Finite Group Databasebeta

← #26316051

Subgroups — #26316051

Order 12032.

690 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#20246099C211yesmaximal, characteristic
2#20244951C211yesmaximal, characteristic, complemented
3#20244946C211yesmaximal, characteristic, complemented
4#20244629C211yesmaximal, characteristic
5#20244629C211yesmaximal, characteristic
6#20244084C211yesmaximal, characteristic
7#20244077C211yesmaximal, characteristic, complemented
8#20244070C211yesmaximal, characteristic
9#20244063C211yesmaximal, characteristic, complemented
10#20243400C211yesmaximal, characteristic, complemented
11#20243395C211yesmaximal, characteristic
12#20243390C211yesmaximal, characteristic, complemented
13#20243385C211yesmaximal, characteristic
14#20243245C211yesmaximal, characteristic, complemented
15C47×#2647C211yesmaximal, characteristic, complemented
16C2×#13094012C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093947C2211yescharacteristic
22C2×#13093947C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689025C2211yescharacteristic
33#15689015—21no—
34#15689015—21no—
35#15688997—21no—
36#15688997—21no—
37#15688993C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C47×#400—21no—
49C47×#400—21no—
50C47×#398—21no—
51C47×#398—21no—
52C47×#389C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#384C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×Dic94C2311yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66#13094012—22no—
67#13094012—22no—
68C4×Dic94—22no—
69C4×Dic94—22no—
70C4×D188—22no—
71C4×D188—22no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
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—
82C2×C4×Dic47—21no—
83C4⋊(C47⋊C8)—22no—
84C4⋊(C47⋊C8)—22no—
85#13093947—22no—
86#13093947—22no—
87#13093947—41no—
88#13093947—41no—
89#13093944—22no—
90#13093944—22no—
91#13093944—41no—
92#13093944—41no—
93#13093935—41no—
94C4×C47⋊C8—41no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C2×C42×C47C2311yescharacteristic
110#95×C47—21no—
111#95×C47—21no—
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47D411yescharacteristic
117#95×C47D411yescharacteristic
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C23×D47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188#4112yes—
141C2×C4×D47—42no—
142C2×C4×D47—81no—
143C2×C4×D47—81no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94#4112yes—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—81no—
159C4⋊C4×C47—81no—
160C4⋊C4×C47—81no—
161C4⋊C4×C47—81no—
162C4⋊C4×C47—81no—
163C4⋊C4×C47—81no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47#4112yes—
169C42×C47#4112yes—
170(C2×D47)⋊C4—24no—
171(C2×D47)⋊C4—24no—
172(C2×D47)⋊C4—42no—
173(C2×D47)⋊C4—81no—
174(C2×D47)⋊C4—81no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—81no—
180C4⋊Dic47—81no—
181Dic47⋊C4—42no—
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×C47C22×D411yescharacteristic
196C22×D47—22no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
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—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
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#12412yes—
232C2×C4×C47#12412yes—
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47H(5,𝔽2)11yescharacteristic
238C2×C4×C472−1+411yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248D188—22no—
249D188—22no—
250D188—41no—
251D188—41no—
252C4×D47—84no—
253Dic94—22no—
254Dic94—22no—
255Dic94—41no—
256Dic94—41no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259C47⋊C8—41no—
260C47⋊C8—41no—
261#29134—471nomaximal, Sylow
262C22×C47#52511yescharacteristic
263C22×C47C2×#12411yescharacteristic
264C22×C47#53711yescharacteristic
265C22×C47#57611yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57811yescharacteristic
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—81no—
273C2×D47—81no—
274C2×D47—81no—
275C2×D47—81no—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—41no—
279C4×C47—41no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47—81no—
283C4×C47—81no—
284C4×C47—81no—
285C4×C47—81no—
286C4×C47#55211yescharacteristic
287C4×C47#55211yescharacteristic
288C4×C47#55311yescharacteristic
289C4×C47#55311yescharacteristic
290Dic47—42no—
291Dic47—81no—
292Dic47—81no—
293#2647D4711yescharacteristic, complemented
294#2326—471no—
295#2100—471no—
296#2099—471no—
297#2042—471no—
298#2042—471no—
299#1936—471no—
300#1935—471no—
301#1934—471no—
302#1933—471no—
303#1805—471no—
304#1804—471no—
305#1803—471no—
306#1802—471no—
307#1768—471no—
308C2×C47#256211yescharacteristic
309C2×C47#311911yescharacteristic
310C2×C47#312011yescharacteristic
311C2×C47#313611yescharacteristic
