Finite Group Databasebeta

← #26225507

Subgroups — #26225507

Order 12032.

1782 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D47C211yesmaximal, characteristic
2C2×#15689177C211yesmaximal, characteristic, complemented
3C2×#15689173C211yesmaximal, characteristic, complemented
4#20244184C214yesmaximal, complemented
5#20244180C214yesmaximal, complemented
6#20243298C211yesmaximal, characteristic, complemented
7#20243297C211yesmaximal, characteristic, complemented
8#20243297C211yesmaximal, characteristic, complemented
9C47×#1779C211yesmaximal, characteristic, complemented
10C23×C22⋊D47C411yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C2×C4×C22⋊D47—24no—
13C2×C4×C22⋊D47—24no—
14C2×C4×C22⋊D47C412yes—
15C2×C4×C22⋊D47C412yes—
16C2×C4×C22⋊D47C2212yes—
17C2×C4×C22⋊D47C2212yes—
18C22×Dic47⋊C4C411yescharacteristic
19C22×C22⋊Dic47C411yescharacteristic
20C22×(C2×D47)⋊C4C411yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22#15689177—24no—
23#15689173—24no—
24C2×Dic47⋊C8C2212yescomplemented
25C2×Dic47⋊C8C2212yescomplemented
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2214yescomplemented
29C2×(C2×D47)⋊C8C2214yescomplemented
30C2×(C2×D47)⋊C8C2214yescomplemented
31C2×#13093934C2212yescomplemented
32C2×C8×Dic47C2212yescomplemented
33C24×C4×C47C2211yescharacteristic
34C2×C22⋊C8×C47C2211yescharacteristic
35C2×C22⋊C8×C47C2211yescharacteristic
36C2×C22⋊C8×C47C2214yescomplemented
37C24×D47C2×C411yescharacteristic
38C22×C22⋊D47—28no—
39C22×C22⋊D47—24no—
40C22×C22⋊D47C2×C412yes—
41C22×C22⋊D47C2×C412yes—
42C23×Dic47C2×C411yescharacteristic
43C23×Dic47C2×C411yescharacteristic
44C23×Dic47C2×C411yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—44no—
48C22×C4×D47C2×C412yes—
49C22×C4×D47C2312yes—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52C4×C22⋊D47—28no—
53C4×C22⋊D47—28no—
54C4×C22⋊D47—24no—
55C4×C22⋊D47—24no—
56C4×C22⋊D47—24no—
57C4×C22⋊D47—24no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4C2×C412yes—
60C2×Dic47⋊C4C2×C412yes—
61C2×C22⋊Dic47—24no—
62C2×C22⋊Dic47C2×C412yes—
63C2×C22⋊Dic47C2×C412yes—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4C2×C412yes—
66C2×(C2×D47)⋊C4C2×C412yes—
67C2×C4×Dic47C2×C411yescharacteristic
68C2×C4×Dic47C2×C411yescharacteristic
69C2×C4×Dic47D414yes—
70C2×C4×Dic47D414yes—
71C2×C4×Dic47C2311yescharacteristic
72C2×C4×Dic47C2311yescharacteristic
73Dic47⋊C8—24no—
74Dic47⋊C8—24no—
75C22⋊(C47⋊C8)—22no—
76C22⋊(C47⋊C8)—22no—
77C22⋊(C47⋊C8)—44no—
78(C2×D47)⋊C8—44no—
79(C2×D47)⋊C8—44no—
80#13093934—24no—
81C8×Dic47—24no—
82C25×C47C2×C411yescharacteristic
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—44no—
86C23×C4×C47C2×C411yescharacteristic
87C23×C4×C47C2×C412yes—
88C23×C4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90C22×C8×C47C2312yes—
91C22×C8×C47C2312yes—
92C22⋊C8×C47—22no—
93C22⋊C8×C47—22no—
94C22⋊C8×C47—44no—
95C24×C47—22no—
96C24×C47—24no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—42no—
100C24×C47C22×C412yes—
101C24×C47C22×C411yescharacteristic
102C23×D47—22no—
103C23×D47—24no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—42no—
107C23×D47C22×C412yes—
108C22×C4×C47—24no—
109C22×C4×C47—22no—
110C22×C4×C47—24no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—24no—
114C22×C4×C47—24no—
115C22×C4×C47—42no—
116C22×C4×C47—42no—
117C22×C4×C47—42no—
118C22×C4×C47—48no—
119C22×C4×C47—48no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2411yescharacteristic
125C2×C22⋊D47—232no—
126C2×C22⋊D47—24no—
127C2×C22⋊D47—24no—
128C2×C22⋊D47—24no—
129C2×C22⋊D47—24no—
130C2×C22⋊D47—24no—
131C2×C22⋊D47M1618yes—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—42no—
137C22×Dic47C22⋊C414yes—
138C22×Dic47C22⋊C414yes—
139C22×Dic47C22×C412yes—
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144C2×C4×D47—24no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—48no—
152C2×C4×D47—48no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C2×C8×C47—24no—
156C2×C8×C47—24no—
157C22⋊Dic47—24no—
