Finite Group Databasebeta

← #27084811

Subgroups — #27084811

Order 12032.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256589C211yesmaximal, characteristic
2#20253683C214yesmaximal, complemented
3#20253635C214yesmaximal, complemented
4#20253596C218yesmaximal, complemented
5#20253332C212yesmaximal, complemented
6#20253206C212yesmaximal, complemented
7#20252871C212yesmaximal, complemented
8#20252646C212yesmaximal
9C4×#13094055C214yesmaximal, complemented
10#20252026C211yesmaximal, characteristic, complemented
11C2×C47×#541C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13094055C2214yes—
14#15689604—22no—
15#15689603C2214yes—
16#15689600C2212yes—
17#15689597C2212yes—
18#15689596—22no—
19C4×#10625350C2212yes—
20C4×#10625350C2212yes—
21#15689591—22no—
22C2×#13093983C2211yescharacteristic
23#117×D47—22no—
24#15689461C2218yes—
25#15689456C2214yescomplemented
26#15689451C2214yescomplemented
27#15689427C2218yescomplemented
28#15689420C2218yescomplemented
29#15689330C2218yescomplemented
30#15689325C2218yescomplemented
31#15689283C2218yes—
32#15689278C2214yes—
33#15689273C2214yes—
34#15689228C2218yescomplemented
35#15689221C2218yescomplemented
36C4×#10625337C2218yescomplemented
37C4×#10625336C2218yescomplemented
38#15689184C2214yes—
39#15689184C2214yescomplemented
40#15689163C2214yescomplemented
41#15689147C2212yescomplemented
42#15689132C2214yescomplemented
43#15689122C2214yes—
44#15689087C2212yes—
45C4×#10625329C2212yescomplemented
46C47×#541—22no—
47C47×#541—22no—
48C47×#541C2214yes—
49C47×#541C2214yescomplemented
50C2×C4⋊Q8×C47C2211yescharacteristic, complemented
51C2×#126×C47—22no—
52C2×#125×C47—22no—
53C2×#125×C47C2211yescharacteristic, complemented
54C2×C22⋊Q8×C47C2212yes—
55C2×C4×Q8×C47C2212yescomplemented
56C2×#117×C47C2211yescharacteristic
57C2×#10625350—21no—
58C2×#10625350—21no—
59C2×#10625350—28no—
60C2×#10625350C2312yes—
61C2×#10625350C2312yes—
62C22×Dic94C2311yescharacteristic
63C22×D188C2311yescharacteristic
64C22×C22⋊D47—21no—
65C22×C4×D47—21no—
66#13094055—216no—
67C2×#10625337C2318yes—
68C2×#10625336C2318yes—
69C2×#10625329C2314yes—
70C4×Dic94C2314yes—
71C4×Dic94C2314yes—
72#13093989—24no—
73C4×D188C2314yes—
74C4×D188C2314yes—
75#13093986—24no—
76C4×C22⋊D47—24no—
77C4×C22⋊D47—24no—
78(C22⋊D47)⋊C4—24no—
79(C22⋊D47)⋊C4—24no—
80#13093983—21no—
81#13093983—21no—
82#13093983C2312yes—
83#13093983C2312yes—
84#13093982—24no—
85#13093981—22no—
86#13093981—22no—
87#13093980—24no—
88C2×C4⋊Dic47C2312yes—
89C4⋊C4×D47—24no—
90C2×C22⋊Dic47—21no—
91C22⋊C4×D47—24no—
92C2×C4×Dic47—21no—
93C42×D47—22no—
94C42×D47—22no—
95#13093957C2318yes—
96#13093955C2318yes—
97C4⋊(C47⋊C8)C2318yes—
98#13093948C2318yes—
99#13093948C2318yes—
100#13093947C2318yes—
101#13093944C2318yes—
102C22⋊(C47⋊C8)C2314yes—
103C22⋊(C47⋊C8)C2314yes—
104#13093935C2314yes—
105C4×C47⋊C8C2314yes—
106C22×Q8×C47C2312yes—
107C23×C4×C47C2311yescharacteristic
108C4⋊Q8×C47—22no—
109C4⋊Q8×C47C2314yes—
110#126×C47—28no—
111#126×C47—28no—
112#125×C47—28no—
113#125×C47—28no—
114#125×C47—22no—
115#125×C47C2314yes—
116C22⋊Q8×C47—22no—
117C22⋊Q8×C47—22no—
