Finite Group Databasebeta

← #26184870

Subgroups — #26184870

Order 12032.

1178 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic
2#20252020C211yesmaximal, characteristic
3#20252016C211yesmaximal, characteristic
4#20242103C214yesmaximal, complemented
5#20242074C211yesmaximal, characteristic, complemented
6#20242073C211yesmaximal, characteristic, complemented
7#20242072C211yesmaximal, characteristic, complemented
8#20242058C214yesmaximal, complemented
9C47×#1549C211yesmaximal, characteristic, complemented
10C22×Q8×D47—21no—
11C2×#13094024C2211yescharacteristic
12C2×#13094023—21no—
13C2×#13094023C412yes—
14C2×#13094022—21no—
15C4⋊Q8×D47—21no—
16C4⋊Q8×D47—21no—
17C4⋊Q8×D47—21no—
18C4⋊Q8×D47—21no—
19C4⋊Q8×D47C412yes—
20C4⋊Q8×D47C412yes—
21C4⋊Q8×D47C2212yes—
22C4⋊Q8×D47C2212yes—
23C2×C4⋊C4×D47—21no—
24C2×C4⋊C4×D47C412yes—
25C2×C42×D47C2211yescharacteristic
26#15689179—21no—
27#15689179C2212yes—
28#15689179C2212yes—
29#15689177—21no—
30#15689176C2211yescharacteristic
31#15689175—21no—
32#15689175C2212yes—
33#15689174—21no—
34#15689147C2211yescharacteristic
35#15689145C2212yes—
36#15689086—21no—
37#15689085—21no—
38M16×Dic47—21no—
39C4×#10625324—21no—
40#15688870C2214yescomplemented
41#15688869C2214yescomplemented
42#15688868C2214yescomplemented
43C2×C4⋊Q8×C47C2211yescharacteristic
44C47×#431C2211yescharacteristic
45C47×#423C2211yescharacteristic
46C47×#332C2214yescomplemented
47C22×Dic94—21no—
48C22×Dic94—22no—
49C2×Q8×D47—22no—
50C2×Q8×D47—22no—
51C2×Q8×D47—22no—
52C2×Q8×D47—22no—
53C2×Q8×D47—48no—
54C22×C4×D47—22no—
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C2×#10625329—21no—
59C2×#10625324—22no—
60#13094024—21no—
61#13094024—21no—
62#13094024C2×C412yes—
63#13094024C2312yes—
64#13094023—22no—
65#13094023—22no—
66#13094023—22no—
67#13094023—22no—
68#13094023—22no—
69#13094023—22no—
70#13094023C2×C414yes—
71#13094023C2×C414yes—
72#13094022—22no—
73#13094022—22no—
74#13094022—22no—
75#13094022—22no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47C2×C412yes—
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—22no—
80C4⋊C4×D47—22no—
81C4⋊C4×D47—22no—
82C4⋊C4×D47—22no—
83C4⋊C4×D47—22no—
84C4⋊C4×D47—22no—
85C4⋊C4×D47—22no—
86C4⋊C4×D47C2×C414yes—
87C4⋊C4×D47C2×C414yes—
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C42×D47C2×C411yescharacteristic
92C42×D47C2×C411yescharacteristic
93C42×D47D411yescharacteristic
94C42×D47D411yescharacteristic
95C42×D47D411yescharacteristic
96C42×D47D411yescharacteristic
97C42×D47C2311yescharacteristic
98C42×D47C2311yescharacteristic
99C4⋊(C47⋊C8)—21no—
100C4⋊(C47⋊C8)C2312yes—
101C4⋊(C47⋊C8)C2312yes—
102Dic47⋊C8—22no—
103C22⋊(C47⋊C8)—21no—
104(C2×D47)⋊C8—22no—
105(C2×D47)⋊C8—22no—
106(C2×D47)⋊C8—22no—
107#13093934—22no—
108#13093934—22no—
109C4×C47⋊C8—21no—
110C8×Dic47—22no—
111C22×Q8×C47—21no—
112C2×M16×C47—21no—
113C4⋊Q8×C47—21no—
114C4⋊Q8×C47—21no—
115C4⋊Q8×C47C2×C412yes—
116C4⋊Q8×C47C2312yes—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47C2×C412yes—
119C2×C42×C47C2311yescharacteristic
120C4⋊C8×C47—21no—
121C4⋊C8×C47C2312yes—
