Finite Group Databasebeta

← #26268351

Subgroups — #26268351

Order 12032.

782 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689420C211yesmaximal, characteristic
2C2×#15689254C211yesmaximal, characteristic, complemented
3#20248306C211yesmaximal, characteristic
4#20244889C211yesmaximal, characteristic, complemented
5#20244879C211yesmaximal, characteristic, complemented
6#20244759C211yesmaximal, characteristic
7#20244759C211yesmaximal, characteristic
8#20244358C211yesmaximal, characteristic, complemented
9#20244151C211yesmaximal, characteristic, complemented
10#20244128C211yesmaximal, characteristic, complemented
11#20244113C211yesmaximal, characteristic, complemented
12#20244084C211yesmaximal, characteristic
13#20244070C211yesmaximal, characteristic
14#20243216C211yesmaximal, characteristic
15C47×#2584C211yesmaximal, characteristic, complemented
16C2×#13094000—21no—
17C2×C4×D188C2211yescharacteristic
18#15689420—24no—
19#15689254—24no—
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093944C2211yescharacteristic
30C2×#13093944C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689045—21no—
35#15689045—21no—
36#15689025C2211yescharacteristic
37#15689025C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15688993C2211yescharacteristic
41#15688980—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50C2×#125×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C47×#398—21no—
53C47×#395—21no—
54C47×#389—21no—
55C47×#389—21no—
56C47×#389C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#389C2211yescharacteristic
59C47×#389C2211yescharacteristic
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094000—44no—
68C4×D188—22no—
69C4×D188—22no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82#13093957—22no—
83#13093955—22no—
84#13093955—22no—
85#13093955—22no—
86C4⋊(C47⋊C8)—22no—
87C4⋊(C47⋊C8)—22no—
88C4⋊(C47⋊C8)D414yes—
89#13093944—22no—
90#13093944—22no—
91#13093944—22no—
92#13093944Q814yescomplemented
93#13093935—22no—
94#13093935—22no—
95C4×C47⋊C8—22no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C22×Q8×C47D411yescharacteristic
99#125×C47—22no—
100#125×C47—22no—
101C4×Q8×C47—22no—
102C4×Q8×C47—22no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47D411yescharacteristic
110C2×C4⋊C4×C47D411yescharacteristic
111C2×C4⋊C4×C47D411yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C23×D47—21no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47—41no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188—21no—
148C2×D188C2×Q812yes—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—42no—
162C4⋊C4×C47—42no—
163C4⋊C4×C47—42no—
164C4⋊C4×C47—42no—
165C4⋊C4×C47—81no—
166C4⋊C4×C47—81no—
167C4⋊C4×C47C2×Q812yes—
168C4⋊C4×C47#4112yes—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—81no—
174C42×C47C2×D412yes—
175C42×C47#4112yes—
176(C2×D47)⋊C4—42no—
177(C2×D47)⋊C4—42no—
178(C2×D47)⋊C4—42no—
179(C2×D47)⋊C4—81no—
180(C2×D47)⋊C4—81no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—42no—
188C4⋊Dic47—81no—
189Dic47⋊C4—42no—
190Dic47⋊C4—42no—
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⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C2×D412yes—
201C2×C47⋊C8C2×D412yes—
202C2×C47⋊C8C2×Q812yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—22no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211Q8×C47—44no—
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—21no—
220C2×C4×C47—21no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
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—41no—
238C2×C4×C47—41no—
239C2×C4×C47#12112yes—
240C2×C4×C47#12112yes—
241C2×C4×C47AGL(1,ℤ/8)12yes—
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×Q811yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C472−1+411yescharacteristic
247C2×C4×C472−1+411yescharacteristic
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—
257D188—22no—
258D188—22no—
259D188—41no—
260D188—41no—
261C4×D47—84no—
262C47⋊C8—22no—
263C47⋊C8—22no—
264C47⋊C8—22no—
265C47⋊C8—41no—
266#25486—471nomaximal, Sylow
267C22×C47C2×#12111yescharacteristic
268C22×C47#54811yescharacteristic
269C22×C47D4×Q811yescharacteristic
270C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C47#57511yescharacteristic
272C22×C47#57711yescharacteristic
273C22×C47#57711yescharacteristic
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—81no—
278C2×D47—81no—
279C2×D47—81no—
280C2×D47—81no—
281C4×C47—22no—
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—42no—
285C4×C47—42no—
286C4×C47—41no—
287C4×C47—41no—
288C4×C47—41no—
289C4×C47—81no—
290C4×C47—81no—
291C4×C47#53311yescharacteristic
292C4×C47#54111yescharacteristic
293C4×C47#54711yescharacteristic
294C4×C47D4×Q811yescharacteristic
295Dic47—81no—
296Dic47—81no—
297Dic47—81no—
298C2×#478—471no—
299C2×#446—471no—
300#2584D4711yescharacteristic, complemented
301#2538—471no—
302#2088—471no—
303#2086—471no—
304#2065—471no—
305#2065—471no—
306#1992—471no—
