Finite Group Databasebeta

← #26227420

Subgroups — #26227420

Order 12032.

1092 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
1C2×#15689262C211yesmaximal, characteristic
2C4×#13093998C211yesmaximal, characteristic
3#20244245C212yesmaximal, complemented
4#20244067C212yesmaximal, complemented
5#20244019C212yesmaximal, complemented
6#20243725C212yesmaximal, complemented
7#20243705C211yesmaximal, characteristic
8#20243581C211yesmaximal, characteristic, complemented
9#20243323C211yesmaximal, characteristic, complemented
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C2×C47×#391C211yesmaximal, characteristic, complemented
12C22×#10625336C2212yes—
13C2×#13093998C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic
15C2×C4×C22⋊D47—21no—
16#15689262C418yescomplemented
17C2×#13093955C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093944C2212yes—
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C4×C47⋊C8C2212yes—
25#15689025C2211yescharacteristic
26#15689021C2211yescharacteristic
27#15689014—21no—
28#15689002—21no—
29#15688983C2211yescharacteristic
30C4×C4⋊Dic47C2211yescharacteristic
31C4×(C2×D47)⋊C4—21no—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688878C2212yes—
35C23×Q8×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C47×#391—21no—
39C47×#391—21no—
40C47×#391C2212yes—
41C47×#391C2212yes—
42C2×#95×C47—21no—
43C2×#95×C47—21no—
44C22×D188C2311yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625336—42no—
48C2×#10625336—42no—
49C2×#10625336C2×C418yescomplemented
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13093998—22no—
53#13093998C2×C414yes—
54C4×D188—42no—
55C4×C22⋊D47—44no—
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×C4⋊Dic47C2311yescharacteristic
59C2×Dic47⋊C4—21no—
60C2×C22⋊Dic47—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—21no—
65C2×C4×Dic47—21no—
66#13093955—42no—
67#13093948—42no—
68#13093948—42no—
69#13093948C2×C418yescomplemented
70#13093944—42no—
71C22⋊(C47⋊C8)—22no—
72C22⋊(C47⋊C8)C2×C414yescomplemented
73C4×C47⋊C8—42no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C22×Q8×C47—21no—
78C22×Q8×C47—22no—
79C22×Q8×C47—42no—
80C22×Q8×C47—42no—
81C22×Q8×C47C2×C414yescomplemented
82C22×Q8×C47C2312yes—
83C23×C4×C47—21no—
84C23×C4×C47—21no—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—42no—
90C2×C4⋊C4×C47C2312yes—
91C2×C42×C47—42no—
92C2×C42×C47D411yescharacteristic
93C2×C42×C47D411yescharacteristic
94C2×C42×C47C2311yescharacteristic
95C2×C42×C47C2311yescharacteristic
96#95×C47—22no—
97#95×C47—22no—
98#95×C47—22no—
99#95×C47—22no—
100C24×C47C2×D411yescharacteristic
101C23×D47—21no—
102C2×Q8×C47—24no—
103C2×Q8×C47—22no—
104C2×Q8×C47—22no—
105C2×Q8×C47—22no—
106C2×Q8×C47—22no—
107C2×Q8×C47—22no—
108C2×Q8×C47—48no—
109C2×Q8×C47—42no—
110C2×Q8×C47—42no—
111C2×Q8×C47—48no—
112C2×Q8×C47C22×C414yes—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—24no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—41no—
121C22×C4×C47—41no—
122C22×C4×C47—41no—
123C22×C4×C47—42no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47SD1614yescomplemented
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47#4112yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—24no—
138C2×C22⋊D47—42no—
139C2×C22⋊D47—42no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—21no—
146C2×D188C22×C412yes—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×C4×D47—42no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—42no—
153C42×C47—22no—
154C42×C47—22no—
155C42×C47—41no—
156C42×C47—41no—
157C22⋊Dic47—42no—
158#10625336—48no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—42no—
