Finite Group Databasebeta

← #26216070

Subgroups — #26216070

Order 12032.

1502 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D47C211yesmaximal, characteristic
2C2×C22⋊C8×D47C211yesmaximal, characteristic
3#20245770C211yesmaximal, characteristic
4#20243484C211yesmaximal, characteristic, complemented
5#20243483C211yesmaximal, characteristic, complemented
6#20243480C211yesmaximal, characteristic, complemented
7#20243477C212yesmaximal, complemented
8#20243473C212yesmaximal, complemented
9#20243103C212yesmaximal, complemented
10#20243103C212yesmaximal, complemented
11C47×#1825C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×C8×D47C2212yes—
14C2×#13093983C2211yescharacteristic
15C2×#13093982C2212yes—
16C2×#13093982C2211yescharacteristic
17C2×#13093982C2211yescharacteristic
18C2×#13093981—21no—
19C2×#13093980—21no—
20#117×D47—24no—
21#117×D47C418yescomplemented
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C22⋊C4×D47—21no—
25C2×C42×D47—21no—
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×(C2×D47)⋊C8C2212yes—
28C22⋊C8×D47—24no—
29#15688985—21no—
30#15688983C2211yescharacteristic
31#15688980C2212yes—
32#15688978—21no—
33C22⋊C4×Dic47—21no—
34#15688970—21no—
35#15688886C2212yescomplemented
36#15688886C2212yescomplemented
37#15688885C2212yescomplemented
38#15688885C2212yescomplemented
39#15688885C2212yescomplemented
40#15688885C2212yescomplemented
41C2×#117×C47C2211yescharacteristic
42C2×C22⋊C8×C47C2211yescharacteristic
43C47×#380C2211yescharacteristic
44C47×#340C2212yescomplemented
45C47×#340C2212yescomplemented
46C24×D47D411yescharacteristic
47C23×Dic47D411yescharacteristic
48C22×C4×D47—28no—
49C22×C4×D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47C2×C414yes—
52C22×C4×D47D411yescharacteristic
53C22×C4×D47D411yescharacteristic
54C22×C4×D47D414yes—
55C22×C4×D47C2311yescharacteristic
56C22×C4×D47C2311yescharacteristic
57C22×C47⋊C8C2312yes—
58C2×C8×D47—24no—
59C2×C8×D47—28no—
60#13093983—22no—
61#13093983C2×C414yes—
62#13093982—22no—
63#13093982—22no—
64#13093982—28no—
65#13093982C2×C414yes—
66#13093982C2×C414yes—
67#13093981—28no—
68#13093980—28no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C4⋊C4×D47—22no—
75C4⋊C4×D47—22no—
76C4⋊C4×D47C2×C414yes—
77C4⋊C4×D47C2×C414yes—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—22no—
84C22⋊C4×D47—28no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—22no—
89C42×D47—28no—
90C22⋊(C47⋊C8)—22no—
91(C2×D47)⋊C8—24no—
92C47⋊#95—21no—
93C47⋊#95—22no—
94C23×C4×C47C2311yescharacteristic
95C22×C8×C47C2312yes—
96#117×C47—22no—
97#117×C47C2×C414yes—
98C2×C4⋊C4×C47C2312yes—
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C42×C47—21no—
104C22⋊C8×C47—22no—
105#95×C47—21no—
106C24×C47C2×D411yescharacteristic
107C23×D47—24no—
108C23×D47—24no—
109C23×D47—41no—
110C23×D47—41no—
111C23×D47C22⋊C414yes—
112C23×D47C2×D411yescharacteristic
113C23×D47C2×D411yescharacteristic
114C22×C4×C47—24no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47C22×C412yes—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D412yes—
121C22×C4×C47C2411yescharacteristic
122C22×Dic47—24no—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C22×Dic47C22⋊C414yes—
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129C2×C4×D47—24no—
130C2×C4×D47—24no—
131C2×C4×D47—24no—
132C2×C4×D47—216no—
133C2×C4×D47—216no—
134C2×C4×D47—24no—
135C2×C4×D47—24no—
136C2×C4×D47—42no—
137C2×C4×D47—42no—
138C2×C4×D47—44no—
139C2×C4×D47C22⋊C414yes—
140C2×C4×D47C22⋊C414yes—
141C2×C4×D47C22×C412yes—
142C2×C4×D47C22×C412yes—
143C2×C4×D47C2×D412yes—
144C2×C4×D47C2×D412yes—
145C2×C4×D47#4118yes—
146C2×C8×C47—22no—
147C2×C8×C47—24no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—24no—
151C4⋊C4×C47C22×C412yes—
152C4⋊C4×C47C22×C412yes—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—42no—
155C42×C47—24no—
156C22⋊Dic47—24no—
