Finite Group Databasebeta

← #26218293

Subgroups — #26218293

Order 12032.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256717C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic, complemented
3C2×#15689096C212yesmaximal, complemented
4#20243928C218yesmaximal, complemented
5#20243121C212yesmaximal, complemented
6C2×C47×#410C211yesmaximal, characteristic, complemented
7C22×#10625350C2211yescharacteristic
8#15689701—22no—
9#15689697—21no—
10#15689694C412yescomplemented
11#15689690—21no—
12#15689687C412yescomplemented
13#15689682C2214yes—
14#15689669—22no—
15#15689634—22no—
16#15689630C414yescomplemented
17#15689606—24no—
18#15689601—21no—
19#15689598—21no—
20#15689598C2214yes—
21#41×Dic47—24no—
22C2×#13093983—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2212yes—
25#15689096—24no—
26#15689096—24no—
27C2×#13093938C2218yes—
28#15688890C2218yes—
29C22×#41×C47C2211yescharacteristic
30C2×#117×C47C2212yes—
31C47×#410—22no—
32C47×#410—22no—
33C2×#99×C47C2214yes—
34C2×#10625350—22no—
35C2×#10625350—22no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C22×Dic94C2×C411yescharacteristic, complemented
43C22×D188C2×C411yescharacteristic, complemented
44C22×C22⋊D47C2312yes—
45C22×C4×D47C2×C412yescomplemented
46#13094022—22no—
47#13094020—22no—
48#13094013—22no—
49#13094011—22no—
50#13094007—24no—
51#13094005C2×C418yescomplemented
52#13094003—24no—
53#13094000—24no—
54#13093998—24no—
55#13093996—28no—
56#13093995—24no—
57#13093993—24no—
58#13093992C2×C418yescomplemented
59Q8×Dic47—28no—
60D4×Dic47—28no—
61D4×Dic47—216no—
62C4×C22⋊D47—28no—
63#13093983—24no—
64#13093983—24no—
65#13093983—22no—
66#13093983—24no—
67#13093983—22no—
68#13093983—21no—
69#13093983—21no—
70#13093983C2×C412yescomplemented
71#13093983C2×C412yescomplemented
72#13093983D414yescomplemented
73#13093983D414yescomplemented
74C2×C4⋊Dic47—24no—
75C2×C4⋊Dic47—22no—
76C2×C4⋊Dic47—21no—
77C2×C22⋊Dic47—24no—
78C2×C22⋊Dic47C2312yes—
79C2×C22⋊Dic47C2314yes—
80C2×C4×Dic47—24no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47—21no—
83#13093938—216no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47C2×C412yes—
89C2×#41×C47C2×C412yes—
90C2×#41×C47D412yes—
91C2×#41×C47D412yes—
92C22×Q8×C47D411yescharacteristic
93C22×D4×C47D411yescharacteristic
94C22×D4×C47C2312yes—
95C23×C4×C47D412yes—
96C23×C4×C47C2311yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99#117×C47—22no—
100#117×C47—22no—
101C2×C4⋊C4×C47—22no—
102C2×C22⋊C4×C47C2314yes—
103C2×C42×C47—22no—
104#99×C47—28no—
105C24×C47C2×D412yes—
106C24×C47C2411yescharacteristic
107C23×D47C22×C412yes—
108#41×C47—24no—
109#41×C47—24no—
110#41×C47—24no—
111#41×C47—24no—
112#41×C47—24no—
113#41×C47—24no—
114#41×C47—24no—
115#41×C47—24no—
116C2×Q8×C47—22no—
117C2×Q8×C47—22no—
118C2×Q8×C47C22⋊C412yes—
119C2×Q8×C47C22⋊C412yes—
120C2×D4×C47—24no—
121C2×D4×C47—24no—
122C2×D4×C47—22no—
123C2×D4×C47—22no—
124C2×D4×C47C22⋊C412yes—
125C2×D4×C47C22⋊C412yes—
126C2×D4×C47C22×C414yes—
127C2×D4×C47C2×D414yes—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—24no—
133C22×C4×C47C22⋊C414yes—
134C22×C4×C47C22⋊C414yes—
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D412yes—
144C22×C4×C47#4112yes—
145C22×C4×C47#4112yes—
