Finite Group Databasebeta

← #26187415

Subgroups — #26187415

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253453C211yesmaximal, characteristic
2#20252406C211yesmaximal, characteristic
3#20242288C211yesmaximal, characteristic, complemented
4#20242278C211yesmaximal, characteristic, complemented
5#20242266C211yesmaximal, characteristic, complemented
6#20242257C211yesmaximal, characteristic, complemented
7#20242236C211yesmaximal, characteristic, complemented
8#20242231C211yesmaximal, characteristic, complemented
9#20242205C211yesmaximal, characteristic, complemented
10C8×#10625337C211yesmaximal, characteristic, complemented
11#20242171C211yesmaximal, characteristic, complemented
12#20241861C211yesmaximal, characteristic, complemented
13#20241773C211yesmaximal, characteristic
14#20241685C211yesmaximal, characteristic, complemented
15C47×#2962C211yesmaximal, characteristic, complemented
16C4×#10625350C2211yescharacteristic
17#15689461C411yescharacteristic
18#15689461C411yescharacteristic
19#15689405C411yescharacteristic
20#15689400C411yescharacteristic
21#15689330C411yescharacteristic
22#15689325C411yescharacteristic
23#15689283C411yescharacteristic
24#15689283C411yescharacteristic
25C8×Dic94C2211yescharacteristic
26C8×Dic94C2211yescharacteristic
27C4×#10625337C2211yescharacteristic
28C4×#10625337C2211yescharacteristic
29C4×#10625336C2211yescharacteristic
30C4×#10625336C2211yescharacteristic
31#15689202C2211yescharacteristic
32#15689202C2211yescharacteristic
33#15689199—21no—
34#15689199—21no—
35#15689180C2211yescharacteristic
36#15689178—21no—
37#15689176C2211yescharacteristic
38#15689176C2211yescharacteristic
39#15689075—21no—
40#15688875C2211yescharacteristic
41#15688875C2211yescharacteristic
42#15688873C2211yescharacteristic
43#15688873C2211yescharacteristic
44Q8⋊(C47⋊C8)C2211yescharacteristic
45Q8⋊(C47⋊C8)C2211yescharacteristic
46Q8⋊(C47⋊C8)C2211yescharacteristic
47Q8⋊(C47⋊C8)C2211yescharacteristic
48#15688853C2211yescharacteristic
49#15688853C2211yescharacteristic
50#15688850C2211yescharacteristic
51#15688850C2211yescharacteristic
52#15688845C2211yescharacteristic
53#15688845C2211yescharacteristic
54#15688844C2211yescharacteristic
55C8×C47⋊C8C2211yescharacteristic
56C47×#533C2211yescharacteristic
57C47×#446—21no—
58C47×#446—21no—
59C47×#446C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C8×Q8×C47C2211yescharacteristic
63C47×#424—21no—
64C2×C4⋊C8×C47C2211yescharacteristic
65C47×#405—21no—
66C4×M16×C47C2211yescharacteristic
67C2×#10625350C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625337C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625336C2×C411yescharacteristic
72C2×#10625324—21no—
73C2×#10625324—21no—
74C4×Dic94C2×C411yescharacteristic
75C4×Dic94C2311yescharacteristic
76C4×D188C2×C411yescharacteristic
77C4×D188C2311yescharacteristic
78C4×C22⋊D47—21no—
79C4×C22⋊D47—21no—
80#13093983C2×C411yescharacteristic
81#13093981—21no—
82C42×D47—21no—
83#13093957C2×C411yescharacteristic
84#13093957C2×C411yescharacteristic
85#13093955C2×C411yescharacteristic
86#13093955C2×C411yescharacteristic
87C4⋊(C47⋊C8)—21no—
88C4⋊(C47⋊C8)C2×C411yescharacteristic
89C4⋊(C47⋊C8)C2×C411yescharacteristic
90C4⋊(C47⋊C8)C2311yescharacteristic
91C4⋊(C47⋊C8)C2311yescharacteristic
92Dic47⋊C8—21no—
93Dic47⋊C8—21no—