118C22⋊Q8×C47C2314yes—
119C22⋊Q8×C47C2314yes—
120C4×Q8×C47—24no—
121C4×Q8×C47C2318yes—
122#117×C47D412yes—
123#117×C47D412yes—
124#117×C47C2312yes—
125#117×C47C2312yes—
126C2×C4⋊C4×C47—22no—
127C2×C4⋊C4×C47—22no—
128C2×C4⋊C4×C47—22no—
129C2×C4⋊C4×C47—22no—
130C2×C4⋊C4×C47D412yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C22⋊C4×C47—22no—
133C2×C22⋊C4×C47D412yes—
134C2×C42×C47—22no—
135C2×C42×C47C2311yescharacteristic
136C2×C42×C47C2311yescharacteristic
137C24×C47C2×D411yescharacteristic
138C23×D47—21no—
139C2×Q8×C47—22no—
140C2×Q8×C47—28no—
141C2×Q8×C47C2414yes—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C22×C4×C47C2411yescharacteristic
153C2×C22⋊D47—28no—
154C2×C22⋊D47—22no—
155C2×C22⋊D47—22no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158#10625350—28no—
159#10625350—28no—
160#10625350—24no—
161#10625350—24no—
162#10625350—24no—
163#10625350#4118yes—
164C2×D188—21no—
165C2×D188—24no—
166C2×D188C2412yes—
167C2×C4×D47—28no—
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×C4×D47—24no—
171C2×C4×D47—24no—
172C2×Dic94—21no—
173C2×Dic94—24no—
174C2×Dic94C2412yes—
175C4⋊C4×C47—28no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—28no—
179C4⋊C4×C47—28no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47C2×D414yes—
182C4⋊C4×C47C2×D414yes—
183C4⋊C4×C47C2414yes—
184C22⋊C4×C47—24no—
185C22⋊C4×C47—24no—
186C22⋊C4×C47C2×D414yes—
187C22⋊C4×C47C2×D414yes—
188C42×C47—28no—
189C42×C47C2×D412yes—
190C42×C47C2×D412yes—
191C42×C47C2412yes—
192C42×C47C2412yes—
193C22⋊Dic47—22no—
194C22⋊Dic47—22no—
195#10625337—216no—
196#10625336—216no—
197(C2×D47)⋊C4—24no—
198(C2×D47)⋊C4—24no—
199C4⋊Dic47—22no—
200C4⋊Dic47C2414yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203C4×Dic47—22no—
204C4×Dic47—24no—
205C4×Dic47—24no—
206C4×Dic47—22no—
207#10625329—28no—
208C2×C47⋊C8C2418yes—
209C23×C47—22no—
210C23×C47—42no—
211C23×C47C22×D411yescharacteristic
212C23×C47C22×D411yescharacteristic
213C23×C47C22×D411yescharacteristic
214C22×D47—22no—
215C22×D47—24no—
216C22×D47—82no—
217Q8×C47—28no—
218Q8×C47—44no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—44no—
229C2×C4×C47—44no—
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C22×D414yes—
232C2×C4×C47C22×D414yes—
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C2×#4114yes—
236C2×C4×C47C2511yescharacteristic
237C22⋊D47—28no—
238C22⋊D47—28no—
239C2×Dic47—24no—
240C2×Dic47—22no—
241C2×Dic47—24no—
242C2×Dic47—82no—
243D188—24no—
244D188—22no—
245D188C2×#4114yes—
246C4×D47—28no—
247C4×D47—28no—
248C4×D47—84no—
249C4×D47—84no—
250Dic94—24no—
251Dic94—22no—
252Dic94C2×#4114yes—
253C47⋊C8—28no—
254#60569—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—82no—
261C22×C47#52512yes—
262C22×C47C23×D411yescharacteristic
263C2×D47—24no—
264C2×D47—44no—
265C2×D47—84no—
266C4×C47—24no—
267C4×C47—24no—
268C4×C47—21no—
269C4×C47—21no—
270C4×C47—44no—
271C4×C47—44no—
272C4×C47#52512yes—
273C4×C47C22×#4112yes—
274Dic47—24no—
275Dic47—44no—
276C2×#541D4711yescharacteristic, complemented
277#3428—471no—
278#3127—474no—
279#3123—474no—
280#3120—478no—
