Finite Group Databasebeta

← #26284903

Subgroups — #26284903

Order 12032.

704 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247926C211yesmaximal, characteristic, complemented
2#20244801C211yesmaximal, characteristic
3#20244762C211yesmaximal, characteristic
4#20244624C211yesmaximal, characteristic
5#20244624C211yesmaximal, characteristic
6#20244067C211yesmaximal, characteristic, complemented
7#20244019C211yesmaximal, characteristic, complemented
8#20243973C211yesmaximal, characteristic, complemented
9#20243966C211yesmaximal, characteristic, complemented
10#20243947C211yesmaximal, characteristic, complemented
11#20243656C211yesmaximal, characteristic, complemented
12#20243649C211yesmaximal, characteristic
13#20243403C211yesmaximal, characteristic, complemented
14C4×#13093945C211yesmaximal, characteristic
15C47×#2367C211yesmaximal, characteristic
16C2×#13093955C2211yescharacteristic
17C2×#13093955C2211yescharacteristic
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×#13093948C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689064—21no—
29#15689054—21no—
30#15689021C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689004—21no—
35#15688989C2211yescharacteristic
36#15688987—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#124×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#390C2211yescharacteristic
53C47×#388—21no—
54C47×#388—21no—
55C47×#385—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C2×C4⋊Dic47C2311yescharacteristic
64C2×C4⋊Dic47C2311yescharacteristic
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×Dic47⋊C4—21no—
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×C4×Dic47—21no—
72#13093955—22no—
73#13093955—22no—
74C4⋊(C47⋊C8)—41no—
75#13093948—22no—
76#13093948—22no—
77#13093948—22no—
78#13093948—41no—
79#13093945—22no—
80#13093945—22no—
81#13093945—22no—
82#13093945—41no—
83C4×C47⋊C8—41no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C22×Q8×C47C2311yescharacteristic
92C22×D4×C47C2311yescharacteristic
93C23×C4×C47—21no—
94#124×C47—22no—
95#124×C47—22no—
96C4×Q8×C47—22no—
97C4×Q8×C47—41no—
98C4×D4×C47—22no—
99C4×D4×C47—41no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
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×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115#95×C47—21no—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47#4112yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47—81no—
139C22×C4×C47C2×D411yescharacteristic
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—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—41no—
155C4⋊C4×C47—42no—
156C4⋊C4×C47#4112yes—
157C22⋊C4×C47—24no—
158C22⋊C4×C47—24no—
159C22⋊C4×C47—42no—
160C22⋊C4×C47—42no—
161C22⋊C4×C47—81no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47—42no—
169C42×C47—81no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47#4112yes—
178Dic47⋊C4—42no—
179Dic47⋊C4—81no—
180Dic47⋊C4—81no—
181Dic47⋊C4—81no—
182C4×Dic47—42no—
183C2×C47⋊C8—21no—
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—
195C23×C47—22no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47C22×D411yescharacteristic
203Q8×C47—22no—
204Q8×C47—22no—
205Q8×C47—41no—
206Q8×C47—41no—
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—41no—
210D4×C47—41no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
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—21no—
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×C47#13512yes—
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47H(5,𝔽2)11yescharacteristic
244C2×C4×C472−1+411yescharacteristic
245C2×Dic47—22no—
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—
257C47⋊C8—41no—
258C47⋊C8—41no—
259C47⋊C8—41no—
260C47⋊C8—41no—
261#26760—471nomaximal, Sylow
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—81no—
268C22×C47—81no—
269C22×C47#52511yescharacteristic
270C22×C47#54611yescharacteristic
271C22×C47#54811yescharacteristic
272C22×C47C2×#13511yescharacteristic
273C22×C47#57511yescharacteristic
274C22×C47#57711yescharacteristic
275C22×C47#57711yescharacteristic
276C4×C47—22no—
277C4×C47—41no—
278C4×C47—41no—
279C4×C47—42no—
280C4×C47—41no—
281C4×C47—42no—
282C4×C47—41no—
283C4×C47—41no—
284C4×C47—81no—
285C4×C47#52911yescharacteristic
286C4×C47#53911yescharacteristic
287C4×C47#55211yescharacteristic
288C4×C47#55311yescharacteristic
289Dic47—42no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293#2508—471no—
294#2367D4711yescharacteristic
295#2073—471no—
296#2065—471no—
297#2042—471no—
298#2042—471no—
299#1933—471no—
300#1927—471no—
301#1920—471no—
302#1919—471no—
303#1916—471no—
304#1865—471no—
305#1864—471no—
306#1805—471no—
307C4×#102—471no—
308C2×C47—82no—
309C2×C47#246011yescharacteristic