157C22⋊Dic47—42no—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—44no—
161(C2×D47)⋊C4—44no—
162C4⋊Dic47—21no—
163C4⋊Dic47—21no—
164C4⋊Dic47—24no—
165C4⋊Dic47C22×C412yes—
166C4⋊Dic47C22×C412yes—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169C4×Dic47—24no—
170C4×Dic47—24no—
171C4×Dic47—24no—
172C4×Dic47—44no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—24no—
175C8×D47—48no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47C2×C22⋊C412yes—
181C23×C47C22×D411yescharacteristic
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—21no—
185C22×D47—21no—
186C22×D47—21no—
187C22×D47—21no—
188C22×D47—28no—
189C22×D47—28no—
190C22×D47—44no—
191C22×D47—48no—
192C22×D47—48no—
193C22×D47C2×C22⋊C412yes—
194C22×D47C2×C22⋊C412yes—
195C22×D47#12314yes—
196C22×D47#13514yes—
197C2×C4×C47—22no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—42no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47C23×C411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C2×#4114yes—
212C2×Dic47—22no—
213C2×Dic47—28no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—42no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47C2×C22⋊C412yes—
223C2×Dic47C2×C22⋊C412yes—
224C2×Dic47#12314yes—
225C4×D47—28no—
226C4×D47—24no—
227C4×D47—24no—
228C4×D47—28no—
229C4×D47—48no—
230C4×D47—48no—
231C4×D47#11718yes—
232C4×D47#12318yes—
233C8×C47—42no—
234C47⋊C8—42no—
235#20699—471nomaximal, Sylow
236C22×C47—22no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47C22×C22⋊C411yescharacteristic
242C22×C47C2×#12312yes—
243C22×C47C2×#13512yes—
244C22×C47#57511yescharacteristic
245C22×C47#57511yescharacteristic
246C2×D47—24no—
247C2×D47—24no—
248C2×D47—24no—
249C2×D47—24no—
250C2×D47—22no—
251C2×D47—22no—
252C2×D47—22no—
253C2×D47—22no—
254C2×D47—416no—
255C2×D47#38114yes—
256C2×D47#41614yes—
257C4×C47—22no—
258C4×C47—42no—
259C4×C47—42no—
260C4×C47—82no—
261C4×C47C2×#11712yes—
262C4×C47C2×#12312yes—
263Dic47—22no—
264Dic47—42no—
265Dic47—42no—
266Dic47—82no—
267Dic47#38114yes—
268C22×#117—471no—
269C22×C22⋊C8—471no—
270#2280—471no—
271#1825—471no—
272#1825—471no—
273#1825—471no—
274#1825D4711yescharacteristic, complemented
275#1824—472no—
276#1823—472no—
277#1735—472no—
278#1735—472no—
279C2×C47—42no—
280C2×C47C2×#38111yescharacteristic
281C2×C47C2×#41611yescharacteristic
282C2×C47#289611yescharacteristic
283C2×C47#309112yes—
284C2×C47#309112yes—
285D47—24no—
286D47—24no—
287C24×C4—471no—
288C23×C8—472no—
289C2×#117—471no—
290C2×#117—478no—
291C2×#117—471no—
292C2×#117—471no—
293C2×#117—472no—
294C2×#117—941no—
295C2×#117—941no—
296C2×#117—944no—
297C2×#117C2×D4711yescharacteristic
298C22×C4⋊C4—471no—
299C22×C4⋊C4—471no—
300C22×C22⋊C4—941no—
301C22×C42—941no—
302C2×C22⋊C8—471no—
303C2×C22⋊C8—472no—
304C2×C22⋊C8—944no—
305C2×C22⋊C8C2×D4711yescharacteristic
306#381—941no—
307#380—471no—
308#380—472no—
309#380C2×D4711yescharacteristic
310#379—941no—
311C4×C22⋊C4—941no—
312C4×C22⋊C4—941no—
313#340—472no—
314#340—472no—
315#340—472no—
316#340—472no—
317#340—472no—
318#340—472no—
319#340C2×D4712yescomplemented
320#340C2×D4712yescomplemented
321C47#2069911yescharacteristic, complemented, Sylow
322C25—471no—
323C23×C4—471no—
324C23×C4—471no—
325C23×C4—474no—
326C23×C4—471no—
327C23×C4—474no—
328C23×C4—471no—
329C23×C4—471no—
330C23×C4—948no—
331C23×C4—941no—
332C23×C4—941no—
333C23×C4C22×D4711yescharacteristic
334C22×C8—472no—
335C22×C8—948no—
336C22×C8—944no—
337C22×C8C22×D4712yes—
338#117—22no—
339#117—474no—
340#117—474no—
341#117—474no—
342#117—948no—
343#117—948no—
344#117—942no—
345#117—942no—
346#117—942no—
347#117—948no—
348#117C4×D4714yes—
349C2×C4⋊C4—471no—
350C2×C4⋊C4—471no—
351C2×C4⋊C4—474no—
352C2×C4⋊C4—474no—
353C2×C4⋊C4—472no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4C22×D4712yes—
359C2×C4⋊C4C22×D4711yescharacteristic
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—948no—