161(C2×D47)⋊C4—42no—
162(C2×D47)⋊C4—42no—
163C4⋊Dic47—21no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47C22×C412yes—
167Dic47⋊C4—42no—
168C4×Dic47—42no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8C22×C414yes—
175C23×C47—21no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47C4×D412yes—
180C23×C47C2×SD1612yes—
181C23×C47#13512yes—
182C23×C47C22×D411yescharacteristic
183C22×D47—22no—
184C22×D47—41no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190Q8×C47—24no—
191Q8×C47—24no—
192Q8×C47—48no—
193Q8×C47—42no—
194Q8×C47—42no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—44no—
210C2×C4×C47—44no—
211C2×C4×C47—48no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—84no—
218C2×C4×C47C4×D412yes—
219C2×C4×C47C2×SD1612yes—
220C2×C4×C47C2×SD1612yes—
221C2×C4×C47#13712yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47H(5,𝔽2)12yes—
225C2×C4×C47H(5,𝔽2)12yes—
226C22⋊D47—48no—
227C2×Dic47—22no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236D188—24no—
237D188—42no—
238C4×D47—84no—
239C47⋊C8—24no—
240C47⋊C8—42no—
241#22041—471nomaximal, Sylow
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47C4×SD1614yes—
249C22×C47C2×C4×D411yescharacteristic
250C22×C47#53411yescharacteristic
251C22×C47#53511yescharacteristic
252C22×C47C22×SD1611yescharacteristic
253C22×C47C2×#13511yescharacteristic
254C22×C47C2×#13711yescharacteristic
255C22×C47#57511yescharacteristic
256C2×D47—42no—
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—81no—
260C2×D47—81no—
261C2×D47—81no—
262C2×D47—81no—
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—22no—
267C4×C47—44no—
268C4×C47—41no—
269C4×C47—82no—
270C4×C47#51812yes—
271C4×C47#53812yes—
272Dic47—42no—
273Dic47—81no—
274Dic47—81no—
275C2×#448—471no—
276C4×#121—471no—
277C2×#391D4711yescharacteristic, complemented
278#1968—472no—
279#1933—472no—
280#1927—472no—
281#1877—472no—
282#1874—471no—
283#1853—471no—
284#1787—471no—
285C4×C22⋊C8—471no—
286C2×C47—24no—
287C2×C47#232111yescharacteristic
288C2×C47C2×C4×SD1611yescharacteristic
289C2×C47C4×#13711yescharacteristic
290C2×C47#318711yescharacteristic
291C2×C47#319011yescharacteristic
292C2×C47#320611yescharacteristic
293C2×C47#321111yescharacteristic
294D47—82no—
295C23×Q8C2×D4711yescharacteristic
296C22×SD16—472no—
297C2×#121—471no—
298C2×C4×D4—471no—
299C2×C4×D4—941no—
300C22×C4⋊C4C2×D4711yescharacteristic
301C22×C42C2×D4711yescharacteristic
302#448—478no—
303C2×#106—472no—
304C2×#103—472no—
305C2×#103—472no—
306C2×#103—472no—
307C2×#102—472no—
308C2×C22⋊C8—471no—
309C2×C22⋊C8—471no—
310C2×C4×C8—472no—
311#391—21no—
312#391—21no—
313#391C2×D4712yes—
314#391C2×D4712yes—
315#390—471no—
316#389—471no—
317#388—941no—
318#386—941no—
319#380—471no—
320C4×C4⋊C4—471no—
321C4×C22⋊C4—941no—
322C2×#95—21no—
323C2×#95—21no—
324#340—472no—
325#340—472no—
326#336—472no—
327C47#2204111yescharacteristic, complemented, Sylow
328C22×Q8—22no—
329C22×Q8—21no—
330C22×Q8—42no—
331C22×Q8—42no—
332C22×Q8C4×D4714yescomplemented
333C22×Q8C22×D4712yes—
334C22×D4—471no—
335C22×D4—941no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—941no—
339C23×C4C22⋊D4711yescharacteristic
340C23×C4C22⋊D4711yescharacteristic
341C23×C4C22×D4711yescharacteristic
342C2×SD16—478no—
343C2×SD16—1882no—
344C2×SD16—1882no—
345C22×C8—472no—
346C22×C8—472no—
347#121—474no—
348#121—942no—
349C4×D4—1882no—
350C4×D4—1884no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4C22×D4712yes—
358C2×C22⋊C4—941no—
359C2×C22⋊C4—941no—
360C2×C22⋊C4—941no—
361C2×C22⋊C4—941no—
362C2×C22⋊C4—941no—
363C2×C42—42no—
364C2×C42—941no—
365C2×C42C22⋊D4711yescharacteristic