158C22⋊Dic47—24no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—24no—
161Dic47⋊C4—24no—
162Dic47⋊C4—24no—
163C4×Dic47C22⋊C414yes—
164C4×Dic47C22⋊C414yes—
165C4×Dic47C2×D412yes—
166C4×Dic47C2×D412yes—
167C4×Dic47C2×D412yes—
168C4×Dic47C2×D412yes—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—24no—
172C2×C47⋊C8—24no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—28no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—48no—
183C23×C47—48no—
184C23×C47—44no—
185C23×C47—44no—
186C23×C47C2×C22⋊C412yes—
187C23×C47C2×M1614yes—
188C23×C47#13112yes—
189C23×C47#13112yes—
190C23×C47C23×C411yescharacteristic
191C22×D47—28no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—48no—
199C22×D47—48no—
200C22×D47—44no—
201C22×D47—44no—
202C22×D47C2×M1614yes—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—48no—
210C2×C4×C47—44no—
211C2×C4×C47—44no—
212C2×C4×C47—44no—
213C2×C4×C47C2×C22⋊C412yes—
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47H(5,𝔽2)14yes—
217C22⋊D47—216no—
218C22⋊D47—216no—
219C2×Dic47—22no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—28no—
225C2×Dic47—28no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—44no—
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47C2×C22⋊C412yes—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×M1614yes—
235C4×D47—48no—
236C4×D47—44no—
237C4×D47—44no—
238C8×C47—42no—
239C8×C47—42no—
240C47⋊C8—42no—
241C47⋊C8—42no—
242#21788—471nomaximal, Sylow
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—24no—
246C22×C47—24no—
247C22×C47—24no—
248C22×C47—24no—
249C22×C47—24no—
250C22×C47—416no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47#40814yes—
256C22×C47C22×C22⋊C411yescharacteristic
257C22×C47#51812yes—
258C22×C47C22×M1612yes—
259C22×C47C2×#13111yescharacteristic
260C22×C47C2×#13111yescharacteristic
261C2×D47—24no—
262C2×D47—24no—
263C2×D47—24no—
264C2×D47—416no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—42no—
269C4×C47—22no—
270C4×C47—42no—
271C4×C47#53512yes—
272C4×C47#53512yes—
273Dic47—22no—
274Dic47—22no—
275Dic47—22no—
276Dic47—42no—
277Dic47#40714yes—
278C22×C4×D4—471no—
279C2×#432—471no—
280C2×#431—471no—
281#1955—474no—
282#1954—474no—
283#1779—471no—
284#1779—471no—
285#1779—471no—
286#1779D4711yescharacteristic, complemented
287C2×C47—24no—
288C2×C47—44no—
289C2×C47#230911yescharacteristic
290C2×C47C2×#40711yescharacteristic
291C2×C47C2×#40811yescharacteristic
292C2×C47#297412yes—
293C2×C47#297412yes—
294D47—44no—
295C23×D4—471no—
296C24×C4—471no—
297C24×C4C2×D4711yescharacteristic
298C2×C4×D4—472no—
299C2×C4×D4—472no—
300C2×C4×D4—472no—
301C2×C4×D4—472no—
302C2×C4×D4—944no—
303C2×C4×D4—944no—
304C22×C4⋊C4—471no—
305C22×C22⋊C4—471no—
306C22×C22⋊C4—471no—
307C22×C42—471no—
308#432—944no—
309#431—944no—
310C2×C4⋊C8—472no—
311C2×C4⋊C8—472no—
312C2×C22⋊C8—471no—
313C2×C22⋊C8—471no—
314C2×C22⋊C8—474no—
315C2×C22⋊C8—474no—
316C2×C22⋊C8—474no—
317C2×C22⋊C8C2×D4711yescharacteristic
318C2×C22⋊C8C2×D4711yescharacteristic
319C2×C22⋊C8C2×D4714yescomplemented
320C2×#97—472no—
321C2×C4×C8—472no—
322C47#2178811yescharacteristic, complemented, Sylow
323C25—471no—
324C25C4×D4711yescharacteristic
325C22×D4—472no—
326C22×D4—472no—
327C22×D4—944no—
328C22×D4—948no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—44no—
332C23×C4—471no—
333C23×C4—471no—
334C23×C4—471no—
335C23×C4—472no—
336C23×C4—472no—
337C23×C4—942no—
338C23×C4—942no—
339C23×C4—1884no—
340C23×C4C4×D4711yescharacteristic
341C23×C4C4×D4712yes—
342C23×C4C22×D4712yes—
343C23×C4C22×D4711yescharacteristic
344C22×C8—472no—
345C22×C8—472no—
346C22×C8C22×D4712yes—
347C22×C8C22×D4712yes—
348C4×D4—948no—
349C4×D4—948no—
350C4×D4—944no—
351C4×D4—944no—
352C4×D4—944no—
353C4×D4—944no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—944no—
357C2×C22⋊C4—472no—
358C2×C22⋊C4—472no—
359C2×C22⋊C4—472no—
360C2×C22⋊C4—472no—
361C2×C22⋊C4—944no—
362C2×C22⋊C4—944no—
363C2×C42—471no—
364C2×C42—471no—
365C2×C42—474no—
366C2×C42—474no—
367C2×C42—471no—