364C2×C22⋊C4—941no—
365C2×C22⋊C4—941no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—941no—
370C2×C42—21no—
371C2×C42—948no—
372C2×C42—941no—
373C2×C42—941no—
374C2×C42—941no—
375C2×C42—942no—
376C22⋊C8—22no—
377C22⋊C8—944no—
378C22⋊C8—942no—
379#95—21no—
380#95—942no—
381#95—941no—
382C24—471no—
383C24—471no—
384C24—474no—
385C24—944no—
386C24—944no—
387C24—1881no—
388C24—1881no—
389C24D4×D4711yescharacteristic
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—471no—
395C22×C4—472no—
396C22×C4—472no—
397C22×C4—478no—
398C22×C4—474no—
399C22×C4—471no—
400C22×C4—474no—
401C22×C4—474no—
402C22×C4—472no—
403C22×C4—472no—
404C22×C4—944no—
405C22×C4—9416no—
406C22×C4—9416no—
407C22×C4—944no—
408C22×C4—944no—
409C22×C4—944no—
410C22×C4—944no—
411C22×C4—944no—
412C22×C4—941no—
413C22×C4—941no—
414C22×C4—942no—
415C22×C4—1884no—
416C22×C4—1882no—
417C22×C4—1882no—
418C22×C4C2×C4×D4712yes—
419C22×C4D4×D4711yescharacteristic
420C22×C4D4×D4712yes—
421C22×C4C23×D4711yescharacteristic
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—942no—
425C2×C8—944no—
426C2×C8—1888no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—472no—
431C4⋊C4—472no—
432C4⋊C4—944no—
433C4⋊C4—944no—
434C4⋊C4—941no—
435C4⋊C4—941no—
436C4⋊C4—944no—
437C4⋊C4C2×C4×D4712yes—
438C4⋊C4C2×C4×D4712yes—
439C22⋊C4—24no—
440C22⋊C4—42no—
441C22⋊C4—944no—
442C22⋊C4—944no—
443C22⋊C4—944no—
444C22⋊C4—1884no—
445C22⋊C4—1884no—
446C22⋊C4—1882no—
447C42—24no—
448C42—944no—
449C42—944no—
450C42—944no—
451C42—1884no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—472no—
457C23—472no—
458C23—474no—
459C23—474no—
460C23—948no—
461C23—948no—
462C23—942no—
463C23—942no—
464C23—941no—
465C23—941no—
466C23—941no—
467C23—941no—
468C23—1884no—
469C23—1888no—
470C23—1888no—
471C23C22⋊C4×D4712yes—
472C23C2×D4×D4711yescharacteristic
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—472no—
485C2×C4—474no—
486C2×C4—478no—
487C2×C4—478no—
488C2×C4—472no—
489C2×C4—948no—
490C2×C4—948no—
491C2×C4—948no—
492C2×C4—942no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—1888no—
498C2×C4—1888no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4C22⋊C4×D4712yes—
506C2×C4C22×C4×D4711yescharacteristic
507C2×C4C2×D4×D4711yescharacteristic
508C2×C4#41×D4714yes—
509C8—42no—
510C8—1882no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—474no—
517C22—474no—
518C22—942no—
519C22—942no—
520C22—942no—
521C22—944no—
522C22—944no—
523C22—944no—
524C22—944no—
525C22—942no—
526C22—18816no—
527C22C2×C22⋊C4×D4711yescharacteristic
528C22#123×D4712yes—
529C22#135×D4712yes—
530C22#1568991911yescharacteristic
531C22#1568992211yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—474no—
537C4—942no—
538C4—3762no—
539C4—1882no—
540C4—1882no—
541C4#117×D4712yes—
542C4#123×D4712yes—
543C2—42no—
544C2—944no—
545C2—944no—
546C2#381×D4711yescharacteristic
547C2#416×D4711yescharacteristic
548C2#2025190411yescharacteristic
549C2#2025330012yes—
550C2#2025330612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1825C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1825D47
hypocenterC47#20699
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #20243484C2semidirect
5. #20243483C2semidirect
6. #20243480C2semidirect
7. #20243477C2semidirect
8. #20243473C2semidirect
9. #20243103C2semidirect
10. #20243103C2semidirect
11. C47×#1825C2semidirect
21. #117×D47C4semidirect
35. #15688886C22semidirect
36. #15688886C22semidirect
37. #15688885C22semidirect
38. #15688885C22semidirect
39. #15688885C22semidirect
40. #15688885C22semidirect
44. C47×#340C22semidirect
45. C47×#340C22semidirect
274. #1825D47semidirect
319. #340C2×D47semidirect
320. #340C2×D47semidirect
321. C47#20699semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D472
C2×C22⋊C8×D472
#202457702
#202434842
#202434832
#202434802
#202434772
#202434732
#202431032
C47×#18252
#2069947