281#3095—472no—
282#3080—472no—
283#3039—472no—
284#3007—472no—
285C4×#137—474no—
286#2921—471no—
287C2×C47—22no—
288C2×C47—41no—
289C2×C47—41no—
290C2×C47C2×#52511yescharacteristic
291C2×C47#358612yes—
292D47—44no—
293C22×#41—471no—
294C2×#137—474no—
295#541—22no—
296#541—22no—
297#541C2×D4714yes—
298#541C2×D4714yescomplemented
299C2×C4⋊Q8C2×D4711yescharacteristic, complemented
300C2×#126—22no—
301C2×#125—22no—
302C2×#125C2×D4711yescharacteristic, complemented
303C2×C22⋊Q8C2×D4712yes—
304#520—474no—
305#520—942no—
306#519—472no—
307#518—472no—
308#518—942no—
309C4×#41—472no—
310C4×#41—472no—
311C4×#41—942no—
312C2×C4×Q8C2×D4712yescomplemented
313C2×#117—471no—
314C2×#117—942no—
315C2×#117C2×D4711yescharacteristic
316#485—478no—
317#484—474no—
318#483—474no—
319#479—478no—
320#478—478no—
321#462—478no—
322#461—478no—
323#452—478no—
324#451—474no—
325#450—474no—
326#441—478no—
327#440—478no—
328C4×Q16—478no—
329C4×SD16—478no—
330#433—474no—
331#433—474no—
332#428—474no—
333#423—472no—
334#419—474no—
335#417—474no—
336#407—472no—
337C4×M16—472no—
338C47#6056911yescharacteristic, complemented, Sylow
339C2×#41—472no—
340C2×#41—472no—
341C2×#41—948no—
342C2×#41—941no—
343C2×#41—941no—
344C22×Q8—471no—
345C22×Q8C22×D4712yes—
346C22×D4—471no—
347C22×D4—941no—
348C23×C4—941no—
349C23×C4C22×D4711yescharacteristic
350#137—9416no—
351C2×Q16—478no—
352C2×SD16—478no—
353C2×M16—474no—
354C4⋊Q8—22no—
355C4⋊Q8C22×D4714yes—
356#126—28no—
357#126—28no—
358#125—28no—
359#125—28no—
360#125—22no—
361#125C22×D4714yes—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8C22×D4714yes—
365C22⋊Q8C22×D4714yes—
366C4×Q8—24no—
367C4×Q8—474no—
368C4×Q8—474no—
369C4×Q8—944no—
370C4×Q8C22×D4718yes—
371C4×D4—474no—
372C4×D4—474no—
373C4×D4—944no—
374C4×D4—944no—
375C4×D4—944no—
376C4×D4—944no—
377C4×D4—944no—
378#117—472no—
379#117—472no—
380#117—941no—
381#117—941no—
382#117—944no—
383#117—944no—
384#117—942no—
385#117—942no—
386#117C22⋊D4712yes—
387#117C22⋊D4712yes—
388#117C22×D4712yes—
389#117C22×D4712yes—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—472no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4C22⋊D4712yes—
397C2×C4⋊C4C22×D4712yes—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4C22⋊D4712yes—
402C2×C42—22no—
403C2×C42—941no—
404C2×C42—942no—
405C2×C42—942no—
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408#107—478no—
409#106—478no—
410C4⋊C8—478no—
411#103—478no—
412#103—478no—
413#103—478no—
414#102—478no—
415C22⋊C8—474no—
416C22⋊C8—474no—
417#97—474no—
418C4×C8—474no—
419C24—941no—
420C24C2×C22⋊D4711yescharacteristic
421#41—478no—
422#41—944no—
423#41—944no—
424#41—948no—
425#41—948no—
426#41—944no—
427C2×Q8—22no—
428C2×Q8—28no—
429C2×Q8—472no—
430C2×Q8—944no—
431C2×Q8—941no—
432C2×Q8C23×D4714yes—
433C2×D4—472no—
434C2×D4—944no—
435C2×D4—948no—
436C2×D4—942no—
437C2×D4—942no—
438C2×D4—941no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—948no—
443C22×C4—941no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4712yes—
453C22×C4C2×C22⋊D4712yes—
454C22×C4C23×D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457Q16—9416no—