307#1948—471no—
308#1944—471no—
309#1941—471no—
310#1936—471no—
311#1934—471no—
312#1762—471no—
313C2×C47#242911yescharacteristic
314C2×C47#305511yescharacteristic
315C2×C47#306011yescharacteristic
316C2×C47#336811yescharacteristic
317C2×C47#337611yescharacteristic
318C2×C47#338411yescharacteristic
319C2×C47#338811yescharacteristic
320D47—82no—
321C2×#125C2×D4711yescharacteristic
322C2×C22⋊Q8—941no—
323C2×C4×Q8C2×D4711yescharacteristic
324C2×C4×D4—471no—
325#478—944no—
326#446—944no—
327C2×#107—471no—
328C2×#106—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×C4⋊C8—471no—
332C2×C4⋊C8—471no—
333C2×C4⋊C8—471no—
334C2×#102—471no—
335C2×#102—471no—
336C2×#102—471no—
337C2×#102—471no—
338C2×#97—471no—
339C2×#97—471no—
340C2×C4×C8—471no—
341#398—21no—
342#396—941no—
343#396—941no—
344#395—21no—
345#390—471no—
346#390—471no—
347#389—21no—
348#389—21no—
349#389—471no—
350#389—471no—
351#389C2×D4711yescharacteristic
352#389C2×D4711yescharacteristic
353#389C2×D4711yescharacteristic
354#389C2×D4711yescharacteristic
355#384—471no—
356#380—941no—
357C4×C4⋊C4C2×D4711yescharacteristic
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366C47#2548611yescharacteristic, complemented, Sylow
367C22×Q8C22⋊D4711yescharacteristic
368C22×D4—471no—
369C23×C4—941no—
370C22×C8—471no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8—471no—
374#125—22no—
375#125—22no—
376C22⋊Q8—1884no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×D4—942no—
380C4×D4—942no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C4⋊C4C22⋊D4711yescharacteristic
396C2×C4⋊C4C22⋊D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C42—21no—
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409C2×C42C22×D4711yescharacteristic
410#107—942no—
411#106—942no—
412#106—942no—
413#106—942no—
414C4⋊C8—474no—
415C4⋊C8—942no—
416C4⋊C8—942no—
417#102—474no—
418#102—942no—
419#102—942no—
420#102—942no—
421#97—942no—
422#97—942no—
423C4×C8—942no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—941no—
430#95—941no—
431C24—941no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C2×D4—472no—
436C2×D4—941no—
437C2×D4—941no—
438C2×D4—941no—
439C2×D4—941no—
440C2×D4—941no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4—1882no—
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C23×D4711yescharacteristic
458C2×C8—472no—
459C2×C8—472no—
460C2×C8—472no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—3761no—
492C4⋊C4—1882no—
493C4⋊C4—1882no—
494C4⋊C4—1882no—
495C4⋊C4Q8×D4712yes—
496C4⋊C4#1062535412yes—
497C22⋊C4—3761no—
498C22⋊C4—3761no—
499C22⋊C4—1882no—
500C22⋊C4—1882no—
501C22⋊C4—1882no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—81no—
507C42D4×D4712yes—
508C42#1062535212yes—
509C23—942no—
510C23—1881no—
511C23—1881no—
512C23—1881no—
513C23—1881no—
514C23—1881no—
515C23—1881no—
516C23C22×C22⋊D4711yescharacteristic
517Q8—44no—
518D4—942no—
519D4—942no—
520D4—1881no—
521D4—1881no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—3764no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4#1309399612yes—
560C2×C4#1309399612yes—
561C2×C4#1309404912yes—
562C2×C4C2×D4×D4711yescharacteristic
563C2×C4C2×Q8×D4711yescharacteristic
564C2×C4C2×#1062535211yescharacteristic
565C2×C4C2×#1062535411yescharacteristic
566C2×C4#1309407211yescharacteristic
567C2×C4#1309407211yescharacteristic
568C8—942no—
569C8—942no—
570C8—942no—
571C8—1881no—
572C22—3761no—
573C22—3761no—
574C22—3761no—
575C22—3761no—
576C22—1882no—
577C22—1882no—
578C22—1882no—
579C22C2×#1309399611yescharacteristic
580C22#1568978411yescharacteristic
581C22Q8×C22⋊D4711yescharacteristic
582C22C2×#1309404911yescharacteristic
583C22#1568992111yescharacteristic
584C22#1568993111yescharacteristic
585C22#1568993411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—3761no—
597C4—3761no—
598C4—3761no—
599C4#1568967811yescharacteristic
600C4#1568973411yescharacteristic
601C4#1568977611yescharacteristic
602C4#1568978811yescharacteristic
603C2—3762no—
604C2#2024699011yescharacteristic
605C2#2025300911yescharacteristic
606C2#2025304711yescharacteristic
607C2#2025611711yescharacteristic
608C2#2025621511yescharacteristic
609C2#2025628711yescharacteristic
610C2#2025632711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2584C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenter#2584D47
hypocenterC47#25486
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×#15689254C2semidirect
4. #20244889C2semidirect
5. #20244879C2semidirect
8. #20244358C2semidirect
9. #20244151C2semidirect
10. #20244128C2semidirect
11. #20244113C2semidirect
15. C47×#2584C2semidirect
92. #13093944Q8semidirect
300. #2584D47semidirect
366. C47#25486semidirect

Maximal subgroups

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

subgroupindex
C2×#156894202
C2×#156892542
#202483062
#202448892
#202448792
#202447592
#202443582
#202441512
#202441282
#202441132
#202440842
#202440702
#202432162
C47×#25842
#2548647