Finite Group Databasebeta

← #26186355

Subgroups — #26186355

Order 12032.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252537C211yesmaximal, characteristic
2#20252469C211yesmaximal, characteristic, complemented
3#20242267C211yesmaximal, characteristic, complemented
4#20242260C211yesmaximal, characteristic, complemented
5#20242239C211yesmaximal, characteristic, complemented
6#20242232C211yesmaximal, characteristic, complemented
7#20242207C211yesmaximal, characteristic, complemented
8SD16⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
9Q16×C47⋊C8C211yesmaximal, characteristic, complemented
10SD16×C47⋊C8C211yesmaximal, characteristic, complemented
11#20242177C211yesmaximal, characteristic, complemented
12#20241817C211yesmaximal, characteristic, complemented
13#20241796C211yesmaximal, characteristic
14#20241688C211yesmaximal, characteristic, complemented
15C47×#3108C211yesmaximal, characteristic
16#15689254C2211yescharacteristic
17Q8×C47⋊C8—21no—
18Q8×C47⋊C8—21no—
19Q8×C47⋊C8C2211yescharacteristic
20Q8×C47⋊C8C2211yescharacteristic
21Q8×C47⋊C8C2211yescharacteristic
22Q8×C47⋊C8C2211yescharacteristic
23Q8×C47⋊C8C2211yescharacteristic
24#15689201—21no—
25#15689201C2211yescharacteristic
26#15689195—21no—
27D4×C47⋊C8C2211yescharacteristic
28#15689184—21no—
29#15689176—21no—
30#15689176—21no—
31#15689176C2211yescharacteristic
32#15689176C2211yescharacteristic
33#15689176C2211yescharacteristic
34#15689150—21no—
35#15689147C2211yescharacteristic
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#15688845C2211yescharacteristic
49#15688845C2211yescharacteristic
50#15688844C2211yescharacteristic
51C8×C47⋊C8C2211yescharacteristic
52C47×#529C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#483C411yescharacteristic
55C47×#480C411yescharacteristic
56C47×#479C411yescharacteristic
57C47×#479C411yescharacteristic
58C47×#464C411yescharacteristic
59C47×#464C411yescharacteristic
60C47×#451C411yescharacteristic
61C47×#448C411yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C4×Q16×C47C2211yescharacteristic
64C4×SD16×C47C2211yescharacteristic
65C4×SD16×C47C2211yescharacteristic
66C47×#431C2211yescharacteristic
67C2×#10625329—21no—
68C2×#10625329—21no—
69C22×C47⋊C8—21no—
70C4⋊(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)—21no—
88C22⋊(C47⋊C8)D411yescharacteristic
89C22⋊(C47⋊C8)D411yescharacteristic
90#13093935—21no—
91C4×C47⋊C8—21no—
92C4×C47⋊C8—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
101C22×Q8×C47C2×C411yescharacteristic
102C2×Q16×C47C2×C411yescharacteristic
103C2×Q16×C47C2×C411yescharacteristic
104C2×SD16×C47C2×C411yescharacteristic
105C2×SD16×C47C2×C411yescharacteristic
106#126×C47—21no—
107#125×C47C2×C411yescharacteristic
108#124×C47C2×C411yescharacteristic
109#123×C47—21no—
110C22⋊Q8×C47C2×C411yescharacteristic
111#121×C47C2×C411yescharacteristic
112C4×Q8×C47—21no—
113C4×Q8×C47—21no—
114C4×Q8×C47C2×C411yescharacteristic
115C4×Q8×C47C2×C411yescharacteristic
116C4×Q8×C47C2311yescharacteristic
117C4×Q8×C47C2311yescharacteristic
118C4×Q8×C47C2311yescharacteristic
119C4×D4×C47—21no—
120C4×D4×C47C2311yescharacteristic
121#117×C47—21no—
122C2×C4⋊C4×C47—21no—
123C2×C4⋊C4×C47C2×C411yescharacteristic
124C2×C42×C47—21no—
125C2×C42×C47C2311yescharacteristic
126#107×C47C2×C411yescharacteristic
127#107×C47C2×C411yescharacteristic
128#106×C47C2×C411yescharacteristic
129#106×C47C2×C411yescharacteristic
130C4⋊C8×C47C2×C411yescharacteristic
131C4⋊C8×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
140C22⋊C8×C47C2×C411yescharacteristic
141C22⋊C8×C47C2×C411yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47—22no—
145C2×Q8×C47—22no—
146C2×Q8×C47—21no—
147C2×Q8×C47C22×C411yescharacteristic
148C2×Q8×C47C22×C411yescharacteristic
149C2×Q8×C47C22×C411yescharacteristic
150C2×D4×C47—21no—
151C2×D4×C47C22×C411yescharacteristic
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47C22×C411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47#4111yescharacteristic
