Finite Group Databasebeta

← #26318940

Subgroups — #26318940

Order 12032.

672 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246136C211yesmaximal, characteristic, complemented
2#20244869C211yesmaximal, characteristic, complemented
3#20244801C211yesmaximal, characteristic
4#20244716C211yesmaximal, characteristic, complemented
5#20244524C211yesmaximal, characteristic
6#20244005C211yesmaximal, characteristic
7#20244005C211yesmaximal, characteristic
8#20243980C211yesmaximal, characteristic, complemented
9#20243977C211yesmaximal, characteristic, complemented
10#20243963C211yesmaximal, characteristic, complemented
11#20243959C211yesmaximal, characteristic, complemented
12#20243649C211yesmaximal, characteristic
13#20243635C211yesmaximal, characteristic
14#20243533C211yesmaximal, characteristic, complemented
15C47×#2319C211yesmaximal, characteristic
16C2×#13094024C2211yescharacteristic
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093945C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25#15689027—21no—
26#15689027—21no—
27#15689021C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689006—21no—
30#15689006—21no—
31#15688989C2211yescharacteristic
32C4×C4⋊Dic47C2211yescharacteristic
33C4×C4⋊Dic47C2211yescharacteristic
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4×D4×C47C2211yescharacteristic
49C47×#390C2211yescharacteristic
50C47×#390C2211yescharacteristic
51C47×#389C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#388—21no—
54C47×#388—21no—
55C47×#381—21no—
56C47×#380—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×Dic94—21no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094024—22no—
65#13094024—22no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47—21no—
75#13093957—22no—
76#13093957—22no—
77#13093957—22no—
78#13093955—22no—
79#13093945—22no—
80#13093945—22no—
81#13093945—22no—
82#13093945—22no—
83C47⋊#95—21no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C22×D4×C47C2311yescharacteristic
90C23×C4×C47—21no—
91C4×D4×C47—22no—
92C4×D4×C47—22no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47C2311yescharacteristic
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C22⋊C4×C47—21no—
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C2×C42×C47C2311yescharacteristic
106C2×C42×C47C2311yescharacteristic
107#95×C47—21no—
108#95×C47—21no—
109#95×C47—21no—
110#95×C47D411yescharacteristic
111#95×C47D411yescharacteristic
112#95×C47D411yescharacteristic
113#95×C47D411yescharacteristic
114C24×C47—21no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47#4112yes—
121C22×C4×C47—21no—
122C22×C4×C47—24no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—81no—
126C22×C4×C47—81no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C2×Dic94—24no—
139C2×Dic94—81no—
140C2×Dic94—81no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—81no—
147C4⋊C4×C47—81no—
148C4⋊C4×C47#4112yes—
149C22⋊C4×C47—24no—
150C22⋊C4×C47—81no—
151C22⋊C4×C47—81no—
152C22⋊C4×C47—81no—
153C22⋊C4×C47—81no—
154C42×C47—21no—
155C42×C47—21no—
156C42×C47—21no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—81no—
161C4⋊Dic47—24no—
162C4⋊Dic47—21no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47#4112yes—
169C4⋊Dic47#4112yes—
170Dic47⋊C4—81no—
171Dic47⋊C4—81no—
172C4×Dic47—81no—
173C4×Dic47—81no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C23×C47—22no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47C22×D411yescharacteristic
194D4×C47—22no—
195D4×C47—22no—
196D4×C47—41no—
197D4×C47—41no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
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—41no—
218C2×C4×C47—41no—
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—84no—
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C472−1+411yescharacteristic
231C2×C4×C472−1+411yescharacteristic
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244Dic94—84no—
245C47⋊C8—41no—
246C47⋊C8—41no—
247C47⋊C8—41no—
248C47⋊C8—41no—
249#29401—471nomaximal, Sylow
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—81no—
254C22×C47—81no—
255C22×C47—81no—
256C22×C47—81no—
257C22×C47#52511yescharacteristic
258C22×C47#52511yescharacteristic
259C22×C47#53611yescharacteristic
260C22×C47#57611yescharacteristic
261C22×C47#57611yescharacteristic
262C22×C47#57711yescharacteristic
263C22×C47#57811yescharacteristic
264C4×C47—41no—
265C4×C47—42no—
266C4×C47—41no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47—41no—
271C4×C47—81no—
272C4×C47—81no—
273C4×C47#55211yescharacteristic
274C4×C47#55211yescharacteristic
275C4×C47#55611yescharacteristic
276C4×C47#55611yescharacteristic
277Dic47—42no—
278Dic47—42no—
279Dic47—81no—
280Dic47—81no—
281#2331—471no—
282#2319D4711yescharacteristic
283#2085—471no—
284#2073—471no—
285#2057—471no—
286#2023—471no—
287#1925—471no—
288#1925—471no—
289#1921—471no—
290#1921—471no—
291#1919—471no—
292#1918—471no—
293#1864—471no—
294#1862—471no—
295#1840—471no—
296C2×C47—82no—
297C2×C47#259911yescharacteristic