122C4⋊C8×C47C2312yes—
123C22⋊C8×C47—21no—
124C4×C8×C47—21no—
125C23×D47C22⋊C411yescharacteristic
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47—44no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×Dic47—22no—
134C22×Dic47C22⋊C411yescharacteristic
135C22×Dic47C22⋊C411yescharacteristic
136C22×Dic47C22⋊C411yescharacteristic
137Q8×D47—48no—
138Q8×D47—48no—
139Q8×D47—48no—
140Q8×D47—48no—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—24no—
146C2×C4×D47—24no—
147C2×C4×D47—48no—
148C2×C4×D47C22⋊C411yescharacteristic
149C2×C4×D47C22⋊C411yescharacteristic
150C2×C4×D47C22⋊C411yescharacteristic
151C2×C4×D47C22⋊C411yescharacteristic
152C2×C4×D47C22⋊C411yescharacteristic
153C2×C4×D47C22⋊C411yescharacteristic
154C2×C4×D47C22×C411yescharacteristic
155C2×C4×D47C22×C411yescharacteristic
156C2×C4×D47C2×D411yescharacteristic
157C2×C4×D47C2×D411yescharacteristic
158C2×C4×D47C2×D411yescharacteristic
159C2×C4×D47C2×D411yescharacteristic
160C2×Dic94—22no—
161C2×Dic94—22no—
162C2×Dic94—24no—
163C2×Dic94—24no—
164C2×Dic94—48no—
165C2×Dic94—44no—
166M16×C47—42no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C22×C414yes—
173C42×C47C22×C411yescharacteristic
174C42×C47C2×D411yescharacteristic
175C42×C47C2×D411yescharacteristic
176C42×C47C2411yescharacteristic
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47C22×C414yes—
181Dic47⋊C4—24no—
182Dic47⋊C4—24no—
183Dic47⋊C4—24no—
184Dic47⋊C4—24no—
185C4×Dic47C22⋊C411yescharacteristic
186C4×Dic47C22⋊C411yescharacteristic
187C4×Dic47C22⋊C411yescharacteristic
188C4×Dic47C22⋊C411yescharacteristic
189C4×Dic47C22⋊C411yescharacteristic
190C4×Dic47C22⋊C411yescharacteristic
191C4×Dic47C22×C411yescharacteristic
192C4×Dic47C22×C411yescharacteristic
193C4×Dic47C2×D411yescharacteristic
194C4×Dic47C2×D411yescharacteristic
195C4×Dic47C2×D411yescharacteristic
196C4×Dic47C2×D411yescharacteristic
197#10625329—42no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200#10625324—44no—
201C23×C47C2×C22⋊C411yescharacteristic
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47C2×C22⋊C411yescharacteristic
206C22×D47C2×C22⋊C411yescharacteristic
207Q8×C47—44no—
208Q8×C47—44no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—44no—
213C2×C4×C47C2×C22⋊C411yescharacteristic
214C2×C4×C47C2×C22⋊C411yescharacteristic
215C2×C4×C47C2×C22⋊C411yescharacteristic
216C2×C4×C47#13712yes—
217C2×C4×C47#13712yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—24no—
225C2×Dic47—44no—
226C2×Dic47C2×C22⋊C411yescharacteristic
227C2×Dic47C2×C22⋊C411yescharacteristic
228C2×Dic47C2×C22⋊C411yescharacteristic
229C2×Dic47C2×C22⋊C411yescharacteristic
230C2×Dic47C2×C22⋊C411yescharacteristic
231C2×Dic47C2×C22⋊C411yescharacteristic
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—21no—
236C4×D47—21no—
237C4×D47—21no—
238C4×D47—21no—
239C4×D47—22no—
240C4×D47—22no—
241C4×D47—22no—
242C4×D47—22no—
243C4×D47—22no—
244C4×D47—48no—
245C4×D47—48no—
246C4×D47#10114yes—
247C4×D47AGL(1,ℤ/8)14yes—
248Dic94—48no—
249Dic94—44no—
250Dic94—44no—
251Dic94—48no—
252C8×C47—42no—
253C8×C47—42no—
254C47⋊C8—42no—
