Finite Group Databasebeta

← #26123630

Subgroups — #26123630

Order 12032.

848 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689171C211yesmaximal, characteristic, complemented
2C2×#15689141C211yesmaximal, characteristic
3#20251905C211yesmaximal, characteristic
4C2×#15689105C211yesmaximal, characteristic
5C2×#15688893C211yesmaximal, characteristic, complemented
6#20243127C211yesmaximal, characteristic, complemented
7C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689171—22no—
13#15689171—22no—
14C2×#13093959—22no—
15#15689141—22no—
16#15689141—22no—
17C2×#13093952C414yescomplemented
18#15689131—21no—
19#15689127—21no—
20#15689122—22no—
21#15689117—22no—
22#15689105—22no—
23#15689105—22no—
24C2×#13093942—21no—
25C2×#13093940—21no—
26#15688893—22no—
27#15688892—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—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329—21no—
60C2×#10625329—21no—
61C2×#10625329C2×C412yes—
62C2×#10625329D412yes—
63C2×#10625329D412yes—
64C2×#10625329D412yes—
65C2×#10625329D412yes—
66C22×C47⋊C8—21no—
67C22×C47⋊C8—21no—
68C22×C47⋊C8—21no—
69#13093959—28no—
70#13093952—28no—
71#13093947—24no—
72#13093944—24no—
73#13093942—24no—
74#13093940—24no—
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×C47D412yes—
82C22×C8×C47—21no—
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×C47—21no—
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×C47C2×C412yes—
97C24×C47#4111yescharacteristic
98C23×D47—21no—
99C22×C4×C47—22no—
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×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
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—21no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188C4⋊C412yes—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—41no—
147C2×C4×D47—41no—
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94C4⋊C412yes—
156M16×C47—22no—
157M16×C47—22no—
158M16×C47—41no—
159M16×C47—41no—
160C2×C8×C47—22no—
161C2×C8×C47—41no—
162C2×C8×C47—41no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—41no—
165C4⋊C4×C47—41no—
166C4⋊C4×C47—41no—
167C22⋊C4×C47—22no—
168C22⋊C4×C47—41no—
169C22⋊C4×C47—41no—
170C22⋊C4×C47—41no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—41no—
175#10625329—22no—
176#10625329—22no—
177#10625329—22no—
178#10625329—22no—
179#10625329—22no—
180#10625329—22no—
181#10625329—22no—
182#10625329—41no—
183#10625329—41no—
184#10625329—41no—
185#10625329—41no—
186#10625329C22⋊C414yes—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
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—81no—
197C23×C47C4×D411yescharacteristic
198C23×C47#12111yescharacteristic
199C23×C47C22⋊Q811yescharacteristic
200C23×C47#12311yescharacteristic
201C23×C47#12411yescharacteristic
202C23×C47#12411yescharacteristic
203C23×C47#12611yescharacteristic
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×C22⋊C411yescharacteristic
213C2×C4×C47C2×C4⋊C411yescharacteristic
214C2×C4×C47#11711yescharacteristic
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×C47#12111yescharacteristic
225C2×C4×C47#12111yescharacteristic
226C2×C4×C47#12111yescharacteristic
227C2×C4×C47#12111yescharacteristic
228C2×C4×C47C22⋊Q811yescharacteristic
229C2×C4×C47C22⋊Q811yescharacteristic
230C2×C4×C47C22⋊Q811yescharacteristic
231C2×C4×C47C22⋊Q811yescharacteristic
232C2×C4×C47C22⋊Q811yescharacteristic
233C2×C4×C47#12311yescharacteristic
234C2×C4×C47#12311yescharacteristic
235C2×C4×C47#12311yescharacteristic
236C2×C4×C47#12311yescharacteristic
237C2×C4×C47#12411yescharacteristic
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—41no—
246C2×Dic47—41no—