310C2×C47#310211yescharacteristic
311C2×C47#312511yescharacteristic
312C2×C47#329611yescharacteristic
313C2×C47#332111yescharacteristic
314C2×C47#336011yescharacteristic
315C2×C47#337111yescharacteristic
316C2×#124C2×D4711yescharacteristic
317C2×C4×Q8C2×D4711yescharacteristic
318C2×C4×D4C2×D4711yescharacteristic
319C2×#106—471no—
320C2×#106—471no—
321C2×C4⋊C8—471no—
322C2×#103—471no—
323C2×#103—471no—
324C2×#103—471no—
325C2×#103—471no—
326C2×#102—471no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#102—471no—
330C2×C4×C8—471no—
331#400—941no—
332#398—941no—
333#391C2×D4711yescharacteristic
334#390C2×D4711yescharacteristic
335#389—471no—
336#389—471no—
337#389—471no—
338#389—471no—
339#388—21no—
340#388—21no—
341#387—941no—
342#385—21no—
343#382—471no—
344#382—941no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4C2×D4711yescharacteristic
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C22⋊C4—21no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#336—471no—
356#336—471no—
357#336—471no—
358#336—471no—
359C47#2676011yescharacteristic, complemented, Sylow
360C22×Q8C22×D4711yescharacteristic
361C22×D4C22×D4711yescharacteristic
362C23×C4—21no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—471no—
367#124—22no—
368#124—22no—
369C4×Q8—22no—
370C4×Q8—41no—
371C4×D4—22no—
372C4×D4—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4C22⋊D4711yescharacteristic
383C2×C4⋊C4C22⋊D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—941no—
394C2×C42C22×D4711yescharacteristic
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397#106—942no—
398#106—942no—
399C4⋊C8—1881no—
400#103—942no—
401#103—942no—
402#103—942no—
403#103—1881no—
404#102—942no—
405#102—942no—
406#102—942no—
407#102—1881no—
408C4×C8—1881no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95C22⋊D4711yescharacteristic
420#95C22⋊D4711yescharacteristic
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1062535012yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#1062535012yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4C2×C22⋊D4711yescharacteristic
445C22×C4C2×C22⋊D4711yescharacteristic
446C22×C4C2×C22⋊D4711yescharacteristic
447C22×C4C2×C22⋊D4711yescharacteristic
448C22×C4C2×C22⋊D4711yescharacteristic
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C23×D4711yescharacteristic
451C2×C8—941no—
452C2×C8—941no—
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—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—472no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—3761no—
478C4⋊C4—3761no—
479C4⋊C4—3761no—
480C4⋊C4—1882no—
481C4⋊C4#1062535212yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—81no—
495C42—1882no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C22⋊D4711yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
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—41no—
538C2×C4—84no—
539C2×C4—942no—
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#1309404412yes—
552C2×C4#41×D4711yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C2×C4C2×#1062535011yescharacteristic
555C2×C4C2×#1062535011yescharacteristic
556C2×C4#1309406911yescharacteristic
557C2×C4#1309407111yescharacteristic
558C8—1881no—
559C8—1881no—
560C8—1881no—
561C8—1881no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1568963411yescharacteristic
570C22#1568976711yescharacteristic
571C22#1568978011yescharacteristic
572C22C2×#1309404411yescharacteristic
573C22#1568992111yescharacteristic
574C22#1568993111yescharacteristic
575C22#1568993411yescharacteristic
576C4—22no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—81no—
585C4—3761no—
586C4—3761no—
587C4—3761no—
588C4—1882no—
589C4#1568965311yescharacteristic
590C4#1568971511yescharacteristic
591C4#1568981311yescharacteristic
592C4#1568982211yescharacteristic
593C2—82no—
594C2#2024733411yescharacteristic
595C2#2025340611yescharacteristic
596C2#2025366511yescharacteristic
597C2#2025522311yescharacteristic
598C2#2025557411yescharacteristic
599C2#2025603311yescharacteristic
600C2#2025615011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2367C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2367D47
hypocenterC47#26760
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #20247926C2semidirect
6. #20244067C2semidirect
7. #20244019C2semidirect
8. #20243973C2semidirect
9. #20243966C2semidirect
10. #20243947C2semidirect
11. #20243656C2semidirect
13. #20243403C2semidirect
359. C47#26760semidirect

Maximal subgroups

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

subgroupindex
#202479262
#202448012
#202447622
#202446242
#202440672
#202440192
#202439732
#202439662
#202439472
#202436562
#202436492
#202434032
C4×#130939452
C47×#23672
#2676047