255C47⋊C8—42no—
256#17742—471nomaximal, Sylow
257C22×C47—21no—
258C22×C47—21no—
259C22×C47#41912yes—
260C22×C47C22×C22⋊C411yescharacteristic
261C2×D47—28no—
262C2×D47—21no—
263C2×D47—21no—
264C2×D47—21no—
265C2×D47—21no—
266C2×D47C2×#10112yes—
267C2×D47#41812yes—
268C4×C47—21no—
269C4×C47—21no—
270C4×C47—44no—
271C4×C47C2×#10112yes—
272C4×C47#41912yes—
273C4×C47C2×AGL(1,ℤ/8)12yes—
274C4×C47C2×#13711yescharacteristic
275C4×C47C2×#13711yescharacteristic
276Dic47—21no—
277Dic47—21no—
278Dic47—21no—
279Dic47—21no—
280Dic47—44no—
281Dic47C2×#10112yes—
282Dic47#41812yes—
283C22×C4⋊Q8—471no—
284#2920—471no—
285#2920—471no—
286#1556—474no—
287#1549—471no—
288#1549—471no—
289#1549—471no—
290#1549D4711yescharacteristic, complemented
291#1546—474no—
292C2×C47—22no—
293C2×C47C22×#10111yescharacteristic
294C2×C47C2×#41811yescharacteristic
295C2×C47C2×#41911yescharacteristic
296D47—22no—
297D47—22no—
298C23×Q8—941no—
299C2×C4⋊Q8—472no—
300C2×C4⋊Q8—472no—
301C2×C4⋊Q8—472no—
302C2×C4⋊Q8—472no—
303C2×C4⋊Q8—472no—
304C2×C4⋊Q8—471no—
305C2×C4⋊Q8—941no—
306C2×C4⋊Q8—941no—
307C2×C4⋊Q8—941no—
308C2×C4⋊Q8—941no—
309C2×C4⋊Q8—941no—
310C2×C4⋊Q8—941no—
311C2×C4⋊Q8C2×D4711yescharacteristic
312C22×C4⋊C4—472no—
313C22×C4⋊C4—941no—
314C22×C42—471no—
315#432—472no—
316#432—472no—
317#432—941no—
318#432—941no—
319#431—471no—
320#431—472no—
321#431—941no—
322#431—941no—
323#431C2×D4711yescharacteristic
324#423—471no—
325#423—472no—
326#423C2×D4711yescharacteristic
327#407—941no—
328#407—941no—
329C4×M16—941no—
330C4×M16—941no—
331#332—474no—
332#332—474no—
333#332—474no—
334#332C2×D4714yescomplemented
335C47#1774211yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8—942no—
338C22×Q8—942no—
339C22×Q8—942no—
340C22×Q8—942no—
341C22×Q8—942no—
342C22×Q8—941no—
343C22×Q8—1888no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—942no—
348C2×M16—21no—
349C2×M16—942no—
350C2×M16—941no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—474no—
354C4⋊Q8—474no—
355C4⋊Q8—472no—
356C4⋊Q8—472no—
357C4⋊Q8—942no—
358C4⋊Q8—942no—
359C4⋊Q8—942no—
360C4⋊Q8—942no—
361C4⋊Q8—942no—
362C4⋊Q8—942no—
363C4⋊Q8—942no—
364C4⋊Q8—942no—
365C4⋊Q8—942no—
366C4⋊Q8—942no—
367C4⋊Q8—941no—
368C4⋊Q8—941no—
369C4⋊Q8C4×D4712yes—
370C4⋊Q8C22×D4712yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—472no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—474no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—942no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4C4×D4712yes—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42C22×D4711yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—472no—
399C4⋊C8—472no—
400C4⋊C8—942no—
401C4⋊C8—941no—
402C4⋊C8C22×D4712yes—
403C4⋊C8C22×D4712yes—
404C22⋊C8—21no—
405C22⋊C8—942no—
406C22⋊C8—942no—
407C22⋊C8—942no—
408C22⋊C8—941no—
409#97—942no—
410#97—942no—
411C4×C8—21no—
412C4×C8—942no—
413C4×C8—941no—
414C24—471no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—44no—
418C2×Q8—944no—
419C2×Q8—944no—
420C2×Q8—942no—
421C2×Q8—942no—
422C2×Q8—1884no—
423C2×Q8—1888no—
424C2×Q8—1888no—
425C2×Q8—1888no—
