Finite Group Databasebeta

← #26123498

Subgroups — #26123498

Order 12032.

882 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689151C211yesmaximal, characteristic, complemented
2#20251908C211yesmaximal, characteristic
3C2×#15689117C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20243120C211yesmaximal, characteristic, complemented
6#20243104C211yesmaximal, characteristic, complemented
7C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C23×C47⋊C8C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689151—22no—
13#15689151—22no—
14C2×Dic47⋊C8—22no—
15#15689133—21no—
16#15689129—21no—
17#15689125—22no—
18#15689120—22no—
19#15689117—22no—
20#15689117—22no—
21C2×#13093947—21no—
22C2×#13093944—21no—
23#15689096—22no—
24#15689096C414yescomplemented
25C2×#13093938—22no—
26#15688889—22no—
27#15688885—22no—
28C22×M16×C47C2211yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C2×#117×C47C2211yescharacteristic
31C47×H(3,ℤ/4)—22no—
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×#10625350—21no—
42C2×#10625350C2×C412yes—
43C2×#10625350Q812yes—
44C22×Dic94D411yescharacteristic
45C22×D188D411yescharacteristic
46C22×C22⋊D47—21no—
47C22×C4×D47—21no—
48C22×C47⋊C8—21no—
49C22×C47⋊C8—21no—
50C22×C47⋊C8—21no—
51C22×C47⋊C8—21no—
52C22×C47⋊C8—21no—
53C22×C47⋊C8—21no—
54C22×C47⋊C8D412yes—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983—21no—
60#13093983—21no—
61#13093983C2×C412yes—
62#13093983D412yes—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47D412yes—
65C2×C22⋊Dic47—21no—
66C2×C22⋊Dic47D412yes—
67C2×C4×Dic47—21no—
68C2×C4×Dic47—21no—
69Dic47⋊C8—28no—
70#13093949—24no—
71#13093947—24no—
72#13093946—24no—
73#13093944—24no—
74#13093938—28no—
75C23×C4×C47C2311yescharacteristic
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47—21no—
81C2×M16×C47—21no—
82C22×C8×C47D412yes—
83#117×C47—21no—
84#117×C47—21no—
85#117×C47—21no—
86#117×C47—21no—
87#117×C47—21no—
88#117×C47—21no—
89#117×C47C2×C412yes—
90#117×C47D412yes—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C24×C47C2×D411yescharacteristic
98C23×D47—21no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×Q811yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C2×C22⋊D47—22no—
116C2×C22⋊D47—22no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121C22×Dic47—22no—
122C22×Dic47—21no—
123C22×Dic47—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188C4⋊C412yes—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94C4⋊C412yes—
158M16×C47—41no—
159M16×C47—41no—
160M16×C47—41no—
161M16×C47—41no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—41no—
168C4⋊C4×C47—41no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—22no—
171C22⋊C4×C47—41no—
172C22⋊C4×C47—41no—
173C42×C47—22no—
174C42×C47—22no—
175C42×C47—41no—
176C42×C47—41no—
177C22⋊Dic47—22no—
178C22⋊Dic47—41no—
179C22⋊Dic47—41no—
180C22⋊Dic47C22⋊C414yes—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—41no—
188C4×Dic47—41no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—41no—
193C2×C47⋊C8—41no—
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—41no—
206C22×D47—41no—
207C22×D47—82no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—82no—
211C2×C4×C47—82no—
212C2×C4×C47C2×C4⋊C411yescharacteristic
213C2×C4×C47#11711yescharacteristic
214C2×C4×C47C4×D411yescharacteristic
215C2×C4×C47C4×D411yescharacteristic
216C2×C4×C47C4×D411yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47#12011yescharacteristic
219C2×C4×C47#12011yescharacteristic
220C2×C4×C47#12011yescharacteristic
221C2×C4×C47#12111yescharacteristic
222C2×C4×C47#12111yescharacteristic
223C2×C4×C47#12111yescharacteristic
224C2×C4×C47C22⋊Q811yescharacteristic
225C2×C4×C47C22⋊Q811yescharacteristic
226C2×C4×C47C22⋊Q811yescharacteristic
227C2×C4×C47C22⋊Q811yescharacteristic
228C2×C4×C47C22⋊Q811yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12311yescharacteristic
232C2×C4×C47#12311yescharacteristic
