Finite Group Databasebeta

← #26123266

Subgroups — #26123266

Order 12032.

942 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251951C211yesmaximal, characteristic, complemented
2C2×#15689141C211yesmaximal, characteristic
3C2×#15689139C211yesmaximal, characteristic
4#20245878C211yesmaximal, characteristic
5#20243120C211yesmaximal, characteristic, complemented
6C2×#15688883C211yesmaximal, characteristic, complemented
7C2×C47×#339C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689151—22no—
13#15689149—22no—
14#15689146—22no—
15#15689141—22no—
16#15689141—22no—
17#15689139—22no—
18#15689139—22no—
19C2×#13093952—22no—
20C2×#13093950—22no—
21#15689029C412yescomplemented
22#15689025C412yescomplemented
23#15689002—22no—
24C4×(C2×D47)⋊C4—22no—
25#15688889—22no—
26#15688883—22no—
27C22×M16×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C47×#339—22no—
31C2×#10625350—21no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350D412yes—
42C2×#10625350Q812yes—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329D412yes—
53C22×C47⋊C8—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983—21no—
60#13093983—21no—
61#13093983D412yes—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47C2×C412yes—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47D412yes—
66C2×(C2×D47)⋊C4—24no—
67C2×C4×Dic47—21no—
68C2×C4×Dic47D412yes—
69Dic47⋊C8—28no—
70#13093952—28no—
71#13093950—28no—
72C47⋊#95—24no—
73C23×C4×C47—21no—
74C23×C4×C47C2311yescharacteristic
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47D412yes—
81C22×C8×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47—21no—
85#117×C47D412yes—
86C2×C4⋊C4×C47—21no—
87C2×C22⋊C4×C47—21no—
88C2×C42×C47—21no—
89C2×C42×C47—21no—
90C2×C42×C47—21no—
91C2×C42×C47—21no—
92C2×C42×C47—21no—
93C2×C42×C47—21no—
94C2×C42×C47C2×C412yes—
95C24×C47C2×D411yescharacteristic
96C23×D47—21no—
97C22×C4×C47—22no—
98C22×C4×C47—44no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47—41no—
102C22×C4×C47C22×C411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×Q811yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C22×C4×C47#4111yescharacteristic
116C2×C22⋊D47—22no—
117C2×C22⋊D47—22no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121C2×C22⋊D47—41no—
122C22×Dic47—21no—
123C22×Dic47—41no—
124C22×Dic47C22⋊C414yes—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188C4⋊C412yes—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×C4×D47—41no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94C4⋊C412yes—
159M16×C47—22no—
160M16×C47—22no—
161M16×C47—41no—
162M16×C47—41no—
163C2×C8×C47—22no—
164C2×C8×C47—41no—
165C2×C8×C47—41no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—41no—
168C22⋊C4×C47—22no—
169C22⋊C4×C47—41no—
170C42×C47—22no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—41no—
174C42×C47—41no—
175C42×C47—41no—
176C22⋊Dic47—22no—
177C22⋊Dic47—22no—
178C22⋊Dic47—22no—
179C22⋊Dic47—41no—
180(C2×D47)⋊C4—48no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—41no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C4×Dic47—41no—
189#10625329—22no—
190#10625329—22no—
191#10625329—41no—
192#10625329—41no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—41no—
195C2×C47⋊C8—41no—
196C23×C47—42no—
197C23×C47C2×C22⋊C411yescharacteristic
198C23×C47#12111yescharacteristic
199C23×C47#12111yescharacteristic
200C23×C47#12111yescharacteristic
201C23×C47#12111yescharacteristic
202C23×C47#12111yescharacteristic
203C23×C47C22⋊Q811yescharacteristic
204C22×D47—22no—
205C22×D47—44no—
206C22×D47—41no—
207C22×D47—41no—
208C2×C4×C47—24no—
209C2×C4×C47—42no—
210C2×C4×C47—44no—
211C2×C4×C47—44no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—82no—
215C2×C4×C47C2×C4⋊C411yescharacteristic
216C2×C4×C47#11711yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C4×D411yescharacteristic
221C2×C4×C47#12011yescharacteristic
222C2×C4×C47#12011yescharacteristic
223C2×C4×C47#12011yescharacteristic
224C2×C4×C47#12111yescharacteristic
225C2×C4×C47#12111yescharacteristic
226C2×C4×C47#12111yescharacteristic
227C2×C4×C47C22⋊Q811yescharacteristic
228C2×C4×C47C22⋊Q811yescharacteristic
229C2×C4×C47C22⋊Q811yescharacteristic
