Finite Group Databasebeta

← #26186925

Subgroups — #26186925

Order 12032.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252537C211yesmaximal, characteristic
2#20252483C211yesmaximal, characteristic, complemented
3#20242297C211yesmaximal, characteristic, complemented
4#20242291C211yesmaximal, characteristic, complemented
5#20242285C211yesmaximal, characteristic, complemented
6#20242279C211yesmaximal, characteristic, complemented
7#20242207C211yesmaximal, characteristic, complemented
8SD16⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
9Q16×C47⋊C8C211yesmaximal, characteristic, complemented
10SD16×C47⋊C8C211yesmaximal, characteristic, complemented
11#20242155C211yesmaximal, characteristic, complemented
12#20241848C211yesmaximal, characteristic
13#20241843C211yesmaximal, characteristic, complemented
14M16×C47⋊C8C211yesmaximal, characteristic, complemented
15C47×#3172C211yesmaximal, characteristic
16#15689254—21no—
17#15689254—21no—
18#15689254C2211yescharacteristic
19#15689254C2211yescharacteristic
20#15689254C2211yescharacteristic
21Q8×C47⋊C8C2211yescharacteristic
22Q8×C47⋊C8C2211yescharacteristic
23Q8×C47⋊C8C2211yescharacteristic
24#15689201—21no—
25#15689201C2211yescharacteristic
26#15689195—21no—
27D4×C47⋊C8C2211yescharacteristic
28#15689184—21no—
29#15689184—21no—
30#15689184—21no—
31#15689180C2211yescharacteristic
32#15689176C2211yescharacteristic
33#15689150—21no—
34#15689147C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36#15688876C2211yescharacteristic
37#15688876C2211yescharacteristic
38#15688874C2211yescharacteristic
39#15688874C2211yescharacteristic
40Q8⋊(C47⋊C8)C2211yescharacteristic
41Q8⋊(C47⋊C8)C2211yescharacteristic
42Q8⋊(C47⋊C8)C2211yescharacteristic
43Q8⋊(C47⋊C8)C2211yescharacteristic
44Q8⋊(C47⋊C8)C2211yescharacteristic
45Q8⋊(C47⋊C8)C2211yescharacteristic
46D4⋊(C47⋊C8)C2211yescharacteristic
47D4⋊(C47⋊C8)C2211yescharacteristic
48#15688844C2211yescharacteristic
49#15688844C2211yescharacteristic
50C8×C47⋊C8C2211yescharacteristic
51C8×C47⋊C8C2211yescharacteristic
52C47×#533C2211yescharacteristic
53C47×#529C2211yescharacteristic
54C47×#501C411yescharacteristic
55C47×#497C411yescharacteristic
56C47×#491C411yescharacteristic
57C47×#489C411yescharacteristic
58C47×#470C411yescharacteristic
59C47×#470C411yescharacteristic
60C47×#468C411yescharacteristic
61C47×#468C411yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C4×Q16×C47C2211yescharacteristic
64C4×SD16×C47C2211yescharacteristic
65C4×SD16×C47C2211yescharacteristic
66C4×M16×C47C2211yescharacteristic
67C2×#10625329—21no—
68C2×#10625329—21no—
69C22×C47⋊C8—21no—
70C22×C47⋊C8—21no—
71C4⋊(C47⋊C8)—21no—
72C4⋊(C47⋊C8)—21no—
73C4⋊(C47⋊C8)—21no—
74C4⋊(C47⋊C8)—21no—
75C4⋊(C47⋊C8)—21no—
76C4⋊(C47⋊C8)—21no—
77C4⋊(C47⋊C8)—21no—
78C4⋊(C47⋊C8)D411yescharacteristic
79C4⋊(C47⋊C8)D411yescharacteristic
80C4⋊(C47⋊C8)C2311yescharacteristic
81C4⋊(C47⋊C8)C2311yescharacteristic
82C4⋊(C47⋊C8)C2311yescharacteristic
83C4⋊(C47⋊C8)C2311yescharacteristic
84C22⋊(C47⋊C8)—21no—
85C22⋊(C47⋊C8)—21no—
86C22⋊(C47⋊C8)—21no—
87C22⋊(C47⋊C8)D411yescharacteristic
88C22⋊(C47⋊C8)D411yescharacteristic
89#13093935—21no—
90#13093935—21no—
91#13093935—21no—
92#13093935—21no—
93C4×C47⋊C8—21no—
94C4×C47⋊C8—21no—
95C4×C47⋊C8—21no—
96C4×C47⋊C8—21no—
97C4×C47⋊C8C2311yescharacteristic
98C4×C47⋊C8C2311yescharacteristic
99C4×C47⋊C8C2311yescharacteristic