247D188—22no—
248D188—22no—
249D188—41no—
250D188—41no—
251C4×D47—42no—
252C4×D47—42no—
253C4×D47—42no—
254C4×D47—42no—
255Dic94—22no—
256Dic94—22no—
257Dic94—41no—
258Dic94—41no—
259C8×C47—42no—
260C47⋊C8—24no—
261C47⋊C8—42no—
262C47⋊C8—42no—
263#11116—471nomaximal, Sylow
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47#38511yescharacteristic
272C22×C47#38811yescharacteristic
273C22×C47#39011yescharacteristic
274C22×C47#39311yescharacteristic
275C22×C47#39411yescharacteristic
276C22×C47#39611yescharacteristic
277C22×C47#39711yescharacteristic
278C2×D47—42no—
279C2×D47—82no—
280C2×D47—82no—
281C2×D47—82no—
282C4×C47—82no—
283C4×C47—82no—
284C4×C47#38011yescharacteristic
285C4×C47#38511yescharacteristic
286C4×C47#38611yescharacteristic
287C4×C47#38811yescharacteristic
288C4×C47#39211yescharacteristic
289C4×C47#39411yescharacteristic
290C4×C47#39611yescharacteristic
291C4×C47#39911yescharacteristic
292Dic47—42no—
293C2×#430—471no—
294C2×#421—471no—
295#2896—471no—
296C2×#413—471no—
297C2×#344—471no—
298#1742—471no—
299C2×H(3,ℤ/4)D4711yescharacteristic, complemented
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47#143911yescharacteristic
308D47—82no—
309C22×#41—471no—
310C22×M16—471no—
311C22×M16—471no—
312C22×M16—471no—
313C22×M16C2×D4711yescharacteristic
314C2×#117C2×D4711yescharacteristic
315C2×#117C2×D4711yescharacteristic
316#430—942no—
317#430—942no—
318C2×#108—942no—
319#421—942no—
320#421—942no—
321C2×#104—474no—
322#419—941no—
323#418—941no—
324#417—942no—
325#416—942no—
326#413—942no—
327#413—942no—
328C2×#101—941no—
329C2×#100—941no—
330#344—942no—
331#343—942no—
332H(3,ℤ/4)—22no—
333C47#1111611yescharacteristic, complemented, Sylow
334C2×#41—472no—
335C2×#41—472no—
336C2×#41—941no—
337C2×#41—941no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C22×Q8—471no—
347C22×D4—471no—
348C22×D4—941no—
349C23×C4—941no—
350C23×C4C22×D4711yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—472no—
357C2×M16—472no—
358C2×M16—472no—
359C2×M16—472no—
360C2×M16—472no—
361C2×M16—941no—
362C2×M16—941no—
363C2×M16—941no—
364C2×M16—941no—
365C2×M16—941no—
366C2×M16—941no—
367C2×M16—941no—
368C2×M16—941no—
369C2×M16—941no—
370C2×M16—941no—
371C2×M16—941no—
372C2×M16—941no—
373C2×M16—941no—
374C2×M16D18812yes—
375C22×C8—21no—
376C22×C8—941no—
377C22×C8—941no—
378C22×C8—941no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117C22⋊D4712yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42C4×D4712yes—
393#108—948no—
394#104—948no—
395#103—944no—
396#102—944no—
397#101—944no—
398#100—944no—
399C24—941no—
400C24#1062535211yescharacteristic
401#41—942no—
402#41—942no—
403#41—942no—
404#41—942no—
405#41—942no—
406#41—942no—
407#41—942no—
408#41—942no—
409#41—1881no—
410#41—1881no—
411#41—1881no—
412#41—1881no—
413#41—1881no—
414#41—1881no—
415#41—1881no—
416#41—1881no—
417C2×Q8—472no—
418C2×Q8—941no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×D4—472no—
424C2×D4—942no—
425C2×D4—942no—
426C2×D4—941no—
427C2×D4—941no—
428C2×D4—941no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—1881no—
432C2×D4—1881no—
433C2×D4—1881no—
434C2×D4—1881no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—941no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—1881no—
441C22×C4—1881no—
442C22×C4—1881no—
443C22×C4—1881no—
444C22×C4C2×C4×D4711yescharacteristic
445C22×C4C2×D18811yescharacteristic
446C22×C4#1062535011yescharacteristic
447C22×C4#1062535011yescharacteristic
448C22×C4#1062535011yescharacteristic
449C22×C4#1062535011yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4D4×D4711yescharacteristic
452C22×C4D4×D4711yescharacteristic