233C2×C4×C47#12311yescharacteristic
234C2×C4×C47#12411yescharacteristic
235C2×C4×C47#12411yescharacteristic
236C2×C4×C47#12411yescharacteristic
237C2×C4×C47#12611yescharacteristic
238C2×C4×C47#12611yescharacteristic
239C2×C4×C47#12711yescharacteristic
240C22⋊D47—42no—
241C22⋊D47—42no—
242C22⋊D47—42no—
243C22⋊D47—42no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—44no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—42no—
250C2×Dic47—82no—
251D188—22no—
252D188—22no—
253D188—41no—
254D188—41no—
255C4×D47—42no—
256C4×D47—42no—
257C4×D47—42no—
258C4×D47—42no—
259Dic94—22no—
260Dic94—22no—
261Dic94—41no—
262Dic94—41no—
263C8×C47—42no—
264C47⋊C8—42no—
265#11092—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47#38011yescharacteristic
274C22×C47#38611yescharacteristic
275C22×C47#39011yescharacteristic
276C22×C47#39411yescharacteristic
277C22×C47#39411yescharacteristic
278C22×C47#39611yescharacteristic
279C22×C47#39611yescharacteristic
280C2×D47—42no—
281C2×D47—82no—
282C2×D47—82no—
283C2×D47—82no—
284C4×C47—82no—
285C4×C47—82no—
286C4×C47#38511yescharacteristic
287C4×C47#38511yescharacteristic
288C4×C47#38811yescharacteristic
289C4×C47#38811yescharacteristic
290C4×C47#39211yescharacteristic
291C4×C47#39311yescharacteristic
292C4×C47#39711yescharacteristic
293C4×C47#39911yescharacteristic
294Dic47—44no—
295Dic47—42no—
296Dic47—82no—
297C2×#424—471no—
298#2896—471no—
299C2×#416—471no—
300C2×#410—471no—
301#1740—471no—
302#1735—471no—
303C2×H(3,ℤ/4)D4711yescharacteristic, complemented
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47—21no—
308C2×C47—21no—
309C2×C47—21no—
310C2×C47#143911yescharacteristic
311C2×C47#205312yes—
312D47—82no—
313C22×#41—471no—
314C22×M16C2×D4711yescharacteristic
315C23×C8—471no—
316C2×#117—471no—
317C2×#117—471no—
318C2×#117C2×D4711yescharacteristic
319C2×#117C2×D4711yescharacteristic
320#424—942no—
321#424—942no—
322C2×C4⋊C8—942no—
323#419—941no—
324#418—941no—
325#417—942no—
326#416—942no—
327#416—942no—
328#416—942no—
329C2×#103—941no—
330C2×#102—941no—
331#410—474no—
332#410—942no—
333C2×#99—942no—
334#342—942no—
335#340—942no—
336H(3,ℤ/4)—22no—
337C47#1109211yescharacteristic, complemented, Sylow
338C2×#41—472no—
339C2×#41—472no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—941no—
347C2×#41—941no—
348C2×#41—941no—
349C2×#41—941no—
350C22×Q8—471no—
351C22×D4—471no—
352C22×D4—941no—
353C23×C4—941no—
354C23×C4C22×D4711yescharacteristic
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C22×C8—472no—
362C22×C8—941no—
363C22×C8—941no—
364C22×C8—941no—
365C22×C8—941no—
366C22×C8—941no—
367C22×C8—941no—
368C22×C8D18812yes—
369#117—21no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—472no—
376#117—472no—
377#117—941no—
378#117—941no—
379#117—941no—
380#117—941no—
381#117—941no—
382#117—941no—
383#117C4×D4712yes—
384#117C22⋊D4712yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—472no—
388C2×C4⋊C4—941no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—472no—
392C2×C22⋊C4—941no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—941no—
396C2×C42—941no—
397C4⋊C8—948no—
398#103—944no—
399#103—944no—
400#102—944no—
401#102—944no—
402#99—948no—
403C24—941no—
404C24D4×D4711yescharacteristic
405#41—942no—
406#41—942no—
407#41—942no—
408#41—942no—
409#41—942no—
410#41—942no—
411#41—942no—
412#41—942no—
413#41—1881no—
414#41—1881no—
415#41—1881no—
416#41—1881no—
417#41—1881no—
418#41—1881no—
419#41—1881no—
420#41—1881no—
421C2×Q8—472no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×D4—472no—
428C2×D4—942no—
429C2×D4—942no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—1881no—
436C2×D4—1881no—
437C2×D4—1881no—
438C2×D4—1881no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—941no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—1881no—
447C22×C4—1881no—
448C22×C4—1881no—
449C22×C4—1881no—
450C22×C4C2×C4×D4711yescharacteristic