94#13093948C2×C411yescharacteristic
95#13093948C2×C411yescharacteristic
96#13093948C2×C411yescharacteristic
97#13093948C2×C411yescharacteristic
98#13093947C2×C411yescharacteristic
99#13093947C2×C411yescharacteristic
100#13093944C2×C411yescharacteristic
101#13093944C2×C411yescharacteristic
102C22⋊(C47⋊C8)—21no—
103C22⋊(C47⋊C8)—21no—
104C22⋊(C47⋊C8)C2×C411yescharacteristic
105C22⋊(C47⋊C8)C2×C411yescharacteristic
106(C2×D47)⋊C8—21no—
107(C2×D47)⋊C8—21no—
108#13093935—21no—
109C4×C47⋊C8C2311yescharacteristic
110C4×C47⋊C8C2311yescharacteristic
111C4×C47⋊C8C2311yescharacteristic
112C4×C47⋊C8C2311yescharacteristic
113C8×Dic47—21no—
114C8×Dic47—21no—
115C2×M16×C47D411yescharacteristic
116C2×M16×C47D411yescharacteristic
117C22×C8×C47—21no—
118C4⋊Q8×C47C2×C411yescharacteristic
119C4⋊Q8×C47C2×C411yescharacteristic
120#125×C47—21no—
121C22⋊Q8×C47—21no—
122C22⋊Q8×C47C2×C411yescharacteristic
123C22⋊Q8×C47C2×C411yescharacteristic
124C4×Q8×C47—21no—
125C4×Q8×C47C2311yescharacteristic
126C4×Q8×C47C2311yescharacteristic
127#117×C47—21no—
128C2×C42×C47C2311yescharacteristic
129C4⋊C8×C47—21no—
130C4⋊C8×C47—21no—
131C4⋊C8×C47—21no—
132C4⋊C8×C47—21no—
133C4⋊C8×C47—21no—
134C4⋊C8×C47C2311yescharacteristic
135C4⋊C8×C47C2311yescharacteristic
136#97×C47—21no—
137#97×C47—21no—
138#97×C47D411yescharacteristic
139#97×C47D411yescharacteristic
140C4×C8×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47—21no—
145C2×Q8×C47C22×C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47#4111yescharacteristic
150C2×C22⋊D47—21no—
151#10625350—22no—
152#10625350—41no—
153C2×D188C22×C411yescharacteristic
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×Dic94C22×C411yescharacteristic
158M16×C47—22no—
159M16×C47—41no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—42no—
163C2×C8×C47C2×D411yescharacteristic
164C2×C8×C47C2×D411yescharacteristic
165C2×C8×C47C2×D411yescharacteristic
166C2×C8×C47C2×D411yescharacteristic
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C22×C411yescharacteristic
175C4⋊C4×C47C22×C411yescharacteristic
176C22⋊C4×C47—21no—
177C22⋊C4×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47C22×C411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C42×C47#4111yescharacteristic
183C42×C47C2411yescharacteristic
184C22⋊Dic47—21no—
185#10625337—22no—
186#10625337—22no—
187#10625336—41no—
188#10625336—41no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191C4⋊Dic47C22×C411yescharacteristic
192C4⋊Dic47C22×C411yescharacteristic
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8C22×C411yescharacteristic
203C2×C47⋊C8C22×C411yescharacteristic
204C2×C47⋊C8C22×C411yescharacteristic
205C2×C47⋊C8C22×C411yescharacteristic
206#10625324—42no—
207#10625324—42no—
208C23×C47C4×D411yescharacteristic
209C22×D47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47—81no—
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—21no—
222C2×C4×C47—21no—
223C2×C4×C47—41no—
224C2×C4×C47C4×D411yescharacteristic
225C2×C4×C47C4×D411yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C22⋊D47—42no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234D188—21no—
235D188—21no—
236C4×D47—42no—
237C4×D47—41no—
238C4×D47—41no—
239C4×D47—81no—
240Dic94—21no—
241Dic94#13112yes—
242C8×C47—21no—
243C8×C47—21no—
244C8×C47—21no—
245C8×C47—41no—
246C8×C47—41no—
