Finite Group Databasebeta

← #26250335

Subgroups — #26250335

Order 12032.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic, complemented
2C2×#15689221C211yesmaximal, characteristic, complemented
3#20248289C211yesmaximal, characteristic
4#20244588C212yesmaximal, complemented
5#20244389C212yesmaximal, complemented
6#20244362C211yesmaximal, characteristic, complemented
7#20244187C211yesmaximal, characteristic, complemented
8#20243705C212yesmaximal, complemented
9#20243576C212yesmaximal, complemented
10#20243403C211yesmaximal, characteristic, complemented
11C2×C47×#554C211yesmaximal, characteristic, complemented
12C22×#10625336C2211yescharacteristic
13C2×#13093998C2212yes—
14C2×C4×D188C2211yescharacteristic
15#15689400—24no—
16#15689221—24no—
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2212yes—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2212yes—
22C2×#13093944C2211yescharacteristic
23C2×#13093944C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×#13093935C2211yescharacteristic
27#15689049—21no—
28#15689031—21no—
29#15688999—22no—
30#15688989C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C47×#554—24no—
36C47×#554—24no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#125×C47—21no—
39C2×C22⋊Q8×C47—22no—
40C2×C22⋊Q8×C47C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4×D4×C47—21no—
43C2×C4×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2212yes—
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—22no—
47C22×C4×D47—21no—
48C2×#10625336—22no—
49C2×#10625336—22no—
50C2×#10625336—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2311yescharacteristic
54#13093998—24no—
55#13093998—42no—
56C4×D188—22no—
57C4×D188—22no—
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2312yes—
60C2×C22⋊Dic47—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—21no—
63#13093957—22no—
64#13093957—24no—
65C4⋊(C47⋊C8)—22no—
66#13093948—22no—
67#13093948—42no—
68#13093944—22no—
69#13093944—24no—
70C22⋊(C47⋊C8)—24no—
71C22⋊(C47⋊C8)—42no—
72#13093935—22no—
73C47⋊#95—22no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C22×Q8×C47—21no—
77C22×Q8×C47C2311yescharacteristic
78C22×D4×C47D411yescharacteristic
79C23×C4×C47—22no—
80C23×C4×C47C2312yes—
81C4⋊Q8×C47—22no—
82C4⋊Q8×C47D414yescomplemented
83#125×C47—28no—
84C22⋊Q8×C47—216no—
85C22⋊Q8×C47—24no—
86C22⋊Q8×C47—24no—
87C4×Q8×C47—22no—
88C4×Q8×C47D414yescomplemented
89C4×D4×C47—28no—
90C4×D4×C47—22no—
91C4×D4×C47—22no—
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—24no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C24×C47C2×D412yes—
106C23×D47—21no—
107C2×Q8×C47—24no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—82no—
114C2×Q8×C47C2×D412yes—
115C2×D4×C47—24no—
116C2×D4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—28no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—82no—
128C22×C4×C47—82no—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—44no—
134C2×C22⋊D47—44no—
135C2×C22⋊D47—82no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188#4112yes—
144C2×C4×D47—42no—
145C2×C4×D47—82no—
146C4⋊C4×C47—28no—
147C4⋊C4×C47—28no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47C2×D412yes—
154C4⋊C4×C47C2×D414yes—
155C22⋊C4×C47—28no—
156C22⋊C4×C47—24no—
157C42×C47—24no—
158C42×C47C2×D412yes—
159C42×C47C2×D412yes—
160C22⋊Dic47—82no—
161#10625336—28no—
162(C2×D47)⋊C4—44no—
163(C2×D47)⋊C4—42no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C2×C47⋊C8—21no—