312C2×C47#313711yescharacteristic
313C2×C47#326211yescharacteristic
314C2×C47#326311yescharacteristic
315D47—82no—
316C2×#124—471no—
317C2×C4×Q8—471no—
318C2×C4×D4—471no—
319C2×C4⋊C8—471no—
320C2×C4⋊C8—471no—
321C2×#103—471no—
322C2×#103—471no—
323C2×#103—471no—
324C2×#103—471no—
325C2×#102—471no—
326C2×#102—471no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#97—471no—
330C2×C4×C8—471no—
331#400—21no—
332#400—21no—
333#398—21no—
334#398—21no—
335#391—471no—
336#390—471no—
337#389C2×D4711yescharacteristic
338#389C2×D4711yescharacteristic
339#389C2×D4711yescharacteristic
340#389C2×D4711yescharacteristic
341#388—941no—
342#388—941no—
343#385—941no—
344#385—941no—
345#384—471no—
346#384C2×D4711yescharacteristic
347#382C2×D4711yescharacteristic
348#376C2×D4711yescharacteristic
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#336—471no—
358#336—471no—
359C47#2913411yescharacteristic, complemented, Sylow
360C22×Q8—471no—
361C22×D4—471no—
362C23×C4—941no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—471no—
367#124—942no—
368#124—942no—
369C4×Q8—942no—
370C4×Q8—942no—
371C4×D4—942no—
372C4×D4—942no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4—941no—
393C2×C42—941no—
394C2×C42C22×D4711yescharacteristic
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397C4⋊C8—942no—
398C4⋊C8—942no—
399#103—942no—
400#103—942no—
401#103—1881no—
402#103—1881no—
403#102—942no—
404#102—942no—
405#102—1881no—
406#102—1881no—
407#97—1881no—
408C4×C8—1881no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—941no—
416#95—941no—
417#95C22⋊D4711yescharacteristic
418#95C22⋊D4711yescharacteristic
419#95C22⋊D4711yescharacteristic
420#95C22⋊D4711yescharacteristic
421C24—941no—
422C2×Q8—472no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×D4—472no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4—3761no—
442C22×C4—3761no—
443C22×C4—1882no—
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—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—3761no—
482C4⋊C4—3761no—
483C4⋊C4—1882no—
484C22⋊C4—944no—
485C22⋊C4—944no—
486C22⋊C4—3761no—
487C22⋊C4—3761no—
488C22⋊C4—1882no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—1882no—
494C42#1062535412yes—
495C42#1062535412yes—
496C23—942no—
497C23—1881no—
498C23—1881no—
499C23—1881no—
500C23—1881no—
501C23—1881no—
502C23—1881no—
503C23C22×C22⋊D4711yescharacteristic
504Q8—942no—
505Q8—942no—
506Q8—1881no—
507Q8—1881no—
508D4—942no—
509D4—942no—
510D4—1881no—
511D4—1881no—
512C2×C4—21no—
513C2×C4—21no—
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—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
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—41no—
540C2×C4—942no—
541C2×C4—3764no—
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#1309401312yes—
551C2×C4#1309401312yes—
552C2×C4#41×D4711yescharacteristic
553C2×C4#41×D4711yescharacteristic
554C2×C4C2×#1062535411yescharacteristic
555C2×C4C2×#1062535411yescharacteristic
556C2×C4#1309407011yescharacteristic
557C2×C4#1309407111yescharacteristic
558C8—1881no—
559C8—1881no—
560C8—1881no—
561C8—1881no—
562C22—3761no—
563C22—3761no—
564C22—3761no—
565C22—3761no—
566C22—1882no—
567C22—1882no—
568C22—1882no—
569C22#1568963011yescharacteristic
570C22C2×#1309401311yescharacteristic
571C22#1568970111yescharacteristic
572C22#1568992611yescharacteristic
573C22#1568993111yescharacteristic
574C22#1568993411yescharacteristic
575C22#1568993811yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—3761no—
587C4—3761no—
588C4—1882no—
589C4#1568981111yescharacteristic
590C4#1568981111yescharacteristic
591C4#1568982611yescharacteristic
592C4#1568982611yescharacteristic
593C2—3762no—
594C2#2024855511yescharacteristic
595C2#2025358111yescharacteristic
596C2#2025359611yescharacteristic
597C2#2025374711yescharacteristic
598C2#2025375811yescharacteristic
599C2#2025484411yescharacteristic
600C2#2025485311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2647C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenter#2647D47
hypocenterC47#29134
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2647

Complements

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

NH ≅ G/Ntype
2. #20244951C2semidirect
3. #20244946C2semidirect
7. #20244077C2semidirect
9. #20244063C2semidirect
10. #20243400C2semidirect
12. #20243390C2semidirect
14. #20243245C2semidirect
15. C47×#2647C2semidirect
293. #2647D47semidirect
359. C47#29134semidirect

Maximal subgroups

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

subgroupindex
#202460992
#202449512
#202449462
#202446292
#202440842
#202440772
#202440702
#202440632
#202434002
#202433952
#202433902
#202433852
#202432452
C47×#26472
#2913447