Finite Group Databasebeta

← #26187406

Subgroups — #26187406

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#20252393C211yesmaximal, characteristic
2Q8×#10625329C211yesmaximal, characteristic, complemented
3#20242297C211yesmaximal, characteristic, complemented
4#20242279C211yesmaximal, characteristic, complemented
5#20242260C211yesmaximal, characteristic, complemented
6#20242246C211yesmaximal, characteristic, complemented
7#20242239C211yesmaximal, characteristic, complemented
8#20242232C211yesmaximal, characteristic, complemented
9SD16⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
10Q16×C47⋊C8C211yesmaximal, characteristic, complemented
11#20242177C211yesmaximal, characteristic, complemented
12#20241853C211yesmaximal, characteristic
13#20241783C211yesmaximal, characteristic, complemented
14#20241683C211yesmaximal, characteristic, complemented
15C47×#3108C211yesmaximal, 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#15689176C2211yescharacteristic
30#15689176C2211yescharacteristic
31#15689147—21no—
32#15689147C2211yescharacteristic
33#15689079—21no—
34C4×#10625329—21no—
35C4×#10625329C2211yescharacteristic
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—
69C2×#10625329D411yescharacteristic
70C2×#10625329D411yescharacteristic
71C22×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)—21no—
79C4⋊(C47⋊C8)—21no—
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—
88#13093935—21no—
89#13093935—21no—
90#13093935—21no—
91#13093935D411yescharacteristic
92#13093935D411yescharacteristic
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—21no—
145C2×Q8×C47—22no—
146C2×Q8×C47—41no—
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—41no—
160SD16×C47—22no—
161SD16×C47—41no—
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—22no—
194#10625329—41no—
195#10625329—42no—
196#10625329—42no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—42no—
204C2×C47⋊C8C2×D411yescharacteristic
205C2×C47⋊C8C2×D411yescharacteristic
206C2×C47⋊C8C2×D411yescharacteristic
207C2×C47⋊C8C2×D411yescharacteristic
208C23×C47—21no—
209C23×C47C4×D411yescharacteristic
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47—21no—
215Q8×C47—42no—
216Q8×C47#13112yes—
217D4×C47—21no—
218D4×C47—21no—
219D4×C47—81no—
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—21no—
231C2×C4×C47—41no—
232C2×C4×C47—42no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—81no—
236C2×C4×C47C4×D411yescharacteristic
237C2×C4×C47C4×D411yescharacteristic
238C2×C4×C47C4×D411yescharacteristic
239C2×C4×C47C23×C411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C8×C47—21no—
243C8×C47—21no—
244C8×C47—21no—
245C8×C47—21no—
246C47⋊C8—21no—
247C47⋊C8—21no—
248C47⋊C8—21no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8—41no—
254C47⋊C8—41no—
255C47⋊C8#13712yes—
256#18012—471nomaximal, Sylow
257C22×C47—21no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47C2×C4×D411yescharacteristic
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×C47—41no—
273C4×C47—41no—
274C4×C47—41no—
275C4×C47C2×#13111yescharacteristic
276C4×C47#56811yescharacteristic
277C4×C47C2×#13711yescharacteristic
278C4×C47#57511yescharacteristic
279#3108D4711yescharacteristic
280#2969—471no—
281Q8×M16—471no—
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—81no—
296C2×C47#293311yescharacteristic
297C2×C47C4×#13711yescharacteristic
298C2×C47#295711yescharacteristic
299#529C2×D4711yescharacteristic
300C2×C4×Q8C2×D4711yescharacteristic
301#483Dic4711yescharacteristic
302#480Dic4711yescharacteristic
303#479Dic4711yescharacteristic
304#479Dic4711yescharacteristic
305#464Dic4711yescharacteristic
306#464Dic4711yescharacteristic
307#451Dic4711yescharacteristic
308#448Dic4711yescharacteristic
309#446—471no—
310#446—471no—
311#446—471no—
312#446—941no—
313#446—941no—
314C8×Q8—471no—
315C8×Q8—471no—
316C8×Q8—471no—
317C4×Q16C2×D4711yescharacteristic
318C4×Q16C2×D4711yescharacteristic
319C4×SD16C2×D4711yescharacteristic
320C4×SD16C2×D4711yescharacteristic
321#436—471no—
322#436—941no—
323#435—941no—
324C8×D4—471no—
325#433—941no—
326#431—471no—
327#431—471no—
328#431C2×D4711yescharacteristic
329#423—471no—
330#423—941no—
331#405—941no—
332C4×M16—471no—
333C4×M16—941no—
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×Q8C2×Dic4711yescharacteristic
352C2×Q16C2×Dic4711yescharacteristic
353C2×Q16C2×Dic4711yescharacteristic
354C2×SD16C2×Dic4711yescharacteristic
355C2×SD16C2×Dic4711yescharacteristic