169C2×C47⋊C8—21no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8#4112yes—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47#13714yes—
184C23×C47C22×D411yescharacteristic
185C22×D47—22no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—42no—
189C22×D47—42no—
190Q8×C47—22no—
191Q8×C47—22no—
192Q8×C47—41no—
193Q8×C47—41no—
194Q8×C47—84no—
195D4×C47—28no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—88no—
214C2×C4×C47#12012yes—
215C2×C4×C47#12012yes—
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47H(5,𝔽2)11yescharacteristic
221C2×C4×C472−1+411yescharacteristic
222C22⋊D47—88no—
223C2×Dic47—41no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228D188—22no—
229D188—22no—
230D188—41no—
231D188—41no—
232C4×D47—84no—
233C47⋊C8—22no—
234C47⋊C8—41no—
235C47⋊C8—42no—
236#24130—471nomaximal, Sylow
237C22×C47—24no—
238C22×C47—24no—
239C22×C47—24no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—42no—
243C22×C47—42no—
244C22×C47C2×#12011yescharacteristic
245C22×C47#54611yescharacteristic
246C22×C47#54711yescharacteristic
247C22×C47C2×#13712yes—
248C22×C47#57612yes—
249C2×D47—42no—
250C2×D47—44no—
251C2×D47—82no—
252C2×D47—81no—
253C2×D47—81no—
254C4×C47—22no—
255C4×C47—22no—
256C4×C47—42no—
257C4×C47—42no—
258C4×C47—42no—
259C4×C47—44no—
260C4×C47#52511yescharacteristic
261C4×C47#53711yescharacteristic
262C4×C47#54811yescharacteristic
263C4×C47#55011yescharacteristic
264Dic47—81no—
265Dic47—82no—
266C2×#554D4711yescharacteristic, complemented
267C2×#474—471no—
268C2×#440—471no—
269#2535—471no—
270#2036—472no—
271#2000—472no—
272#1993—471no—
273#1955—471no—
274#1874—472no—
275#1852—472no—
276#1805—471no—
277C2×C47—44no—
278C2×C47#240711yescharacteristic
279C2×C47#300411yescharacteristic
280C2×C47#300811yescharacteristic
281C2×C47#331412yes—
282C2×C47#334612yes—
283D47—82no—
284C22×SD16—471no—
285#554—24no—
286#554—24no—
287C2×C4⋊Q8C2×D4711yescharacteristic
288C2×#125—21no—
289C2×C22⋊Q8—22no—
290C2×C22⋊Q8C2×D4712yes—
291C2×#121—472no—
292C2×C4×Q8C2×D4711yescharacteristic
293C2×C4×D4—21no—
294C2×C4×D4—471no—
295C2×C4×D4C2×D4711yescharacteristic
296C22×C4⋊C4C2×D4712yes—
297#474—944no—
298#440—944no—
299C2×#107—471no—
300C2×#107—472no—
301C2×C4⋊C8—471no—
302C2×#103—471no—
303C2×#103—472no—
304C2×#102—471no—
305C2×#102—472no—
306C2×C22⋊C8—472no—
307C2×C22⋊C8—472no—
308C2×#97—471no—
309#397—941no—
310#392—941no—
311#386—942no—
312#382—471no—
313#380—472no—
314#340—472no—
315#340—472no—
316#340—472no—
317C47#2413011yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8C22×D4711yescharacteristic
320C22×D4—471no—
321C22×D4—942no—
322C22×D4C22⋊D4711yescharacteristic
323C23×C4—22no—
324C23×C4—941no—
325C23×C4C22×D4712yes—
326C2×SD16—942no—
327C2×SD16—942no—
328C2×SD16—942no—
329C22×C8—471no—
330C22×C8—471no—
331C22×C8—472no—
332C4⋊Q8—22no—
333C4⋊Q8C22⋊D4714yescomplemented
334#125—28no—
335C22⋊Q8—216no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338#121—944no—
339#121—1882no—
340C4×Q8—22no—
341C4×Q8C22⋊D4714yescomplemented
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—942no—
346C4×D4—942no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4C22⋊D4711yescharacteristic
356C2×C4⋊C4C22×D4711yescharacteristic
357C2×C4⋊C4C22×D4712yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—942no—
360C2×C22⋊C4—941no—
361C2×C22⋊C4—942no—
362C2×C22⋊C4C22⋊D4712yes—
363C2×C42—21no—
364C2×C42C22×D4711yescharacteristic
365#107—942no—
366#107—944no—
