Finite Group Databasebeta

← #26187404

Subgroups — #26187404

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#20253468C211yesmaximal, characteristic
2#20252392C211yesmaximal, characteristic
3#20242295C211yesmaximal, characteristic, complemented
4#20242278C211yesmaximal, characteristic, complemented
5#20242259C211yesmaximal, characteristic, complemented
6#20242243C211yesmaximal, characteristic, complemented
7#20242236C211yesmaximal, characteristic, complemented
8#20242231C211yesmaximal, characteristic, complemented
9#20242198C211yesmaximal, characteristic, complemented
10C8×#10625337C211yesmaximal, characteristic, complemented
11#20242179C211yesmaximal, characteristic, complemented
12#20241851C211yesmaximal, characteristic
13#20241783C211yesmaximal, characteristic, complemented
14#20241685C211yesmaximal, characteristic, complemented
15Q8×M16×C47C211yesmaximal, characteristic, complemented
16#15689657C2211yescharacteristic
17#15689453C411yescharacteristic
18#15689436C411yescharacteristic
19#15689430C411yescharacteristic
20#15689430C411yescharacteristic
21#15689345C411yescharacteristic
22#15689345C411yescharacteristic
23#15689279C411yescharacteristic
24#15689262C411yescharacteristic
25#15689253C2211yescharacteristic
26C8×Dic94C2211yescharacteristic
27C4×#10625337C2211yescharacteristic
28C4×#10625337C2211yescharacteristic
29C4×#10625336C2211yescharacteristic
30C4×#10625336C2211yescharacteristic
31#15689202C2211yescharacteristic
32#15689199—21no—
33#15689192—21no—
34C8×D188C2211yescharacteristic
35#15689185—21no—
36#15689176C2211yescharacteristic
37#15689176C2211yescharacteristic
38#15689176C2211yescharacteristic
39#15689080—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
56C2×C4×Q8×C47C2211yescharacteristic
57C47×#446—21no—
58C47×#446—21no—
59C47×#446C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C8×Q8×C47C2211yescharacteristic
63C47×#423—21no—
64C47×#423C2211yescharacteristic
65C4×M16×C47—21no—
66C4×M16×C47C2211yescharacteristic
67C2×#10625337C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625336C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625324—21no—
72C2×C8×D47—21no—
73#13094018—21no—
74#13094016C2×C411yescharacteristic
75#13094012C2×C411yescharacteristic
76#13094007—21no—
77#13094003C2×C411yescharacteristic
78#13093998C2×C411yescharacteristic
79C4×Dic94C2311yescharacteristic
80C4×D188C2311yescharacteristic
81C4×C22⋊D47—21no—
82#13093981—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#13093934—21no—
109C4×C47⋊C8—21no—
110C4×C47⋊C8C2311yescharacteristic
111C4×C47⋊C8C2311yescharacteristic
112C4×C47⋊C8C2311yescharacteristic
113C4×C47⋊C8C2311yescharacteristic
114C8×Dic47—21no—
115C22×Q8×C47C2×C411yescharacteristic
116C2×M16×C47—21no—
117C2×M16×C47D411yescharacteristic
118C2×M16×C47D411yescharacteristic
119C4×Q8×C47—21no—
120C4×Q8×C47—21no—
121C4×Q8×C47C2×C411yescharacteristic
122C4×Q8×C47C2×C411yescharacteristic
123C4×Q8×C47C2311yescharacteristic
124C4×Q8×C47C2311yescharacteristic
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47C2×C411yescharacteristic
127C2×C42×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×C47—22no—
146C2×Q8×C47—41no—
147C2×Q8×C47C22×C411yescharacteristic
148C2×Q8×C47C22×C411yescharacteristic
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47C22×C411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47#4111yescharacteristic
154C2×C22⋊D47—21no—
155C2×D188C22×C411yescharacteristic
156C2×C4×D47—21no—
157C2×Dic94C22×C411yescharacteristic
158M16×C47—22no—
159M16×C47—41no—
160M16×C47—42no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
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×C47C22×C411yescharacteristic
173C4⋊C4×C47C22×C411yescharacteristic
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47C22×C411yescharacteristic
179C42×C47C2×D411yescharacteristic
180C42×C47#4111yescharacteristic
181C42×C47C2411yescharacteristic
182C22⋊Dic47—21no—
183#10625337—22no—
184#10625337—41no—
185#10625336—22no—
186#10625336—41no—
187(C2×D47)⋊C4—21no—
188(C2×D47)⋊C4—21no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191C4⋊Dic47C22×C411yescharacteristic
192C4⋊Dic47C22×C411yescharacteristic
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195Dic47⋊C4—21no—
196Dic47⋊C4—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—
207C8×D47—42no—
208C23×C47C4×D411yescharacteristic
209C22×D47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47—42no—
214Q8×C47#13112yes—
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×C47—42no—
225C2×C4×C47—81no—
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C22⋊D47—81no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236D188—21no—
237D188—21no—
238C4×D47—41no—
239C4×D47—41no—
240Dic94—21no—
241Dic94—21no—