458SD16—9416no—
459M16—948no—
460C2×C8—478no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—474no—
468C4⋊C4—944no—
469C4⋊C4—944no—
470C4⋊C4—942no—
471C4⋊C4C2×C22⋊D4714yes—
472C4⋊C4C2×C22⋊D4714yes—
473C4⋊C4C23×D4714yes—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—942no—
477C22⋊C4—942no—
478C22⋊C4—944no—
479C22⋊C4—944no—
480C22⋊C4C2×C22⋊D4714yes—
481C22⋊C4C2×C22⋊D4714yes—
482C42—28no—
483C42—942no—
484C42—942no—
485C42—944no—
486C42—944no—
487C42C2×C22⋊D4712yes—
488C42C2×C22⋊D4712yes—
489C42C23×D4712yes—
490C42C23×D4712yes—
491C23—22no—
492C23—42no—
493C23—944no—
494C23—942no—
495C23—3762no—
496C23C22×C22⋊D4711yescharacteristic
497C23C22×C22⋊D4711yescharacteristic
498C23C22×C22⋊D4711yescharacteristic
499Q8—28no—
500Q8—44no—
501Q8—474no—
502Q8—944no—
503Q8—942no—
504D4—474no—
505D4—944no—
506D4—948no—
507D4—948no—
508D4—942no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—948no—
521C2×C4—944no—
522C2×C4—942no—
523C2×C4—948no—
524C2×C4—944no—
525C2×C4—3764no—
526C2×C4—3764no—
527C2×C4—3762no—
528C2×C4C22×C22⋊D4714yes—
529C2×C4C22×C22⋊D4711yescharacteristic
530C2×C4C22×C22⋊D4714yes—
531C2×C4C22×C22⋊D4711yescharacteristic
532C2×C4C22×C22⋊D4711yescharacteristic
533C2×C4#41×D4714yes—
534C2×C4C24×D4711yescharacteristic
535C8—948no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—82no—
542C22—944no—
543C22—3764no—
544C22—1884no—
545C22#1568963012yes—
546C22C23×C22⋊D4711yescharacteristic
547C4—21no—
548C4—21no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—944no—
554C4—1884no—
555C4#1568963012yes—
556C4C2×#41×D4712yes—
557C2—22no—
558C2—41no—
559C2—41no—
560C2—1884no—
561C2C2×#1568963011yescharacteristic
562C2#2025817512yes—
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#541C2
socleC22×C47C23×D4
prime radicalC22×#10625350C22
hypercenterC2×#541D47
hypocenterC47#60569
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
2. #20253683C2semidirect
3. #20253635C2semidirect
4. #20253596C2semidirect
5. #20253332C2semidirect
6. #20253206C2semidirect
7. #20252871C2semidirect
9. C4×#13094055C2semidirect
10. #20252026C2semidirect
11. C2×C47×#541C2semidirect
25. #15689456C22semidirect
26. #15689451C22semidirect
27. #15689427C22semidirect
28. #15689420C22semidirect
29. #15689330C22semidirect
30. #15689325C22semidirect
34. #15689228C22semidirect
35. #15689221C22semidirect
36. C4×#10625337C22semidirect
37. C4×#10625336C22semidirect
39. #15689184C22semidirect
40. #15689163C22semidirect
41. #15689147C22semidirect
42. #15689132C22semidirect
45. C4×#10625329C22semidirect
49. C47×#541C22semidirect
50. C2×C4⋊Q8×C47C22semidirect
53. C2×#125×C47C22semidirect
55. C2×C4×Q8×C47C22semidirect
276. C2×#541D47semidirect
298. #541C2×D47semidirect
299. C2×C4⋊Q8C2×D47semidirect
302. C2×#125C2×D47semidirect
312. C2×C4×Q8C2×D47semidirect
338. C47#60569semidirect

Maximal subgroups

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

subgroupindex
#202565892
#202536832
#202536352
#202535962
#202533322
#202532062
#202528712
#202526462
C4×#130940552
#202520262
C2×C47×#5412
#6056947