100C4×C47⋊C8C2311yescharacteristic
101C2×Q16×C47C2×C411yescharacteristic
102C2×Q16×C47C2×C411yescharacteristic
103C2×SD16×C47C2×C411yescharacteristic
104C2×SD16×C47C2×C411yescharacteristic
105C2×M16×C47C2×C411yescharacteristic
106C2×M16×C47C2×C411yescharacteristic
107C4⋊Q8×C47C2×C411yescharacteristic
108C4⋊Q8×C47C2×C411yescharacteristic
109#126×C47—21no—
110#125×C47—21no—
111#125×C47C2×C411yescharacteristic
112#124×C47C2×C411yescharacteristic
113#123×C47—21no—
114C22⋊Q8×C47—21no—
115C22⋊Q8×C47C2×C411yescharacteristic
116C22⋊Q8×C47C2×C411yescharacteristic
117C22⋊Q8×C47C2×C411yescharacteristic
118#121×C47C2×C411yescharacteristic
119C4×Q8×C47—21no—
120C4×Q8×C47C2311yescharacteristic
121C4×Q8×C47C2311yescharacteristic
122C4×Q8×C47C2311yescharacteristic
123C4×D4×C47—21no—
124C4×D4×C47C2311yescharacteristic
125#117×C47—21no—
126#117×C47—21no—
127C2×C42×C47C2311yescharacteristic
128#107×C47C2×C411yescharacteristic
129#107×C47C2×C411yescharacteristic
130#106×C47C2×C411yescharacteristic
131#106×C47C2×C411yescharacteristic
132#103×C47C2×C411yescharacteristic
133#103×C47C2×C411yescharacteristic
134#103×C47C2×C411yescharacteristic
135#103×C47C2×C411yescharacteristic
136#103×C47C2×C411yescharacteristic
137#103×C47C2×C411yescharacteristic
138#102×C47C2×C411yescharacteristic
139#102×C47C2×C411yescharacteristic
140#97×C47C2×C411yescharacteristic
141#97×C47C2×C411yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47—21no—
145C2×Q8×C47C22×C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147C2×Q8×C47C22×C411yescharacteristic
148C2×D4×C47—21no—
149C2×D4×C47C22×C411yescharacteristic
150C22×C4×C47—21no—
151C22×C4×C47C22×C411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47#4111yescharacteristic
154Q16×C47—22no—
155Q16×C47—22no—
156SD16×C47—22no—
157SD16×C47—22no—
158M16×C47—22no—
159M16×C47—22no—
160C2×C8×C47C22×C411yescharacteristic
161C2×C8×C47C22×C411yescharacteristic
162C2×C8×C47C22×C411yescharacteristic
163C2×C8×C47C22×C411yescharacteristic
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
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×C47—21no—
175C4⋊C4×C47C22×C411yescharacteristic
176C4⋊C4×C47C22×C411yescharacteristic
177C4⋊C4×C47C22×C411yescharacteristic
178C4⋊C4×C47C22×C411yescharacteristic
179C22⋊C4×C47—21no—
180C22⋊C4×C47—21no—
181C22⋊C4×C47—21no—
182C22⋊C4×C47—21no—
183C22⋊C4×C47—21no—
184C22⋊C4×C47—21no—
185C22⋊C4×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47C22×C411yescharacteristic
190C42×C47C2×D411yescharacteristic
191C42×C47#4111yescharacteristic
192C42×C47C2411yescharacteristic
193#10625329—81no—
194#10625329—81no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—42no—
202C2×C47⋊C8—81no—
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205C2×C47⋊C8C2×D411yescharacteristic
206C2×C47⋊C8C2×D411yescharacteristic
207C23×C47—21no—
208C23×C47C4×D411yescharacteristic
209Q8×C47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47—21no—
215Q8×C47—81no—
216D4×C47—21no—
217D4×C47—21no—
218D4×C47—81no—
219C2×C4×C47—21no—
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×C47C4×D411yescharacteristic
234C2×C4×C47C4×D411yescharacteristic
235C2×C4×C47C4×D411yescharacteristic
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C8×C47—21no—
240C8×C47—21no—
241C8×C47#13112yes—
242C8×C47#13112yes—
243C47⋊C8—22no—
244C47⋊C8—21no—
245C47⋊C8—21no—