451C22×C4C2×D18811yescharacteristic
452C22×C4#1062535011yescharacteristic
453C22×C4#1062535011yescharacteristic
454C22×C4#1062535011yescharacteristic
455C22×C4#1062535011yescharacteristic
456C22×C4D4×D4711yescharacteristic
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4#1062535211yescharacteristic
461C22×C4Q8×D4711yescharacteristic
462C22×C4#1062535411yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—1881no—
475C2×C8—1881no—
476C2×C8—1881no—
477C2×C8—1881no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—942no—
483C4⋊C4—942no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—474no—
491C22⋊C4—942no—
492C22⋊C4—1881no—
493C22⋊C4—1881no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—942no—
499C42—942no—
500C42—1881no—
501C42—1881no—
502C23—42no—
503C23—942no—
504C23—3762no—
505C23—1881no—
506C23—1881no—
507C23C22⋊C4×D4711yescharacteristic
508C23#1309399611yescharacteristic
509C23#1309399711yescharacteristic
510C23#1309399711yescharacteristic
511C23#1309399711yescharacteristic
512C23#1309399911yescharacteristic
513C23#1309400211yescharacteristic
514Q8—942no—
515Q8—942no—
516Q8—1881no—
517Q8—1881no—
518D4—942no—
519D4—942no—
520D4—1882no—
521D4—1882no—
522D4—1882no—
523D4—1882no—
524D4—1881no—
525D4—1881no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—942no—
531C2×C4—944no—
532C2×C4—3762no—
533C2×C4—1882no—
534C2×C4—1884no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4C4⋊C4×D4711yescharacteristic
542C2×C4#1309398111yescharacteristic
543C2×C4(C22⋊D47)⋊C411yescharacteristic
544C2×C4C4×C22⋊D4711yescharacteristic
545C2×C4#1309398611yescharacteristic
546C2×C4C4×D18811yescharacteristic
547C2×C4#1309399211yescharacteristic
548C2×C4#1309399411yescharacteristic
549C2×C4#1309399411yescharacteristic
550C2×C4#1309399711yescharacteristic
551C2×C4#1309399711yescharacteristic
552C2×C4#1309399811yescharacteristic
553C2×C4#1309400011yescharacteristic
554C2×C4#1309400211yescharacteristic
555C2×C4#1309400211yescharacteristic
556C2×C4#1309400211yescharacteristic
557C2×C4#1309400411yescharacteristic
558C2×C4#1309400611yescharacteristic
559C2×C4#1309400711yescharacteristic
560C2×C4#1309400911yescharacteristic
561C2×C4#1309400911yescharacteristic
562C2×C4#1309400911yescharacteristic
563C2×C4#1309401011yescharacteristic
564C2×C4#1309401011yescharacteristic
565C2×C4#1309401211yescharacteristic
566C2×C4#1309401811yescharacteristic
567C2×C4#1309401911yescharacteristic
568C2×C4#1309402011yescharacteristic
569C8—42no—
570C8—1882no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3762no—
579C22—3762no—
580C22—3762no—
581C22—1882no—
582C22#1568898211yescharacteristic
583C22#1568900111yescharacteristic
584C22#1568902311yescharacteristic
585C22#1568903811yescharacteristic
586C22#1568904111yescharacteristic
587C22#1568904511yescharacteristic
588C22#1568904711yescharacteristic
589C4—82no—
590C4—82no—
591C4—3762no—
592C4—1884no—
593C4—1882no—
594C4#1568899411yescharacteristic
595C4#1568899711yescharacteristic
596C4#1568901211yescharacteristic
597C4#1568901411yescharacteristic
598C4#1568903211yescharacteristic
599C4#1568903611yescharacteristic
600C4#1568905211yescharacteristic
601C4#1568905811yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—3762no—
609C2#2024163211yescharacteristic
610C2#2024469512yes—
subgroupG/H
centerC2×C4#13094012
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC22×C47#394
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47#11092
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#15689151C2semidirect
5. #20243120C2semidirect
6. #20243104C2semidirect
7. C2×C47×H(3,ℤ/4)C2semidirect
24. #15689096C4semidirect
303. C2×H(3,ℤ/4)D47semidirect
337. C47#11092semidirect

Maximal subgroups

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

subgroupindex
C2×#156891512
#202519082
C2×#156891172
C2×#156890962
#202431202
#202431042
C2×C47×H(3,ℤ/4)2
#1109247