367C4⋊C8—942no—
368#103—942no—
369#103—1882no—
370#102—942no—
371#102—944no—
372C22⋊C8—944no—
373C22⋊C8—1882no—
374#97—942no—
375#95—942no—
376#95—941no—
377#95—941no—
378C24—941no—
379C24C2×C22⋊D4712yes—
380C2×Q8—24no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—82no—
387C2×Q8C2×C22⋊D4712yes—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—472no—
391C2×D4—941no—
392C2×D4—941no—
393C2×D4—941no—
394C2×D4—941no—
395C2×D4—941no—
396C2×D4—3762no—
397C2×D4—1884no—
398C2×D4—1884no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—942no—
412C22×C4—941no—
413C22×C4—3762no—
414C22×C4—1882no—
415C22×C4C2×C22⋊D4712yes—
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C2×C22⋊D4711yescharacteristic
418C22×C4C23×D4711yescharacteristic
419SD16—948no—
420C2×C8—472no—
421C2×C8—941no—
422C2×C8—941no—
423C2×C8—941no—
424C2×C8—942no—
425C2×C8—942no—
426C2×C8—941no—
427C2×C8—942no—
428C2×C8—941no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—941no—
437C4⋊C4—941no—
438C4⋊C4—942no—
439C4⋊C4—942no—
440C4⋊C4C2×C22⋊D4712yes—
441C4⋊C4C2×C22⋊D4714yes—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—3762no—
445C22⋊C4—1884no—
446C22⋊C4—1882no—
447C42—24no—
448C42C2×C22⋊D4712yes—
449C42C2×C22⋊D4712yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—942no—
457C23—1881no—
458C23—1881no—
459C23—1882no—
460C23—1882no—
461C23#1309405514yes—
462C23C22×C22⋊D4711yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—84no—
468D4—28no—
469D4—942no—
470D4—942no—
471D4—3768no—
472D4—1881no—
473D4—1881no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—88no—
492C2×C4—3764no—
493C2×C4—1882no—
494C2×C4—1881no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4—1882no—
498C2×C4#1309399312yes—
499C2×C4#1309399312yes—
500C2×C4C22×C22⋊D4711yescharacteristic
501C2×C4C22×C22⋊D4711yescharacteristic
502C2×C4#41×D4711yescharacteristic
503C2×C4#41×D4711yescharacteristic
504C2×C4#1309407011yescharacteristic
505C2×C4#1309407111yescharacteristic
506C8—942no—
507C8—1882no—
508C8—1881no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—3762no—
517C22—3761no—
518C22—3761no—
519C22—1882no—
520C22—1884no—
521C22C2×#1309399311yescharacteristic
522C22#1568976111yescharacteristic
523C22#1568977211yescharacteristic
524C22C2×#1309405512yes—
525C22#1568992612yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—3762no—
533C4—3761no—
534C4#1568963011yescharacteristic
535C4#1568970111yescharacteristic
536C4#1568978011yescharacteristic
537C4#1568979211yescharacteristic
538C2—44no—
539C2—3762no—
540C2#2024672711yescharacteristic
541C2#2025262211yescharacteristic
542C2#2025265611yescharacteristic
543C2#2025546312yes—
544C2#2025586712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#554C2
socleC23×C47C22×D4
prime radical#20248289C2
hypercenterC2×#554D47
hypocenterC47#24130
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#15689400C2semidirect
2. C2×#15689221C2semidirect
4. #20244588C2semidirect
5. #20244389C2semidirect
6. #20244362C2semidirect
7. #20244187C2semidirect
8. #20243705C2semidirect
9. #20243576C2semidirect
10. #20243403C2semidirect
11. C2×C47×#554C2semidirect
82. C4⋊Q8×C47D4semidirect
88. C4×Q8×C47D4semidirect
266. C2×#554D47semidirect
317. C47#24130semidirect
333. C4⋊Q8C22⋊D47semidirect
341. C4×Q8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C2×#156892212
#202482892
#202445882
#202443892
#202443622
#202441872
#202437052
#202435762
#202434032
C2×C47×#5542
#2413047