247C8×C47#13712yes—
248C47⋊C8—21no—
249C47⋊C8—21no—
250C47⋊C8—21no—
251C47⋊C8—21no—
252C47⋊C8—41no—
253C47⋊C8—41no—
254C47⋊C8—41no—
255C47⋊C8—41no—
256#18013—471nomaximal, Sylow
257C22×C47—21no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47C2×C4×D411yescharacteristic
261C2×D47—41no—
262C2×D47—41no—
263C2×D47—41no—
264C4×C47—21no—
265C4×C47—21no—
266C4×C47—21no—
267C4×C47—21no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47C2×#13111yescharacteristic
273C4×C47#56811yescharacteristic
274C4×C47C2×#13711yescharacteristic
275C4×C47#57511yescharacteristic
276Dic47—22no—
277Dic47—41no—
278Dic47—41no—
279#3107—471no—
280#2971—471no—
281#2962D4711yescharacteristic, complemented
282#1592—471no—
283#1590—471no—
284#1588—471no—
285#1587—471no—
286#1584—471no—
287#1583—471no—
288#1579—471no—
289C8×Q16—471no—
290#1573—471no—
291#1497—471no—
292#1477—471no—
293#1453—471no—
294C2×C47—41no—
295C2×C47#293311yescharacteristic
296C2×C47C4×#13711yescharacteristic
297C2×C47#295711yescharacteristic
298D47—81no—
299#533C2×D4711yescharacteristic
300C4×#41—471no—
301#485—471no—
302#485—471no—
303#475—471no—
304#474—471no—
305#462—471no—
306#461—471no—
307#452—471no—
308#452—471no—
309#446—21no—
310#446—21no—
311#446C2×D4711yescharacteristic
312#446C2×D4711yescharacteristic
313C8×Q8—471no—
314C8×Q8—471no—
315C8×Q8C2×D4711yescharacteristic
316C8×Q8C2×D4711yescharacteristic
317C4×Q16—471no—
318C4×Q16—471no—
319C4×SD16—471no—
320C4×SD16—471no—
321#436—471no—
322#436—471no—
323#436—941no—
324#436—941no—
325#432—471no—
326#432—941no—
327#431—471no—
328#431—471no—
329#424—21no—
330C2×C4⋊C8C2×D4711yescharacteristic
331#405—21no—
332C4×M16—941no—
333C4×M16C2×D4711yescharacteristic
334#335—471no—
335#335—471no—
336#334—471no—
337#334—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8—471no—
342Q8⋊C8—471no—
343Q8⋊C8—471no—
344D4⋊C8—471no—
345D4⋊C8—471no—
346#322—471no—
347#322—471no—
348#322—471no—
349C82—471no—
350C47#1801311yescharacteristic, complemented, Sylow
351C2×#41—471no—
352C2×Q16—471no—
353C2×Q16—471no—
354C2×SD16—471no—
355C2×SD16—471no—
356C2×M16—941no—
357C2×M16—941no—
358C2×M16C22⋊D4711yescharacteristic
359C2×M16C22⋊D4711yescharacteristic
360C22×C8—21no—
361C4⋊Q8C4×D4711yescharacteristic
362C4⋊Q8C4×D4711yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C4×D4711yescharacteristic
366C22⋊Q8C4×D4711yescharacteristic
367C4×Q8—21no—
368C4×Q8—471no—
369C4×Q8—471no—
370C4×Q8C22×D4711yescharacteristic
371C4×Q8C22×D4711yescharacteristic
372C4×D4—471no—
373C4×D4—471no—
374C4×D4—941no—
375C4×D4—941no—
376#117—21no—
377#117—471no—
378#117—941no—
379C2×C42—941no—
380C2×C42C22×D4711yescharacteristic
381#107—471no—
382#107—471no—
383#106—471no—
384#106—471no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—471no—
391C4⋊C8—471no—
392C4⋊C8—471no—
393C4⋊C8—471no—
394C4⋊C8—941no—
395C4⋊C8—941no—
396C4⋊C8—941no—
397C4⋊C8C22×D4711yescharacteristic
398C4⋊C8C22×D4711yescharacteristic
399#103—471no—
400#103—471no—
401#103—471no—
402#103—471no—
403#103—471no—
404#103—471no—
405#102—471no—
406#102—471no—
407C22⋊C8—471no—
408C22⋊C8—471no—
409C22⋊C8—941no—
410C22⋊C8—941no—
411C22⋊C8—941no—
412C22⋊C8—941no—
413#97—21no—
414#97—21no—
415#97—941no—
416#97C22⋊D4711yescharacteristic
417#97C22⋊D4711yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—471no—