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#18012—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—22no—
266C4×C47—21no—
267C4×C47—21no—
268C4×C47—21no—
269C4×C47—41no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47C2×#13111yescharacteristic
273C4×C47#56811yescharacteristic
274C4×C47C2×#13711yescharacteristic
275C4×C47#57511yescharacteristic
276Dic47—41no—
277Dic47—41no—
278Dic47—41no—
279#3108—471no—
280#2969—471no—
281Q8×M16D4711yescharacteristic, complemented
282#1593—471no—
283#1590—471no—
284#1587—471no—
285#1585—471no—
286#1584—471no—
287#1583—471no—
288SD16⋊C8—471no—
289C8×Q16—471no—
290#1574—471no—
291#1495—471no—
292#1479—471no—
293#1453—471no—
294C2×C47—41no—
295C2×C47#293311yescharacteristic
296C2×C47C4×#13711yescharacteristic
297C2×C47#295711yescharacteristic
298D47—81no—
299#529—471no—
300C2×C4×Q8C2×D4711yescharacteristic
301#483—471no—
302#480—471no—
303#479—471no—
304#479—471no—
305#464—471no—
306#464—471no—
307#451—471no—
308#448—471no—
309#446—21no—
310#446—21no—
311#446—471no—
312#446C2×D4711yescharacteristic
313#446C2×D4711yescharacteristic
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—941no—
323#435—941no—
324C8×D4—471no—
325#433—941no—
326#431—471no—
327#431—471no—
328#431—471no—
329#423—21no—
330#423C2×D4711yescharacteristic
331#405—941no—
332C4×M16—21no—
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#1801211yescharacteristic, complemented, Sylow
351C22×Q8C4×D4711yescharacteristic
352C2×Q16—471no—
353C2×Q16—471no—
354C2×SD16—471no—
355C2×SD16—471no—
356C2×M16—21no—
357C2×M16—941no—
358C2×M16C22⋊D4711yescharacteristic
359C2×M16C22⋊D4711yescharacteristic
360C22×C8—941no—
361#126—941no—
362#125—471no—
363#124—471no—
364#123—941no—
365C22⋊Q8—471no—
366#121—471no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—471no—
370C4×Q8C4×D4711yescharacteristic
371C4×Q8C4×D4711yescharacteristic
372C4×Q8C22×D4711yescharacteristic
373C4×Q8C22×D4711yescharacteristic
374C4×D4—471no—
375C4×D4—941no—
376#117—941no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D4711yescharacteristic
379C2×C42—21no—
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
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—471no—
432C2×Q8C2×C4×D4711yescharacteristic
433C2×Q8C2×C4×D4711yescharacteristic
434C2×D4—471no—
435C2×D4—941no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—941no—
439C22×C4C2×C4×D4711yescharacteristic
440C22×C4#1062535011yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442Q16—942no—
443Q16—1881no—
444SD16—942no—
445SD16—1881no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—1882no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—471no—
453C2×C8—471no—
454C2×C8—471no—
455C2×C8—471no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—1882no—
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—471no—
471C4⋊C4—471no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4C2×C4×D4711yescharacteristic
477C4⋊C4C2×C4×D4711yescharacteristic
478C22⋊C4—941no—
479C22⋊C4—941no—
480C22⋊C4—941no—
481C22⋊C4—941no—
482C22⋊C4—941no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—941no—
488C42C2×C4×D4711yescharacteristic
489C42#1062535011yescharacteristic
490C42C2×C22⋊D4711yescharacteristic
491C42C23×D4711yescharacteristic
492C23—941no—
493C23C4×C22⋊D4711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—941no—
499Q8—941no—
500Q8#1309403212yes—
501D4—941no—
502D4—941no—
503D4—3761no—
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—42no—
514C2×C4—81no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
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—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—1881no—
556C4—1881no—
557C4—1881no—
558C4C2×#1309403211yescharacteristic
559C4#1568988911yescharacteristic
560C4C2×#1309405511yescharacteristic
561C4#1568992111yescharacteristic
562C2—41no—
563C2—3761no—
564C2#2025211311yescharacteristic
565C2C4×#1309405511yescharacteristic
566C2#2025230311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingQ8×M16×C47C2
socleC22×C47C2×C4×D4
prime radical#13093998C2×C4
cosocleC42×C47C24
hypercenterQ8×M16D47
hypocenterC47#18012
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #20242295C2semidirect
4. #20242278C2semidirect
5. #20242259C2semidirect
6. #20242243C2semidirect
7. #20242236C2semidirect
8. #20242231C2semidirect
9. #20242198C2semidirect
10. C8×#10625337C2semidirect
11. #20242179C2semidirect
13. #20241783C2semidirect
14. #20241685C2semidirect
15. Q8×M16×C47C2semidirect
281. Q8×M16D47semidirect
350. C47#18012semidirect

Maximal subgroups

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

subgroupindex
#202534682
#202523922
#202422952
#202422782
#202422592
#202422432
#202422362
#202422312
#202421982
C8×#106253372
#202421792
#202418512
#202417832
#202416852
Q8×M16×C472
#1801247