no—
332#400—941no—
333#396—941no—
334#391C2×D4711yescharacteristic
335#389—21no—
336#389—471no—
337#389—471no—
338#389C2×D4711yescharacteristic
339#388—941no—
340#387—21no—
341#387—941no—
342#385—21no—
343#384—471no—
344#380—471no—
345#380—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4C2×D4711yescharacteristic
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#336—471no—
359#336—471no—
360C47#2703611yescharacteristic, complemented, Sylow
361C22×Q8C22×D4711yescharacteristic
362C23×C4—21no—
363C23×C4C22×D4711yescharacteristic
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—471no—
367C22×C8—471no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C4×Q8—22no—
371C4×Q8—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4C22⋊D4711yescharacteristic
393C2×C22⋊C4C22⋊D4711yescharacteristic
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—941no—
397C2×C42C22×D4711yescharacteristic
398C2×C42C22×D4711yescharacteristic
399#107—942no—
400#107—942no—
401#107—942no—
402#107—1881no—
403C4⋊C8—942no—
404#103—942no—
405#103—942no—
406#103—942no—
407#103—1881no—
408C22⋊C8—942no—
409C22⋊C8—1881no—
410C4×C8—1881no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95C22⋊D4711yescharacteristic
421#95C22⋊D4711yescharacteristic
422C24C2×C22⋊D4711yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#1062535012yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4C2×C22⋊D4711yescharacteristic
447C22×C4C2×C22⋊D4711yescharacteristic
448C22×C4C2×C22⋊D4711yescharacteristic
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C23×D4711yescharacteristic
452C2×C8—472no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—3761no—
481C4⋊C4—3761no—
482C4⋊C4—3761no—
483C4⋊C4—1882no—
484C4⋊C4#1062535012yes—
485C4⋊C4#1062535212yes—
486C22⋊C4—24no—
487C22⋊C4—41no—
488C22⋊C4—3761no—
489C22⋊C4—3761no—
490C22⋊C4#1062533714yes—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—81no—
497C42—3761no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×#1062533712yes—
505C23C22×C22⋊D4711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—1881no—
541C2×C4—1881no—
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×C4C2×#1062533712yes—
553C2×C4C2×#1062533712yes—
554C2×C4#41×D4711yescharacteristic
555C2×C4C2×#1062535211yescharacteristic
556C2×C4C2×#1062535011yescharacteristic
557C2×C4C2×#1062535011yescharacteristic
558C2×C4#1309406911yescharacteristic
559C2×C4#1309407111yescharacteristic
560C8—942no—
561C8—1881no—
562C8—1881no—
563C8—1881no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1568963411yescharacteristic
572C22#1568976711yescharacteristic
573C22#1568977211yescharacteristic
574C22C22×#1062533711yescharacteristic
575C22#1568992111yescharacteristic
576C22#1568992611yescharacteristic
577C22#1568993111yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—3761no—
588C4—3761no—
589C4—3761no—
590C4—3761no—
591C4#1568965311yescharacteristic
592C4#1568971511yescharacteristic
593C4#1568982311yescharacteristic
594C4#1568984311yescharacteristic
595C2—42no—
596C2#2024741811yescharacteristic
597C2#2025342711yescharacteristic
598C2#2025358711yescharacteristic
599C2#2025526411yescharacteristic
600C2#2025554411yescharacteristic
601C2#2025577611yescharacteristic
602C2#2025594811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2360C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2360D47
hypocenterC47#27036
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20244796C2semidirect
3. #20244783C2semidirect
6. #20244146C2semidirect
7. #20244118C2semidirect
9. #20243980C2semidirect
10. #20243977C2semidirect
11. #20243642C2semidirect
14. C4×#13093948C2semidirect
360. C47#27036semidirect

Maximal subgroups

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

subgroupindex
#202478212
#202447962
#202447832
#202445882
#202441462
#202441182
#202439962
#202439802
#202439772
#202436422
#202435762
#202434782
C4×#130939482
C47×#23602
#2703647