421C4×C8—471no—
422C4×C8—471no—
423C4×C8—471no—
424C4×C8—941no—
425C4×C8—941no—
426C4×C8C22×D4711yescharacteristic
427C4×C8C22×D4711yescharacteristic
428#41—942no—
429#41—1881no—
430C2×Q8—21no—
431C2×Q8—471no—
432C2×Q8C2×C4×D4711yescharacteristic
433C2×Q8C2×C4×D4711yescharacteristic
434C2×D4—471no—
435C2×D4—941no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4C2×C4×D4711yescharacteristic
440C22×C4#1062535011yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442Q16—942no—
443Q16—942no—
444SD16—1881no—
445SD16—1881no—
446M16—22no—
447M16—41no—
448M16—1882no—
449M16—1882no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—471no—
454C2×C8—471no—
455C2×C8—471no—
456C2×C8—471no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8C2×C22⋊D4711yescharacteristic
462C2×C8C2×C22⋊D4711yescharacteristic
463C2×C8C2×C22⋊D4711yescharacteristic
464C2×C8C2×C22⋊D4711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—471no—
473C4⋊C4—471no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4C2×C4×D4711yescharacteristic
477C4⋊C4C2×C4×D4711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—941no—
481C22⋊C4—941no—
482C22⋊C4—941no—
483C42—21no—
484C42—21no—
485C42—941no—
486C42—941no—
487C42—941no—
488C42C2×C4×D4711yescharacteristic
489C42#1062535011yescharacteristic
490C42C2×C22⋊D4711yescharacteristic
491C42C23×D4711yescharacteristic
492C23—941no—
493C23C4×C22⋊D4711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—472no—
500Q8—941no—
501D4—941no—
502D4—941no—
503D4—1882no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—3761no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4C4×C22⋊D4711yescharacteristic
521C2×C4C4×C22⋊D4711yescharacteristic
522C2×C4C4×C22⋊D4711yescharacteristic
523C2×C4C22×C4×D4711yescharacteristic
524C2×C4C22×C22⋊D4711yescharacteristic
525C2×C4C2×#1062535011yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—941no—
532C8—941no—
533C8—941no—
534C8—941no—
535C8—1881no—
536C8—1881no—
537C8—1881no—
538C8—1881no—
539C8#1309405512yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—1881no—
544C22—1881no—
545C22—1881no—
546C22C2×C4×C22⋊D4711yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—942no—
556C4—1881no—
557C4—1881no—
558C4#131×D4711yescharacteristic
559C4#1568988811yescharacteristic
560C4C2×#1309405511yescharacteristic
561C4#1568992111yescharacteristic
562C2—41no—
563C2—3761no—
564C2#2025211011yescharacteristic
565C2C4×#1309405511yescharacteristic
566C2#2025230311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2962C2
socleC22×C47C2×C4×D4
prime radicalC2×#10625350C2×C4
cosocleC42×C47C24
hypercenter#2962D47
hypocenterC47#18013
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #20242288C2semidirect
4. #20242278C2semidirect
5. #20242266C2semidirect
6. #20242257C2semidirect
7. #20242236C2semidirect
8. #20242231C2semidirect
9. #20242205C2semidirect
10. C8×#10625337C2semidirect
11. #20242171C2semidirect
12. #20241861C2semidirect
14. #20241685C2semidirect
15. C47×#2962C2semidirect
281. #2962D47semidirect
350. C47#18013semidirect

Maximal subgroups

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

subgroupindex
#202534532
#202524062
#202422882
#202422782
#202422662
#202422572
#202422362
#202422312
#202422052
C8×#106253372
#202421712
#202418612
#202417732
#202416852
C47×#29622
#1801347