146C2×C22⋊D47—24no—
147C2×C22⋊D47—22no—
148C2×C22⋊D47—22no—
149C2×C22⋊D47—24no—
150C2×C22⋊D47—28no—
151C22×Dic47—28no—
152C22×Dic47—24no—
153C22×Dic47—22no—
154C22×Dic47C22×C412yes—
155#10625350—41no—
156#10625350—42no—
157#10625350—41no—
158#10625350—42no—
159#10625350—42no—
160#10625350—42no—
161#10625350—42no—
162#10625350—41no—
163#10625350—41no—
164#10625350—42no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188—22no—
168C2×D188—22no—
169C2×C4×D47—22no—
170C2×C4×D47—22no—
171C2×C4×D47—24no—
172C2×C4×D47—24no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—22no—
176C2×Dic94—22no—
177C4⋊C4×C47—42no—
178C4⋊C4×C47—42no—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—24no—
181C42×C47—42no—
182C42×C47—42no—
183C22⋊Dic47—28no—
184C22⋊Dic47—28no—
185C22⋊Dic47—24no—
186C22⋊Dic47—22no—
187C22⋊Dic47C22×C414yes—
188C22⋊Dic47C2×D418yes—
189(C2×D47)⋊C4—28no—
190C4⋊Dic47—28no—
191C4⋊Dic47—22no—
192C4⋊Dic47—22no—
193C4⋊Dic47—24no—
194C4⋊Dic47—24no—
195Dic47⋊C4—28no—
196C4×Dic47—28no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47—24no—
200C4×Dic47—24no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—81no—
204C23×C47—82no—
205C23×C47C2×C22⋊C414yes—
206C23×C47C23×C411yescharacteristic
207C23×C47C22×D412yes—
208C23×C47C22×D411yescharacteristic
209C23×C47C22×D411yescharacteristic
210C23×C47C2×#4112yes—
211C22×D47—22no—
212C22×D47—24no—
213C22×D47—82no—
214Q8×C47—24no—
215Q8×C47—24no—
216D4×C47—28no—
217D4×C47—28no—
218D4×C47—24no—
219D4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—44no—
227C2×C4×C47—84no—
228C2×C4×C47C2×C22⋊C414yes—
229C2×C4×C47C2×C22⋊C411yescharacteristic
230C2×C4×C47C2×C22⋊C411yescharacteristic
231C2×C4×C47C2×C22⋊C411yescharacteristic
232C2×C4×C47C2×C22⋊C411yescharacteristic
233C2×C4×C47C4×D412yes—
234C2×C4×C47C4×D412yes—
235C2×C4×C47C4×D412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47#12011yescharacteristic
238C2×C4×C47#12012yes—
239C2×C4×C47#12011yescharacteristic
240C2×C4×C47#12112yes—
241C2×C4×C47#12112yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12112yes—
244C2×C4×C47#12311yescharacteristic
245C2×C4×C47#12311yescharacteristic
246C2×C4×C47#12312yes—
247C22⋊D47—42no—
248C22⋊D47—44no—
249C22⋊D47—42no—
250C22⋊D47—88no—
251C2×Dic47—24no—
252C2×Dic47—22no—
253C2×Dic47—24no—
254C2×Dic47—28no—
255C2×Dic47—84no—
256C2×Dic47—88no—
257C2×Dic47—82no—
258D188—41no—
259D188—41no—
260D188—42no—
261C4×D47—42no—
262C4×D47—44no—
263C4×D47—42no—
264Dic94—41no—
265Dic94—41no—
266Dic94—42no—
267#21014—471nomaximal, Sylow
268C22×C47—24no—
269C22×C47—42no—
270C22×C47—41no—
271C22×C47—42no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—82no—
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×C4×D412yes—
279C22×C47C2×#12011yescharacteristic
280C22×C47C2×#12112yes—
281C22×C47C2×#12311yescharacteristic
282C2×D47—44no—
283C2×D47—42no—
284C2×D47—84no—
285C4×C47—24no—
286C4×C47—84no—
287C4×C47#38611yescharacteristic
288C4×C47#38611yescharacteristic
289C4×C47#38612yes—
290C4×C47#38611yescharacteristic
291C4×C47#38612yes—
292C4×C47#38611yescharacteristic
293Dic47—48no—
294Dic47—44no—
295Dic47—42no—
296#3449—471no—
297#3428—471no—
298C2×#410—472no—
299C2×#410D4711yescharacteristic, complemented