230C2×C4×C47C22⋊Q811yescharacteristic
231C2×C4×C47C22⋊Q811yescharacteristic
232C2×C4×C47#12311yescharacteristic
233C2×C4×C47#12311yescharacteristic
234C2×C4×C47#12311yescharacteristic
235C2×C4×C47#12311yescharacteristic
236C2×C4×C47#12311yescharacteristic
237C2×C4×C47#12411yescharacteristic
238C2×C4×C47#12411yescharacteristic
239C2×C4×C47#12411yescharacteristic
240C2×C4×C47#12611yescharacteristic
241C2×C4×C47#12611yescharacteristic
242C2×C4×C47#12711yescharacteristic
243C22⋊D47—42no—
244C22⋊D47—42no—
245C22⋊D47—42no—
246C22⋊D47—42no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—44no—
252C2×Dic47—42no—
253C2×Dic47—82no—
254D188—22no—
255D188—22no—
256D188—41no—
257D188—41no—
258C4×D47—42no—
259C4×D47—42no—
260C4×D47—42no—
261C4×D47—42no—
262Dic94—22no—
263Dic94—22no—
264Dic94—41no—
265Dic94—41no—
266C8×C47—42no—
267C47⋊C8—42no—
268#11056—471nomaximal, Sylow
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47#38011yescharacteristic
277C22×C47#38611yescharacteristic
278C22×C47#39011yescharacteristic
279C22×C47#39411yescharacteristic
280C22×C47#39411yescharacteristic
281C22×C47#39611yescharacteristic
282C22×C47#39611yescharacteristic
283C2×D47—44no—
284C2×D47—42no—
285C2×D47—82no—
286C2×D47—82no—
287C4×C47—44no—
288C4×C47—82no—
289C4×C47#38511yescharacteristic
290C4×C47#38511yescharacteristic
291C4×C47#38811yescharacteristic
292C4×C47#38811yescharacteristic
293C4×C47#39211yescharacteristic
294C4×C47#39311yescharacteristic
295C4×C47#39711yescharacteristic
296C4×C47#39911yescharacteristic
297Dic47—44no—
298Dic47—42no—
299Dic47—82no—
300#2905—471no—
301C2×#421—471no—
302C2×#421—471no—
303#2297—471no—
304#1740—471no—
305C2×#339—471no—
306C2×#339D4711yescharacteristic, complemented
307C2×C47—21no—
308C2×C47—21no—
309C2×C47—21no—
310C2×C47—21no—
311C2×C47—21no—
312C2×C47—21no—
313C2×C47#143911yescharacteristic
314C2×C47#188412yes—
315D47—82no—
316C22×#41—471no—
317C22×M16—471no—
318C22×M16C2×D4711yescharacteristic
319C2×#117—471no—
320C2×#117—471no—
321C2×#117C2×D4711yescharacteristic
322C22×C42C2×D4711yescharacteristic
323#424—942no—
324#424—942no—
325#423—942no—
326#421—942no—
327#421—942no—
328#421—942no—
329#421—942no—
330C2×#104—942no—
331C2×#104—942no—
332#391—472no—
333#390—472no—
334#386—942no—
335C4×C22⋊C4—942no—
336#342—942no—
337#339—22no—
338#339—942no—
339C47#1105611yescharacteristic, complemented, Sylow
340C2×#41—472no—
341C2×#41—472no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—941no—
347C2×#41—941no—
348C2×#41—941no—
349C2×#41—941no—
350C2×#41—941no—
351C2×#41—941no—
352C22×Q8—471no—
353C22×D4—471no—
354C22×D4—941no—
355C23×C4—21no—
356C23×C4—941no—
357C23×C4C22×D4711yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—472no—
364C2×M16—941no—
365C2×M16—941no—
366C2×M16—941no—
367C2×M16—941no—
368C2×M16—941no—
369C2×M16D18812yes—
370C22×C8—21no—
371C22×C8—941no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—472no—
376#117—941no—
377#117—941no—
378#117—941no—
379#117—941no—
380#117—941no—
381#117—941no—
382#117—941no—
383#117C22⋊D4712yes—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—472no—
386C2×C4⋊C4—941no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—472no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—944no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—472no—
398C2×C42—941no—
399C2×C42C4×D4712yes—
400C4⋊C8—948no—
401#104—948no—
402#104—948no—
403#95—944no—
404C24—941no—
405C24D4×D4711yescharacteristic
406#41—942no—
407#41—942no—
408#41—942no—
409#41—942no—
410#41—942no—
411#41—942no—
412#41—942no—
413#41—942no—
414#41—1881no—
415#41—1881no—
416#41—1881no—
417#41—1881no—
418#41—1881no—
419#41—1881no—
420#41—1881no—
421#41—1881no—
422C2×Q8—472no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×D4—472no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—1881no—
437C2×D4—1881no—
438C2×D4—1881no—
439C2×D4—1881no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—474no—
446C22×C4—941no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—1881no—
450C22×C4—1881no—
451C22×C4—1881no—
452C22×C4—1881no—
453C22×C4—1881no—
454C22×C4C2×C4×D4711yescharacteristic