426C2×Q8—1888no—
427C2×Q8—1888no—
428C22×C4—22no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—471no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—471no—
444C22×C4—944no—
445C22×C4—944no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—1888no—
452C22×C4C2×C4×D4711yescharacteristic
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4711yescharacteristic
455M16—42no—
456M16—1884no—
457M16—1882no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—942no—
461C2×C8—942no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—474no—
466C4⋊C4—944no—
467C4⋊C4—944no—
468C4⋊C4—944no—
469C4⋊C4—944no—
470C4⋊C4—942no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4C2×C4×D4714yes—
474C42—471no—
475C42—471no—
476C42—471no—
477C42—471no—
478C42—471no—
479C42—471no—
480C42—471no—
481C42—471no—
482C42—471no—
483C42—471no—
484C42—471no—
485C42—471no—
486C42C2×C4×D4711yescharacteristic
487C42D4×D4711yescharacteristic
488C42D4×D4711yescharacteristic
489C42C23×D4711yescharacteristic
490C23—471no—
491C23—471no—
492C23—942no—
493C23—942no—
494C23—942no—
495C23C22⋊C4×D4711yescharacteristic
496Q8—44no—
497Q8—44no—
498Q8—1884no—
499Q8—1888no—
500Q8—1888no—
501Q8—1884no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—471no—
507C2×C4—471no—
508C2×C4—474no—
509C2×C4—474no—
510C2×C4—471no—
511C2×C4—471no—
512C2×C4—471no—
513C2×C4—471no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—944no—
530C2×C4—1888no—
531C2×C4—1888no—
532C2×C4—1884no—
533C2×C4C22⋊C4×D4711yescharacteristic
534C2×C4C22⋊C4×D4711yescharacteristic
535C2×C4C22⋊C4×D4711yescharacteristic
536C2×C4#1309405312yes—
537C2×C4#1309405612yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C2×D4×D4711yescharacteristic
540C2×C4C2×D4×D4711yescharacteristic
541C8—42no—
542C8—42no—
543C8—1882no—
544C8—1882no—
545C22—21no—
546C22—21no—
547C22—472no—
548C22—472no—
549C22—948no—
550C22—941no—
551C22—941no—
552C22—941no—
553C22—941no—
554C22#1568913012yes—
555C22C2×C22⋊C4×D4711yescharacteristic
556C4—21no—
557C4—21no—
558C4—44no—
559C4—472no—
560C4—472no—
561C4—941no—
562C4—941no—
563C4—941no—
564C4—941no—
565C4—1884no—
566C4#101×D4712yes—
567C4#1568913012yes—
568C4AGL(1,ℤ/8)×D4712yes—
569C4C2×#1309405311yescharacteristic
570C4C2×#1309405611yescharacteristic
571C2—22no—
572C2—942no—
573C2—942no—
574C2C2×#101×D4711yescharacteristic
575C2#418×D4711yescharacteristic
576C2C2×#1568913011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1549C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1549D47
hypocenterC47#17742
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #20242103C2semidirect
5. #20242074C2semidirect
6. #20242073C2semidirect
7. #20242072C2semidirect
8. #20242058C2semidirect
9. C47×#1549C2semidirect
40. #15688870C22semidirect
41. #15688869C22semidirect
42. #15688868C22semidirect
46. C47×#332C22semidirect
290. #1549D47semidirect
334. #332C2×D47semidirect
335. C47#17742semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
#202520202
#202520162
#202421032
#202420742
#202420732
#202420722
#202420582
C47×#15492
#1774247