368C2×C42—471no—
369C4⋊C8—944no—
370C4⋊C8—944no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—44no—
374C22⋊C8—942no—
375C22⋊C8—942no—
376C22⋊C8—1884no—
377C22⋊C8—1884no—
378C22⋊C8—1884no—
379#97—944no—
380C4×C8—944no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—22no—
385C24—42no—
386C24—472no—
387C24—944no—
388C24—942no—
389C24—942no—
390C24—942no—
391C24—1882no—
392C24C2×C4×D4712yes—
393C24C2×C4×D4711yescharacteristic
394C2×D4—478no—
395C2×D4—944no—
396C2×D4—944no—
397C2×D4—944no—
398C2×D4—944no—
399C2×D4—9432no—
400C2×D4—944no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—471no—
414C22×C4—474no—
415C22×C4—474no—
416C22×C4—471no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—472no—
420C22×C4—944no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—944no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—944no—
428C22×C4—944no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4—1888no—
432C22×C4—1888no—
433C22×C4—1882no—
434C22×C4C2×C4×D4712yes—
435C22×C4C2×C4×D4711yescharacteristic
436C22×C4D4×D4712yes—
437C22×C4D4×D4712yes—
438C22×C4C23×D4711yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—944no—
446C2×C8—944no—
447C4⋊C4—944no—
448C4⋊C4—944no—
449C22⋊C4—944no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C22⋊C4—944no—
453C42—474no—
454C42—474no—
455C42—472no—
456C42—472no—
457C42—472no—
458C42—472no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—48no—
469C23—48no—
470C23—44no—
471C23—44no—
472C23—474no—
473C23—948no—
474C23—942no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—942no—
479C23—942no—
480C23—1888no—
481C23—1888no—
482C23—1884no—
483C23—1884no—
484C23C22⋊C4×D4712yes—
485C23M16×D4714yes—
486C23#1309403012yes—
487C23#1309403212yes—
488C23C22×C4×D4711yescharacteristic
489D4—9416no—
490D4—9416no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—48no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—472no—
502C2×C4—474no—
503C2×C4—472no—
504C2×C4—472no—
505C2×C4—472no—
506C2×C4—948no—
507C2×C4—948no—
508C2×C4—942no—
509C2×C4—941no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—1888no—
517C2×C4—1884no—
518C2×C4—1884no—
519C2×C4—1884no—
520C2×C4C22⋊C4×D4712yes—
521C2×C4C2×D4×D4711yescharacteristic
522C2×C4C2×D4×D4711yescharacteristic
523C2×C4#1309406914yes—
524C8—42no—
525C8—42no—
526C8—1882no—
527C8—1882no—
528C22—24no—
529C22—24no—
530C22—22no—
531C22—22no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—416no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—944no—
541C22—944no—
542C22—944no—
543C22—18816no—
544C22—1882no—
545C22—1882no—
546C22—1882no—
547C22—1882no—
548C22#1568908814yes—
549C22C2×C22⋊C4×D4711yescharacteristic
550C22#1568959612yes—
551C22C2×M16×D4712yes—
552C22C2×#1309403011yescharacteristic
553C22C2×#1309403211yescharacteristic
554C4—22no—
555C4—42no—
556C4—474no—
557C4—942no—
558C4—942no—
559C4—942no—
560C4—1882no—
561C4#1568968812yes—
562C4#1568968812yes—
563C2—24no—
564C2—44no—
565C2—1884no—
566C2#2024596911yescharacteristic
567C2#407×D4711yescharacteristic
568C2C2×#1568908811yescharacteristic
569C2#2025241912yes—
570C2#2025242412yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1779C2
prime radicalC23×C22⋊D47C4
cosocleC22×C4×C47C24
hypercenter#1779D47
hypocenterC47#21788
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#15689177C2semidirect
3. C2×#15689173C2semidirect
4. #20244184C2semidirect
5. #20244180C2semidirect
6. #20243298C2semidirect
7. #20243297C2semidirect
8. #20243297C2semidirect
9. C47×#1779C2semidirect
24. C2×Dic47⋊C8C22semidirect
25. C2×Dic47⋊C8C22semidirect
28. C2×C22⋊(C47⋊C8)C22semidirect
29. C2×(C2×D47)⋊C8C22semidirect
30. C2×(C2×D47)⋊C8C22semidirect
31. C2×#13093934C22semidirect
32. C2×C8×Dic47C22semidirect
36. C2×C22⋊C8×C47C22semidirect
286. #1779D47semidirect
319. C2×C22⋊C8C2×D47semidirect
322. C47#21788semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D472
C2×#156891772
C2×#156891732
#202441842
#202441802
#202432982
#202432972
C47×#17792
#2178847