300#1913—478no—
301#1740—472no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—22no—
305C2×C47—22no—
306C2×C47—21no—
307C2×C47—44no—
308C2×C47C2×#38611yescharacteristic
309D47—82no—
310C22×#41—471no—
311C22×#41C2×D4711yescharacteristic
312#537—942no—
313#536—472no—
314#536—941no—
315#535—472no—
316#535—941no—
317#534—474no—
318#532—942no—
319#525—474no—
320#525—942no—
321#520—944no—
322#519—941no—
323#518—474no—
324#518—941no—
325C4×#41—944no—
326C2×#117—472no—
327C2×#117—471no—
328C2×#117—942no—
329C2×#117C2×D4712yes—
330#410—22no—
331#410—22no—
332#410—944no—
333#410—944no—
334C2×#99—478no—
335C2×#99C2×D4714yes—
336#342—478no—
337C47#2101411yescharacteristic, complemented, Sylow
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—942no—
345C2×#41—942no—
346C2×#41—942no—
347C2×#41—942no—
348C2×#41—941no—
349C2×#41—941no—
350C2×#41C4×D4712yes—
351C2×#41C4×D4712yes—
352C2×#41C22⋊D4712yes—
353C2×#41C22⋊D4712yes—
354C22×Q8—471no—
355C22×Q8D18811yescharacteristic
356C22×D4—472no—
357C22×D4—471no—
358C22×D4D18811yescharacteristic
359C22×D4C22×D4712yes—
360C23×C4—472no—
361C23×C4D18812yes—
362C23×C4C22×D4711yescharacteristic
363C4⋊Q8—942no—
364#127—942no—
365#124—942no—
366#124—942no—
367#123—478no—
368#123—944no—
369C22⋊Q8—944no—
370C22⋊Q8—944no—
371#121—944no—
372#121—948no—
373#121—944no—
374#120—478no—
375#120—944no—
376C4×Q8—948no—
377C4×D4—9416no—
378C4×D4—948no—
379C4×D4—948no—
380#117—22no—
381#117—22no—
382#117—22no—
383#117—22no—
384#117—472no—
385#117—472no—
386#117—474no—
387#117—474no—
388#117—944no—
389#117—944no—
390#117—942no—
391#117—942no—
392#117—941no—
393#117—941no—
394#117—944no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—944no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—941no—
399C2×C22⋊C4—474no—
400C2×C22⋊C4—472no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4C22×D4714yes—
403C2×C42—22no—
404C2×C42—944no—
405C2×C42—942no—
406C2×C42—941no—
407#99—28no—
408#99—9416no—
409C24—472no—
410C24C2×D18812yes—
411C24C23×D4711yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41—24no—
420#41—1882no—
421#41—1881no—
422#41—1881no—
423#41—1882no—
424#41—1882no—
425#41—1882no—
426#41—1882no—
427#41—1881no—
428#41—1882no—
429#41—1881no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×Q8—942no—
435C2×Q8—942no—
436C2×Q8(C2×D47)⋊C412yes—
437C2×Q8(C2×D47)⋊C412yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—948no—
443C2×D4—944no—
444C2×D4—942no—
445C2×D4—942no—
446C2×D4—944no—
447C2×D4—941no—
448C2×D4—941no—
449C2×D4—942no—
450C2×D4—942no—
451C2×D4(C2×D47)⋊C412yes—
452C2×D4(C2×D47)⋊C412yes—
453C2×D4C2×C4×D4714yes—
454C2×D4C2×C22⋊D4714yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—472no—
461C22×C4—948no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4(C2×D47)⋊C414yes—
469C22×C4(C2×D47)⋊C414yes—
470C22×C4C2×C4×D4711yescharacteristic
471C22×C4C2×C4×D4711yescharacteristic
472C22×C4C2×D18812yes—
473C22×C4C2×D18811yescharacteristic
474C22×C4C2×D18811yescharacteristic
475C22×C4D4×D4712yes—
476C22×C4D4×D4712yes—
477C22×C4#1062535212yes—
478C22×C4#1062535212yes—
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—948no—
484C4⋊C4—948no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—944no—