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4711yescharacteristic
455C22×C4Q8×D4711yescharacteristic
456C22×C4#1062535411yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—474no—
463M16—942no—
464M16—942no—
465M16—942no—
466M16—942no—
467M16—942no—
468M16—942no—
469M16—942no—
470M16—1881no—
471M16—1881no—
472M16—1881no—
473M16—1881no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—942no—
478C2×C8—942no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8—942no—
482C2×C8—1881no—
483C2×C8—1881no—
484C2×C8—1881no—
485C2×C8—1881no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—41no—
498C23—81no—
499C23—942no—
500C23—3762no—
501C23—1881no—
502C23—1881no—
503C23(C22⋊D47)⋊C411yescharacteristic
504C23#1309399611yescharacteristic
505C23#1309400411yescharacteristic
506C23#1309400611yescharacteristic
507C23#1309401011yescharacteristic
508C23#1309401011yescharacteristic
509C23#1309401911yescharacteristic
510Q8—942no—
511Q8—942no—
512Q8—1881no—
513Q8—1881no—
514D4—942no—
515D4—942no—
516D4—1882no—
517D4—1882no—
518D4—1882no—
519D4—1882no—
520D4—1881no—
521D4—1881no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—942no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4C22⋊C4×D4711yescharacteristic
534C2×C4C4⋊C4×D4711yescharacteristic
535C2×C4#1309398111yescharacteristic
536C2×C4C4×C22⋊D4711yescharacteristic
537C2×C4#1309398611yescharacteristic
538C2×C4C4×D18811yescharacteristic
539C2×C4#1309399211yescharacteristic
540C2×C4#1309399411yescharacteristic
541C2×C4#1309399411yescharacteristic
542C2×C4#1309399711yescharacteristic
543C2×C4#1309399711yescharacteristic
544C2×C4#1309399711yescharacteristic
545C2×C4#1309399711yescharacteristic
546C2×C4#1309399711yescharacteristic
547C2×C4#1309399811yescharacteristic
548C2×C4#1309399911yescharacteristic
549C2×C4#1309400011yescharacteristic
550C2×C4#1309400211yescharacteristic
551C2×C4#1309400211yescharacteristic
552C2×C4#1309400211yescharacteristic
553C2×C4#1309400211yescharacteristic
554C2×C4#1309400711yescharacteristic
555C2×C4#1309400911yescharacteristic
556C2×C4#1309400911yescharacteristic
557C2×C4#1309400911yescharacteristic
558C2×C4#1309401211yescharacteristic
559C2×C4#1309401811yescharacteristic
560C2×C4#1309402011yescharacteristic
561C8—42no—
562C8—944no—
563C8—1882no—
564C8—1882no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3762no—
573C22—3762no—
574C22—3762no—
575C22—1882no—
576C22#1568899711yescharacteristic
577C22#1568901211yescharacteristic
578C22#1568902311yescharacteristic
579C22#1568903611yescharacteristic
580C22#1568903811yescharacteristic
581C22#1568904511yescharacteristic
582C22#1568905211yescharacteristic
583C4—82no—
584C4—82no—
585C4—1882no—
586C4#1568898211yescharacteristic
587C4#1568899411yescharacteristic
588C4#1568900111yescharacteristic
589C4#1568901411yescharacteristic
590C4#1568903211yescharacteristic
591C4#1568904111yescharacteristic
592C4#1568904711yescharacteristic
593C4#1568905811yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—3762no—
602C2#2024163211yescharacteristic
subgroupG/H
centerC4#15689014
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC2×C47#1439
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47#11116
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#15689171C2semidirect
5. C2×#15688893C2semidirect
6. #20243127C2semidirect
7. C2×C47×H(3,ℤ/4)C2semidirect
17. C2×#13093952C4semidirect
299. C2×H(3,ℤ/4)D47semidirect
333. C47#11116semidirect

Maximal subgroups

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

subgroupindex
C2×#156891712
C2×#156891412
#202519052
C2×#156891052
C2×#156888932
#202431272
C2×C47×H(3,ℤ/4)2
#1111647