246C47⋊C8—21no—
247C47⋊C8—21no—
248C47⋊C8—41no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253#17969—471nomaximal, Sylow
254C22×C47—21no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47C2×C4×D411yescharacteristic
261C4×C47—21no—
262C4×C47—21no—
263C4×C47—21no—
264C4×C47—21no—
265C4×C47—41no—
266C4×C47—41no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47C2×#13111yescharacteristic
273C4×C47C2×#13111yescharacteristic
274C4×C47#57711yescharacteristic
275C4×C47#57811yescharacteristic
276#3172D4711yescharacteristic
277#2993—471no—
278#2985—471no—
279#1593—471no—
280#1592—471no—
281#1591—471no—
282#1590—471no—
283#1579—471no—
284SD16⋊C8—471no—
285C8×Q16—471no—
286C8×SD16—471no—
287#1569—471no—
288#1494—471no—
289#1493—471no—
290C8×M16—471no—
291C2×C47—41no—
292C2×C47—81no—
293C2×C47#293111yescharacteristic
294C2×C47#294711yescharacteristic
295C2×C47#295611yescharacteristic
296#533C2×D4711yescharacteristic
297#529C2×D4711yescharacteristic
298#501Dic4711yescharacteristic
299#497Dic4711yescharacteristic
300#491Dic4711yescharacteristic
301#489Dic4711yescharacteristic
302#470Dic4711yescharacteristic
303#470Dic4711yescharacteristic
304#468Dic4711yescharacteristic
305#468Dic4711yescharacteristic
306#446—471no—
307#446—471no—
308#446—471no—
309#446—941no—
310#446—941no—
311C8×Q8—471no—
312C8×Q8—471no—
313C8×Q8—471no—
314C4×Q16C2×D4711yescharacteristic
315C4×Q16C2×D4711yescharacteristic
316C4×SD16C2×D4711yescharacteristic
317C4×SD16C2×D4711yescharacteristic
318#436—471no—
319#436—941no—
320#435—941no—
321C8×D4—471no—
322#433—941no—
323#433—941no—
324#433—941no—
325#432—471no—
326#431—471no—
327#424—941no—
328#423—471no—
329C4×M16C2×D4711yescharacteristic
330C2×C4×C8—471no—
331#335—471no—
332#335—471no—
333#334—471no—
334#334—471no—
335Q8⋊C8—471no—
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341D4⋊C8—471no—
342D4⋊C8—471no—
343#322—471no—
344#322—471no—
345C82—471no—
346C82—471no—
347C47#1796911yescharacteristic, complemented, Sylow
348C2×Q16C2×Dic4711yescharacteristic
349C2×Q16C2×Dic4711yescharacteristic
350C2×SD16C2×Dic4711yescharacteristic
351C2×SD16C2×Dic4711yescharacteristic
352C2×M16—941no—
353C2×M16—941no—
354C2×M16C2×Dic4711yescharacteristic
355C2×M16C2×Dic4711yescharacteristic
356C22×C8—941no—
357C22×C8—941no—
358C4⋊Q8C2×Dic4711yescharacteristic
359C4⋊Q8C2×Dic4711yescharacteristic
360#126—21no—
361#125—21no—
362#125C2×Dic4711yescharacteristic
363#124C2×Dic4711yescharacteristic
364#123—21no—
365C22⋊Q8—21no—
366C22⋊Q8C2×Dic4711yescharacteristic
367C22⋊Q8C2×Dic4711yescharacteristic
368C22⋊Q8C2×Dic4711yescharacteristic
369#121C2×Dic4711yescharacteristic
370C4×Q8—21no—
371C4×Q8C22×D4711yescharacteristic
372C4×Q8C22×D4711yescharacteristic
373C4×Q8C22×D4711yescharacteristic
374C4×D4—21no—
375C4×D4C22×D4711yescharacteristic
376#117—21no—
377#117—21no—
378C2×C42C22×D4711yescharacteristic
379#107C2×Dic4711yescharacteristic
380#107C2×Dic4711yescharacteristic
381#106C2×Dic4711yescharacteristic
382#106C2×Dic4711yescharacteristic
383C4⋊C8—471no—
384C4⋊C8—471no—
385C4⋊C8—471no—
386C4⋊C8—471no—
387C4⋊C8—471no—
388C4⋊C8—471no—
389C4⋊C8—941no—
390C4⋊C8—941no—
391C4⋊C8—941no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8—941no—
396#103C2×Dic4711yescharacteristic
397#103C2×Dic4711yescharacteristic
398#103C2×Dic4711yescharacteristic
399#103C2×Dic4711yescharacteristic