298C2×C47#311811yescharacteristic
299C2×C47#312111yescharacteristic
300C2×C47#312211yescharacteristic
301C2×C47#312411yescharacteristic
302C2×C47#324111yescharacteristic
303C2×C47#324311yescharacteristic
304C2×C4⋊Q8—471no—
305C2×C4×D4C2×D4711yescharacteristic
306C2×#107—471no—
307C2×#107—471no—
308C2×#107—471no—
309C2×#106—471no—
310C2×#102—471no—
311C2×#102—471no—
312C2×#102—471no—
313C2×#102—471no—
314#391—941no—
315#391—941no—
316#390C2×D4711yescharacteristic
317#390C2×D4711yescharacteristic
318#389—471no—
319#389—471no—
320#389C2×D4711yescharacteristic
321#389C2×D4711yescharacteristic
322#388—21no—
323#388—21no—
324#387—941no—
325#387—941no—
326#382—471no—
327#381—21no—
328#380—21no—
329C4×C4⋊C4—471no—
330C4×C4⋊C4—471no—
331C4×C4⋊C4C2×D4711yescharacteristic
332#376C2×D4711yescharacteristic
333#340—471no—
334#340—471no—
335#340—471no—
336#340—471no—
337#340—471no—
338#340—471no—
339#340—471no—
340#340—471no—
341#336—471no—
342#336—471no—
343#336—471no—
344#336—471no—
345#336—471no—
346#336—471no—
347C47#2940111yescharacteristic, complemented, Sylow
348C22×Q8—941no—
349C22×D4C22×D4711yescharacteristic
350C23×C4—21no—
351C22×C8—471no—
352C22×C8—471no—
353C22×C8—471no—
354C22×C8—471no—
355C4⋊Q8—942no—
356C4⋊Q8—942no—
357C4×D4—22no—
358C4×D4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—471no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4C22×D4711yescharacteristic
369C2×C4⋊C4C22×D4711yescharacteristic
370C2×C4⋊C4C22×D4711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C42—21no—
377C2×C42—941no—
378C2×C42—941no—
379C2×C42C22×D4711yescharacteristic
380C2×C42C22×D4711yescharacteristic
381C2×C42C22×D4711yescharacteristic
382#107—942no—
383#107—942no—
384#107—942no—
385#106—942no—
386#102—942no—
387#102—942no—
388#102—942no—
389#102—942no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—941no—
394#95—941no—
395#95—941no—
396#95—941no—
397#95—941no—
398#95—941no—
399#95C22⋊D4711yescharacteristic
400#95C22⋊D4711yescharacteristic
401#95C22⋊D4711yescharacteristic
402#95C22⋊D4711yescharacteristic
403C24—21no—
404C2×Q8—944no—
405C2×Q8—3761no—
406C2×Q8—3761no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4#1062535012yes—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4C2×C22⋊D4711yescharacteristic
424C22×C4C2×C22⋊D4711yescharacteristic
425C22×C4C2×C22⋊D4711yescharacteristic
426C22×C4C2×C22⋊D4711yescharacteristic
427C22×C4C2×C22⋊D4711yescharacteristic
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C23×D4711yescharacteristic
430C2×C8—941no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—941no—
434C2×C8—941no—
435C2×C8—941no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—472no—
450C4⋊C4—472no—
451C4⋊C4—944no—
452C4⋊C4—941no—
453C4⋊C4—941no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—3761no—
459C4⋊C4—3761no—
460C4⋊C4#1062535012yes—
461C22⋊C4—24no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—81no—
473C42—3761no—
474C42—3761no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C22×C22⋊D4711yescharacteristic
483Q8—3764no—
484D4—22no—
485D4—22no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—84no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1881no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4#41×D4711yescharacteristic
529C2×C4#41×D4711yescharacteristic
530C2×C4C2×#1062535011yescharacteristic
531C2×C4C2×#1062535011yescharacteristic
532C2×C4#1309407111yescharacteristic
533C2×C4#1309407111yescharacteristic
534C8—1881no—
535C8—1881no—
536C8—1881no—
537C8—1881no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22#1568963011yescharacteristic
546C22#1568963411yescharacteristic
547C22#1568969411yescharacteristic
548C22#1568992611yescharacteristic
549C22#1568992611yescharacteristic
550C22#1568993411yescharacteristic
551C22#1568993811yescharacteristic
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C4—3761no—
562C4—3761no—
563C4—1882no—
564C4—1882no—
565C4#1568980611yescharacteristic
566C4#1568980611yescharacteristic
567C4#1568984111yescharacteristic
568C4#1568984111yescharacteristic
569C2—82no—
570C2#2024896511yescharacteristic
571C2#2025357211yescharacteristic
572C2#2025361111yescharacteristic
573C2#2025362611yescharacteristic
574C2#2025365011yescharacteristic
575C2#2025463511yescharacteristic
576C2#2025466511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2319C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2319D47
hypocenterC47#29401
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. #20246136C2semidirect
2. #20244869C2semidirect
4. #20244716C2semidirect
8. #20243980C2semidirect
9. #20243977C2semidirect
10. #20243963C2semidirect
11. #20243959C2semidirect
14. #20243533C2semidirect
347. C47#29401semidirect

Maximal subgroups

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

subgroupindex
#202461362
#202448692
#202448012
#202447162
#202445242
#202440052
#202439802
#202439772
#202439632
#202439592
#202436492
#202436352
#202435332
C47×#23192
#2940147