158Q16×C47—22no—
159Q16×C47—22no—
160SD16×C47—22no—
161SD16×C47—22no—
162C2×C8×C47C22×C411yescharacteristic
163C2×C8×C47C22×C411yescharacteristic
164C2×C8×C47C22×C411yescharacteristic
165C2×C8×C47C22×C411yescharacteristic
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—
184C42×C47—21no—
185C42×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—81no—
202C2×C47⋊C8C2×D411yescharacteristic
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205C2×C47⋊C8C2×D411yescharacteristic
206C23×C47—21no—
207C23×C47C4×D411yescharacteristic
208Q8×C47—24no—
209Q8×C47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47#13112yes—
214Q8×C47#13112yes—
215D4×C47—21no—
216D4×C47—21no—
217D4×C47—81no—
218C2×C4×C47—24no—
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×C47—81no—
234C2×C4×C47C4×D411yescharacteristic
235C2×C4×C47C4×D411yescharacteristic
236C2×C4×C47C4×D411yescharacteristic
237C2×C4×C47C23×C411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C8×C47—21no—
241C8×C47—21no—
242C8×C47—21no—
243C8×C47—21no—
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—
253C47⋊C8—41no—
254#17919—471nomaximal, Sylow
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—21no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47C2×C4×D411yescharacteristic
262C4×C47—22no—
263C4×C47—22no—
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×C47—41no—
273C4×C47C2×#13111yescharacteristic
274C4×C47C2×#13111yescharacteristic
275C4×C47#57711yescharacteristic
276C4×C47#57811yescharacteristic
277#3108D4711yescharacteristic
278#2993—471no—
279#2982—471no—
280#1588—471no—
281#1587—471no—
282#1584—471no—
283#1583—471no—
284#1579—471no—
285SD16⋊C8—471no—
286C8×Q16—471no—
287C8×SD16—471no—
288#1574—471no—
289#1487—471no—
290#1482—471no—
291#1454—471no—
292C2×C47—41no—
293C2×C47—81no—
294C2×C47#292911yescharacteristic
295C2×C47#294711yescharacteristic
296C2×C47#295711yescharacteristic
297#529C2×D4711yescharacteristic
298C2×C4×Q8C2×D4711yescharacteristic
299#483Dic4711yescharacteristic
300#480Dic4711yescharacteristic
301#479Dic4711yescharacteristic
302#479Dic4711yescharacteristic
303#464Dic4711yescharacteristic
304#464Dic4711yescharacteristic
305#451Dic4711yescharacteristic
306#448Dic4711yescharacteristic
307#446—471no—
308C8×Q8—471no—
309C8×Q8—471no—
310C8×Q8—471no—
311C8×Q8—471no—
312C8×Q8—471no—
313C8×Q8—941no—
314C8×Q8—941no—
315C4×Q16C2×D4711yescharacteristic
316C4×Q16C2×D4711yescharacteristic
317C4×SD16C2×D4711yescharacteristic
318C4×SD16C2×D4711yescharacteristic
319#436—471no—
320#436—941no—
321#435—941no—
322C8×D4—471no—
323#433—941no—
324#431—471no—
325#431—471no—
326#431—471no—
327#431—941no—
328#431—941no—
329#431C2×D4711yescharacteristic
330#424—941no—
331#423—471no—
332#335—471no—
333#335—471no—
334#334—471no—
335#334—471no—
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8—471no—
342D4⋊C8—471no—
343D4⋊C8—471no—
344#322—471no—
345#322—471no—
346#322—471no—
347C82—471no—
348C47#1791911yescharacteristic, complemented, Sylow
349C22×Q8C2×Dic4711yescharacteristic
350C2×Q16C2×Dic4711yescharacteristic
351C2×Q16C2×Dic4711yescharacteristic
352C2×SD16C2×Dic4711yescharacteristic
353C2×SD16C2×Dic4711yescharacteristic
354C2×M16—941no—
355C2×M16—941no—
356C22×C8—941no—
357#126—21no—
358#125C2×Dic4711yescharacteristic
359#124C2×Dic4711yescharacteristic
360#123—21no—
361C22⋊Q8C2×Dic4711yescharacteristic
362#121C2×Dic4711yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8C2×Dic4711yescharacteristic
366C4×Q8C2×Dic4711yescharacteristic
367C4×Q8C22×D4711yescharacteristic
368C4×Q8C22×D4711yescharacteristic
369C4×Q8C22×D4711yescharacteristic
370C4×D4—21no—
371C4×D4C22×D4711yescharacteristic
372#117—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C2×Dic4711yescharacteristic
375C2×C42—21no—
376C2×C42C22×D4711yescharacteristic
377#107C2×Dic4711yescharacteristic
378#107C2×Dic4711yescharacteristic
379#106C2×Dic4711yescharacteristic