366C2×C42C22⋊D4711yescharacteristic
367C2×C42C22×D4711yescharacteristic
368C2×C42C22×D4711yescharacteristic
369#106—1882no—
370#103—478no—
371#103—1882no—
372#103—1882no—
373#102—1882no—
374C22⋊C8—474no—
375C22⋊C8—942no—
376C4×C8—1882no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—941no—
382#95—941no—
383#95—941no—
384C24—941no—
385C24C2×C22⋊D4711yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D4714yes—
397C2×D4—472no—
398C2×D4—944no—
399C2×D4—941no—
400C2×D4—942no—
401C2×D4—942no—
402C2×D4—1882no—
403C2×D4—1882no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—1882no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4#1062533614yescomplemented
426C22×C4C2×C4×D4712yes—
427C22×C4#1062535012yes—
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C23×D4711yescharacteristic
434SD16—1888no—
435C2×C8—474no—
436C2×C8—942no—
437C2×C8—942no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8—942no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—472no—
445C4⋊C4—941no—
446C4⋊C4—942no—
447C4⋊C4—942no—
448C4⋊C4—1882no—
449C22⋊C4—944no—
450C22⋊C4—1882no—
451C22⋊C4—1882no—
452C22⋊C4—1882no—
453C22⋊C4—1882no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—1882no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—942no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23—1881no—
470C23C4×C22⋊D4712yes—
471C23C2×#1062533612yes—
472C23#1309404412yes—
473C23C22×C22⋊D4711yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—944no—
480D4—1888no—
481D4—1882no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—942no—
506C2×C4—3764no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4C4×C22⋊D4712yes—
516C2×C4C2×#1062533612yes—
517C2×C4C2×#1062533612yes—
518C2×C4#1309405512yes—
519C2×C4C22×C4×D4711yescharacteristic
520C2×C4C2×#1062535011yescharacteristic
521C2×C4#1309406912yes—
522C2×C4#1309406912yes—
523C8—944no—
524C8—1882no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—3761no—
532C22—3761no—
533C22—3761no—
534C22—3761no—
535C22—1882no—
536C22—1882no—
537C22—1882no—
538C22C4×#1062533614yes—
539C22C2×C4×C22⋊D4711yescharacteristic
540C22#1568968211yescharacteristic
541C22#1568969011yescharacteristic
542C22C22×#1062533611yescharacteristic
543C22C2×#1309404411yescharacteristic
544C22C2×#1309405511yescharacteristic
545C22#1568992111yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—41no—
552C4—82no—
553C4—3761no—
554C4—3761no—
555C4—1882no—
556C4#1568959612yes—
557C4#1568971112yes—
558C2—24no—
559C2—3762no—
560C2#2024604611yescharacteristic
561C2C2×C4×#1062533611yescharacteristic
562C2C4×#1309405511yescharacteristic
563C2#2025414111yescharacteristic
564C2#2025416911yescharacteristic
565C2#2025431311yescharacteristic
566C2#2025435311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#391C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenterC2×#391D47
hypocenterC47#22041
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #20244245C2semidirect
4. #20244067C2semidirect
5. #20244019C2semidirect
6. #20243725C2semidirect
8. #20243581C2semidirect
9. #20243323C2semidirect
10. C4×C22⋊(C47⋊C8)C2semidirect
11. C2×C47×#391C2semidirect
16. #15689262C4semidirect
49. C2×#10625336C2×C4semidirect
69. #13093948C2×C4semidirect
72. C22⋊(C47⋊C8)C2×C4semidirect
81. C22×Q8×C47C2×C4semidirect
128. C22×C4×C47SD16semidirect
277. C2×#391D47semidirect
327. C47#22041semidirect
332. C22×Q8C4×D47semidirect
425. C22×C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156892622
C4×#130939982
#202442452
#202440672
#202440192
#202437252
#202437052
#202435812
#202433232
C4×C22⋊(C47⋊C8)2
C2×C47×#3912
#2204147