400#103C2×Dic4711yescharacteristic
401#103C2×Dic4711yescharacteristic
402#102C2×Dic4711yescharacteristic
403#102C2×Dic4711yescharacteristic
404C22⋊C8—471no—
405C22⋊C8—471no—
406C22⋊C8—941no—
407C22⋊C8—941no—
408C22⋊C8—941no—
409#97—941no—
410#97—941no—
411#97—941no—
412#97—941no—
413#97C2×Dic4711yescharacteristic
414#97C2×Dic4711yescharacteristic
415C4×C8—471no—
416C4×C8—471no—
417C4×C8—471no—
418C4×C8—471no—
419C4×C8—941no—
420C4×C8—941no—
421C4×C8—941no—
422C4×C8—941no—
423C4×C8C22×D4711yescharacteristic
424C4×C8C22×D4711yescharacteristic
425C2×Q8—21no—
426C2×Q8C22×Dic4711yescharacteristic
427C2×Q8C22×Dic4711yescharacteristic
428C2×Q8C22×Dic4711yescharacteristic
429C2×D4—21no—
430C2×D4C22×Dic4711yescharacteristic
431C22×C4—21no—
432C22×C4D4×D4711yescharacteristic
433C22×C4#1062535211yescharacteristic
434C22×C4C22×Dic4711yescharacteristic
435Q16—22no—
436Q16—22no—
437SD16—22no—
438SD16—22no—
439M16—22no—
440M16—22no—
441M16—3761no—
442M16—3761no—
443C2×C8—471no—
444C2×C8—471no—
445C2×C8—471no—
446C2×C8—471no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—3761no—
454C2×C8—1882no—
455C2×C8C22×Dic4711yescharacteristic
456C2×C8C22×Dic4711yescharacteristic
457C2×C8C22×Dic4711yescharacteristic
458C2×C8C22×Dic4711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×Dic4711yescharacteristic
471C4⋊C4C22×Dic4711yescharacteristic
472C4⋊C4C22×Dic4711yescharacteristic
473C4⋊C4C22×Dic4711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42D4×D4711yescharacteristic
485C42#1062535211yescharacteristic
486C42C22×Dic4711yescharacteristic
487C42C23×D4711yescharacteristic
488C23—21no—
489C23D4×Dic4711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—81no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4D4×Dic4711yescharacteristic
515C2×C4D4×Dic4711yescharacteristic
516C2×C4D4×Dic4711yescharacteristic
517C2×C4C23×Dic4711yescharacteristic
518C2×C4C2×D4×D4711yescharacteristic
519C2×C4C2×#1062535211yescharacteristic
520C8—21no—
521C8—21no—
522C8—942no—
523C8—941no—
524C8—941no—
525C8—941no—
526C8—941no—
527C8—1881no—
528C8—1881no—
529C8—1881no—
530C8—1881no—
531C8—1881no—
532C8#1309403112yes—
533C8#1309403112yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×D4×Dic4711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4C2×#1309403111yescharacteristic
553C4C2×#1309403111yescharacteristic
554C4#1568993211yescharacteristic
555C4#1568993911yescharacteristic
556C2—41no—
557C2—81no—
558C2#2025210011yescharacteristic
559C2#2025223211yescharacteristic
560C2#2025229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#3172C2
socleC22×C47C2×C4×D4
prime radical#121×C47C2×C4
cosocleC42×C47C24
hypercenter#3172D47
hypocenterC47#17969
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #20252483C2semidirect
3. #20242297C2semidirect
4. #20242291C2semidirect
5. #20242285C2semidirect
6. #20242279C2semidirect
7. #20242207C2semidirect
8. SD16⋊(C47⋊C8)C2semidirect
9. Q16×C47⋊C8C2semidirect
10. SD16×C47⋊C8C2semidirect
11. #20242155C2semidirect
13. #20241843C2semidirect
14. M16×C47⋊C8C2semidirect
347. C47#17969semidirect

Maximal subgroups

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

subgroupindex
#202525372
#202524832
#202422972
#202422912
#202422852
#202422792
#202422072
SD16⋊(C47⋊C8)2
Q16×C47⋊C82
SD16×C47⋊C82
#202421552
#202418482
#202418432
M16×C47⋊C82
C47×#31722
#1796947