380#106C2×Dic4711yescharacteristic
381C4⋊C8—471no—
382C4⋊C8—471no—
383C4⋊C8—471no—
384C4⋊C8—471no—
385C4⋊C8—471no—
386C4⋊C8—471no—
387C4⋊C8—941no—
388C4⋊C8—941no—
389C4⋊C8—941no—
390C4⋊C8—941no—
391C4⋊C8—941no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8C2×Dic4711yescharacteristic
396C4⋊C8C2×Dic4711yescharacteristic
397#103C2×Dic4711yescharacteristic
398#103C2×Dic4711yescharacteristic
399#103C2×Dic4711yescharacteristic
400#103C2×Dic4711yescharacteristic
401#103C2×Dic4711yescharacteristic
402#103C2×Dic4711yescharacteristic
403#102C2×Dic4711yescharacteristic
404#102C2×Dic4711yescharacteristic
405C22⋊C8—471no—
406C22⋊C8—471no—
407C22⋊C8—941no—
408C22⋊C8—941no—
409C22⋊C8—941no—
410C22⋊C8—941no—
411C22⋊C8C2×Dic4711yescharacteristic
412C22⋊C8C2×Dic4711yescharacteristic
413#97—941no—
414C4×C8—471no—
415C4×C8—471no—
416C4×C8—471no—
417C4×C8—471no—
418C4×C8—941no—
419C4×C8—941no—
420C4×C8—941no—
421C4×C8—941no—
422C4×C8—941no—
423C4×C8—941no—
424C4×C8C22×D4711yescharacteristic
425C4×C8C22×D4711yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8C22×Dic4711yescharacteristic
430C2×Q8C22×Dic4711yescharacteristic
431C2×Q8C22×Dic4711yescharacteristic
432C2×D4—21no—
433C2×D4C22×Dic4711yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4D4×D4711yescharacteristic
438C22×C4#1062535211yescharacteristic
439C22×C4C22×Dic4711yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—22no—
443SD16—22no—
444M16—3761no—
445M16—3761no—
446C2×C8—471no—
447C2×C8—471no—
448C2×C8—471no—
449C2×C8—471no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—3761no—
457C2×C8C22×Dic4711yescharacteristic
458C2×C8C22×Dic4711yescharacteristic
459C2×C8C22×Dic4711yescharacteristic
460C2×C8C22×Dic4711yescharacteristic
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—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42D4×D4711yescharacteristic
485C42#1062535211yescharacteristic
486C42C22×Dic4711yescharacteristic
487C42C23×D4711yescharacteristic
488C23—21no—
489C23D4×Dic4711yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8#1309403112yes—
496Q8#1309403112yes—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—24no—
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—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4D4×Dic4711yescharacteristic
517C2×C4D4×Dic4711yescharacteristic
518C2×C4D4×Dic4711yescharacteristic
519C2×C4C23×Dic4711yescharacteristic
520C2×C4C2×D4×D4711yescharacteristic
521C2×C4C2×#1062535211yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—941no—
527C8—941no—
528C8—941no—
529C8—941no—
530C8—1881no—
531C8—1881no—
532C8—1881no—
533C8—1881no—
534C8—1881no—
535C8—1881no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×D4×Dic4711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4C2×#1309403111yescharacteristic
555C4C2×#1309403111yescharacteristic
556C4#1568993211yescharacteristic
557C4#1568993911yescharacteristic
558C2—41no—
559C2—81no—
560C2#2025208311yescharacteristic
561C2#2025223211yescharacteristic
562C2#2025230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#3108C2
socleC22×C47C2×C4×D4
prime radical#121×C47C2×C4
cosocleC42×C47C24
hypercenter#3108D47
hypocenterC47#17919
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3108

Complements

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

NH ≅ G/Ntype
2. #20252469C2semidirect
3. #20242267C2semidirect
4. #20242260C2semidirect
5. #20242239C2semidirect
6. #20242232C2semidirect
7. #20242207C2semidirect
8. SD16⋊(C47⋊C8)C2semidirect
9. Q16×C47⋊C8C2semidirect
10. SD16×C47⋊C8C2semidirect
11. #20242177C2semidirect
12. #20241817C2semidirect
14. #20241688C2semidirect
348. C47#17919semidirect

Maximal subgroups

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

subgroupindex
#202525372
#202524692
#202422672
#202422602
#202422392
#202422322
#202422072
SD16⋊(C47⋊C8)2
Q16×C47⋊C82
SD16×C47⋊C82
#202421772
#202418172
#202417962
#202416882
C47×#31082
#1791947