455C22×C4C2×D18811yescharacteristic
456C22×C4#1062535011yescharacteristic
457C22×C4#1062535011yescharacteristic
458C22×C4#1062535011yescharacteristic
459C22×C4#1062535011yescharacteristic
460C22×C4D4×D4711yescharacteristic
461C22×C4D4×D4711yescharacteristic
462C22×C4D4×D4711yescharacteristic
463C22×C4D4×D4711yescharacteristic
464C22×C4#1062535211yescharacteristic
465C22×C4Q8×D4711yescharacteristic
466C22×C4#1062535411yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468M16—22no—
469M16—22no—
470M16—41no—
471M16—41no—
472M16—942no—
473M16—942no—
474M16—1881no—
475M16—1881no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—942no—
480C2×C8—1881no—
481C2×C8—1881no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—1881no—
488C22⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—942no—
491C22⋊C4—942no—
492C22⋊C4—942no—
493C22⋊C4—1888no—
494C22⋊C4—1881no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—942no—
502C42—942no—
503C42—942no—
504C42—1881no—
505C23—42no—
506C23—942no—
507C23—1884no—
508C23—1881no—
509C23—1881no—
510C23C22⋊C4×D4711yescharacteristic
511C23#1309399611yescharacteristic
512C23#1309399711yescharacteristic
513C23#1309399711yescharacteristic
514C23#1309399711yescharacteristic
515C23#1309399911yescharacteristic
516C23#1309400211yescharacteristic
517Q8—942no—
518Q8—942no—
519Q8—1881no—
520Q8—1881no—
521D4—942no—
522D4—942no—
523D4—1882no—
524D4—1882no—
525D4—1882no—
526D4—1882no—
527D4—1881no—
528D4—1881no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—942no—
537C2×C4—944no—
538C2×C4—3762no—
539C2×C4—1882no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1884no—
547C2×C4C4⋊C4×D4711yescharacteristic
548C2×C4#1309398111yescharacteristic
549C2×C4(C22⋊D47)⋊C411yescharacteristic
550C2×C4C4×C22⋊D4711yescharacteristic
551C2×C4#1309398611yescharacteristic
552C2×C4C4×D18811yescharacteristic
553C2×C4#1309399211yescharacteristic
554C2×C4#1309399411yescharacteristic
555C2×C4#1309399411yescharacteristic
556C2×C4#1309399711yescharacteristic
557C2×C4#1309399711yescharacteristic
558C2×C4#1309399811yescharacteristic
559C2×C4#1309400011yescharacteristic
560C2×C4#1309400211yescharacteristic
561C2×C4#1309400211yescharacteristic
562C2×C4#1309400211yescharacteristic
563C2×C4#1309400411yescharacteristic
564C2×C4#1309400611yescharacteristic
565C2×C4#1309400711yescharacteristic
566C2×C4#1309400911yescharacteristic
567C2×C4#1309400911yescharacteristic
568C2×C4#1309400911yescharacteristic
569C2×C4#1309401011yescharacteristic
570C2×C4#1309401011yescharacteristic
571C2×C4#1309401211yescharacteristic
572C2×C4#1309401811yescharacteristic
573C2×C4#1309401911yescharacteristic
574C2×C4#1309402011yescharacteristic
575C8—42no—
576C8—1882no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—3762no—
585C22—3762no—
586C22—1884no—
587C22—1882no—
588C22#1568898211yescharacteristic
589C22#1568900111yescharacteristic
590C22#1568902311yescharacteristic
591C22#1568903811yescharacteristic
592C22#1568904111yescharacteristic
593C22#1568904511yescharacteristic
594C22#1568904711yescharacteristic
595C4—44no—
596C4—82no—
597C4—3762no—
598C4—1884no—
599C4—1882no—
600C4#1568899411yescharacteristic
601C4#1568899711yescharacteristic
602C4#1568901211yescharacteristic
603C4#1568901411yescharacteristic
604C4#1568903211yescharacteristic
605C4#1568903611yescharacteristic
606C4#1568905211yescharacteristic
607C4#1568905811yescharacteristic
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—3762no—
615C2#2024163211yescharacteristic
616C2#2024377012yes—
subgroupG/H
centerC2×C4#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#339C2
socleC22×C47#386
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×#339D47
hypocenterC47#11056
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. #20251951C2semidirect
5. #20243120C2semidirect
6. C2×#15688883C2semidirect
7. C2×C47×#339C2semidirect
21. #15689029C4semidirect
22. #15689025C4semidirect
306. C2×#339D47semidirect
339. C47#11056semidirect

Maximal subgroups

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

subgroupindex
#202519512
C2×#156891412
C2×#156891392
#202458782
#202431202
C2×#156888832
C2×C47×#3392
#1105647