356C2×M16—471no—
357C2×M16—471no—
358C2×M16—941no—
359C2×M16—941no—
360C22×C8—941no—
361#126—21no—
362#125C2×Dic4711yescharacteristic
363#124C2×Dic4711yescharacteristic
364#123—21no—
365C22⋊Q8C2×Dic4711yescharacteristic
366#121C2×Dic4711yescharacteristic
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8C2×Dic4711yescharacteristic
370C4×Q8C2×Dic4711yescharacteristic
371C4×Q8C22×D4711yescharacteristic
372C4×Q8C22×D4711yescharacteristic
373C4×Q8C22×D4711yescharacteristic
374C4×D4—21no—
375C4×D4C22×D4711yescharacteristic
376#117—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C2×Dic4711yescharacteristic
379C2×C42—21no—
380C2×C42C22×D4711yescharacteristic
381#107C2×Dic4711yescharacteristic
382#107C2×Dic4711yescharacteristic
383#106C2×Dic4711yescharacteristic
384#106C2×Dic4711yescharacteristic
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—
396C4⋊C8—941no—
397C4⋊C8C2×Dic4711yescharacteristic
398C4⋊C8C2×Dic4711yescharacteristic
399#103C2×Dic4711yescharacteristic
400#103C2×Dic4711yescharacteristic
401#103C2×Dic4711yescharacteristic
402#103C2×Dic4711yescharacteristic
403#103C2×Dic4711yescharacteristic
404#103C2×Dic4711yescharacteristic
405#102C2×Dic4711yescharacteristic
406#102C2×Dic4711yescharacteristic
407C22⋊C8—941no—
408C22⋊C8—941no—
409C22⋊C8—941no—
410C22⋊C8—941no—
411C22⋊C8C2×Dic4711yescharacteristic
412C22⋊C8C2×Dic4711yescharacteristic
413#97—471no—
414#97—471no—
415#97—941no—
416#97—941no—
417#97—941no—
418C4×C8—471no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—471no—
422C4×C8—941no—
423C4×C8—941no—
424C4×C8—941no—
425C4×C8—941no—
426C4×C8C22×D4711yescharacteristic
427C4×C8C22×D4711yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8C22×Dic4711yescharacteristic
432C2×Q8C22×Dic4711yescharacteristic
433C2×Q8C22×Dic4711yescharacteristic
434C2×D4—21no—
435C2×D4C22×Dic4711yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4D4×D4711yescharacteristic
440C22×C4#1062535211yescharacteristic
441C22×C4C22×Dic4711yescharacteristic
442Q16—22no—
443Q16—41no—
444SD16—22no—
445SD16—41no—
446M16—942no—
447M16—1882no—
448M16—1881no—
449M16—1882no—
450C2×C8—471no—
451C2×C8—471no—
452C2×C8—471no—
453C2×C8—471no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—1882no—
461C2×C8C22×Dic4711yescharacteristic
462C2×C8C22×Dic4711yescharacteristic
463C2×C8C22×Dic4711yescharacteristic
464C2×C8C22×Dic4711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4C22×Dic4711yescharacteristic
475C4⋊C4C22×Dic4711yescharacteristic
476C4⋊C4C22×Dic4711yescharacteristic
477C4⋊C4C22×Dic4711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42D4×D4711yescharacteristic
489C42#1062535211yescharacteristic
490C42C22×Dic4711yescharacteristic
491C42C23×D4711yescharacteristic
492C23—21no—
493C23D4×Dic4711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—42no—
500Q8#1309403112yes—
501D4—21no—
502D4—21no—
503D4—81no—
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—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4D4×Dic4711yescharacteristic
521C2×C4D4×Dic4711yescharacteristic
522C2×C4D4×Dic4711yescharacteristic
523C2×C4C23×Dic4711yescharacteristic
524C2×C4C2×D4×D4711yescharacteristic
525C2×C4C2×#1062535211yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—472no—
531C8—941no—
532C8—941no—
533C8—941no—
534C8—1881no—
535C8—1881no—
536C8—1881no—
537C8—1881no—
538C8—1881no—
539C8—1881no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×Dic4711yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4C2×#1309403111yescharacteristic
559C4#1568989011yescharacteristic
560C4#137×D4711yescharacteristic
561C4#1568992211yescharacteristic
562C2—41no—
563C2—81no—
564C2#2025211511yescharacteristic
565C2#137×Dic4711yescharacteristic
566C2#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#18012
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. Q8×#10625329C2semidirect
3. #20242297C2semidirect
4. #20242279C2semidirect
5. #20242260C2semidirect
6. #20242246C2semidirect
7. #20242239C2semidirect
8. #20242232C2semidirect
9. SD16⋊(C47⋊C8)C2semidirect
10. Q16×C47⋊C8C2semidirect
11. #20242177C2semidirect
13. #20241783C2semidirect
14. #20241683C2semidirect
350. C47#18012semidirect

Maximal subgroups

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

subgroupindex
#202523932
Q8×#106253292
#202422972
#202422792
#202422602
#202422462
#202422392
#202422322
SD16⋊(C47⋊C8)2
Q16×C47⋊C82
#202421772
#202418532
#202417832
#202416832
C47×#31082
#1801247