488C4⋊C4—944no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—474no—
492C22⋊C4—478no—
493C22⋊C4—948no—
494C22⋊C4—948no—
495C22⋊C4—944no—
496C22⋊C4—942no—
497C22⋊C4—948no—
498C42—42no—
499C42—42no—
500C42—948no—
501C42—942no—
502C42—942no—
503C42—944no—
504C42—944no—
505C23—22no—
506C23—22no—
507C23—81no—
508C23—82no—
509C23—944no—
510C23—942no—
511C23—3762no—
512C23C2×(C2×D47)⋊C414yes—
513C23C22×C4×D4711yescharacteristic
514C23C2×D4×D4712yes—
515C23C22×C22⋊D4711yescharacteristic
516C23C22×D18811yescharacteristic
517C23C2×#1062535212yes—
518Q8—24no—
519Q8—24no—
520Q8—1881no—
521Q8—1881no—
522Q8—1882no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—3768no—
528D4—1882no—
529D4—1884no—
530D4—1882no—
531D4—1881no—
532D4—1881no—
533D4—1882no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—44no—
541C2×C4—84no—
542C2×C4—944no—
543C2×C4—942no—
544C2×C4—944no—
545C2×C4—948no—
546C2×C4—3764no—
547C2×C4—3768no—
548C2×C4—3762no—
549C2×C4—1882no—
550C2×C4—1884no—
551C2×C4—1882no—
552C2×C4C2×(C2×D47)⋊C414yes—
553C2×C4C2×(C2×D47)⋊C411yescharacteristic
554C2×C4C2×(C2×D47)⋊C411yescharacteristic
555C2×C4C2×(C2×D47)⋊C411yescharacteristic
556C2×C4C2×(C2×D47)⋊C411yescharacteristic
557C2×C4(C22⋊D47)⋊C412yes—
558C2×C4(C22⋊D47)⋊C412yes—
559C2×C4(C22⋊D47)⋊C412yes—
560C2×C4(C22⋊D47)⋊C412yes—
561C2×C4#1309399412yes—
562C2×C4#1309399411yescharacteristic
563C2×C4#1309399411yescharacteristic
564C2×C4#1309399612yes—
565C2×C4#1309399612yes—
566C2×C4#1309399612yes—
567C2×C4#1309399612yes—
568C2×C4#1309400811yescharacteristic
569C2×C4#1309400812yes—
570C2×C4#1309400811yescharacteristic
571C22—24no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—82no—
580C22—3764no—
581C22—1884no—
582C22—1882no—
583C22C22×(C2×D47)⋊C411yescharacteristic
584C22C2×(C22⋊D47)⋊C412yes—
585C22C2×#1309399411yescharacteristic
586C22C2×#1309399612yes—
587C22C2×#1309400811yescharacteristic
588C4—24no—
589C4—84no—
590C4—1888no—
591C4—1884no—
592C4—1882no—
593C4#1568900011yescharacteristic
594C4#1568900011yescharacteristic
595C4#1568900011yescharacteristic
596C4#1568900012yes—
597C4#1568900012yes—
598C4#1568900011yescharacteristic
599C2—22no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—22no—
604C2—44no—
605C2—3762no—
606C2C2×#1568900011yescharacteristic
subgroupG/H
centerC4#15689000
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#410C2
socleC2×C47C2×#386
prime radical#20256717C2
cosocleC24×C47C24
hypercenterC2×#410D47
hypocenterC47#21014
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. #20256591C2semidirect
3. C2×#15689096C2semidirect
4. #20243928C2semidirect
5. #20243121C2semidirect
6. C2×C47×#410C2semidirect
10. #15689694C4semidirect
12. #15689687C4semidirect
16. #15689630C4semidirect
42. C22×Dic94C2×C4semidirect
43. C22×D188C2×C4semidirect
45. C22×C4×D47C2×C4semidirect
51. #13094005C2×C4semidirect
58. #13093992C2×C4semidirect
70. #13093983C2×C4semidirect
71. #13093983C2×C4semidirect
72. #13093983D4semidirect
73. #13093983D4semidirect
299. C2×#410D47semidirect
337. C47#21014semidirect

Maximal subgroups

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

subgroupindex
#202567172
#202565912
C2×#156890962
#202439282
#202431212
C2×C47×#4102
#2101447