Finite Group Databasebeta

← #26122621

Subgroups — #26122621

Order 12032.

904 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689105C211yesmaximal, characteristic
2C2×#15689096C211yesmaximal, characteristic
3#20251832C211yesmaximal, characteristic
4#20251807C211yesmaximal, characteristic, complemented
5C2×#15688895C211yesmaximal, characteristic, complemented
6#20243126C211yesmaximal, characteristic, complemented
7C2×C47×#344C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689105—22no—
13#15689105—22no—
14C2×#13093942C412yescomplemented
15C2×#13093940C412yescomplemented
16#15689096—22no—
17#15689096—22no—
18C2×#13093938C414yes—
19#15689091—22no—
20#15689089—22no—
21#15689086C414yescomplemented
22#15689079—22no—
23#15689079—22no—
24C4×#10625329—22no—
25#15688895—22no—
26#15688891—22no—
27C22×M16×C47C2211yescharacteristic
28C22×M16×C47C2211yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C47×#344—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×#10625350—21no—
42C2×#10625350—21no—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47C2×C412yes—
46C22×C4×D47C2×C412yes—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329C2×C412yes—
55C2×#10625329C2×C412yes—
56C2×#10625329D412yes—
57C2×#10625329D412yes—
58C2×#10625329D412yes—
59C22×C47⋊C8—21no—
60C22×C47⋊C8D412yes—
61#13093983—21no—
62#13093983—21no—
63#13093983—21no—
64#13093983—21no—
65C2×C4⋊Dic47—21no—
66C2×C22⋊Dic47C2×C412yes—
67C2×C4×Dic47—21no—
68#13093942—24no—
69#13093940—24no—
70#13093938—28no—
71(C2×D47)⋊C8—28no—
72#13093935—24no—
73C4×C47⋊C8—24no—
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×C47—21no—
81C2×M16×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47—21no—
84C2×M16×C47—21no—
85C2×M16×C47C2×C412yes—
86C2×M16×C47C2×C412yes—
87C22×C8×C47—21no—
88C22×C8×C47—21no—
89#117×C47—21no—
90#117×C47—21no—
91#117×C47D412yes—
92#117×C47D412yes—
93C2×C4⋊C4×C47—21no—
94C2×C22⋊C4×C47C2×C412yes—
95C2×C42×C47—21no—
96C24×C47C22×C411yescharacteristic
97C23×D47C4212yes—
98C22×C4×C47—22no—
99C22×C4×C47C22×C411yescharacteristic
100C22×C4×C47C22×C411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C2×C22⋊D47—22no—
114C2×C22⋊D47—22no—
115C2×C22⋊D47—22no—
116C2×C22⋊D47—22no—
117C2×C22⋊D47—22no—
118C2×C22⋊D47—22no—
119C22×Dic47—22no—
120C22×Dic47C4212yes—
121#10625350—41no—
122#10625350—41no—
123#10625350—41no—
124#10625350—41no—
125#10625350—41no—
126#10625350—41no—
127#10625350—41no—
128#10625350—41no—
129#10625350—41no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155M16×C47—22no—
156M16×C47—22no—
157M16×C47—22no—
158M16×C47—22no—
159M16×C47—41no—
160M16×C47—41no—
161M16×C47—41no—
162M16×C47—41no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—41no—
166C2×C8×C47—41no—
167C2×C8×C47—41no—
168C2×C8×C47—41no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47C22⋊C414yes—
173C42×C47—22no—
174C42×C47—22no—
175C22⋊Dic47—22no—
176C22⋊Dic47—22no—
177C4⋊Dic47—41no—
178C4⋊Dic47—41no—
179C4×Dic47—41no—
180C4×Dic47—41no—
181#10625329—22no—
182#10625329—22no—
183#10625329—22no—
184#10625329—22no—
185#10625329—22no—
186#10625329—22no—
187#10625329—22no—
188#10625329C22⋊C414yes—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8C22⋊C414yes—
195C23×C47—81no—
196C23×C47C2×C4211yescharacteristic
197C23×C47C2×C22⋊C411yescharacteristic
198C23×C47#11711yescharacteristic
199C23×C47C4×D411yescharacteristic
200C23×C47C4×D411yescharacteristic
201C23×C47C4×D411yescharacteristic
202C23×C47C4×D411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—82no—
207C2×C4×C47—24no—
208C2×C4×C47—82no—
209C2×C4×C47C2×C22⋊C411yescharacteristic
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47#11711yescharacteristic
212C2×C4×C47#11711yescharacteristic
213C2×C4×C47C4×D411yescharacteristic
214C2×C4×C47C4×D411yescharacteristic
215C2×C4×C47C4×D411yescharacteristic
216C2×C4×C47C4×D411yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C4×D411yescharacteristic
221C2×C4×C47#12011yescharacteristic
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×C47#12311yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12411yescharacteristic
234C2×C4×C47#12611yescharacteristic
235C2×C4×C47#12611yescharacteristic
236C2×C4×C47#12711yescharacteristic
237C22⋊D47—42no—
238C22⋊D47—42no—
239C22⋊D47—42no—
240C22⋊D47—42no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—42no—
245C2×Dic47—82no—
246D188—41no—
247D188—41no—
248D188—41no—
249D188—41no—
250C4×D47—42no—
251C4×D47—42no—
252C4×D47—42no—
253C4×D47—42no—
254Dic94—41no—
255Dic94—41no—
256Dic94—41no—
257Dic94—41no—
258C8×C47—42no—
259C8×C47—42no—
260C47⋊C8—24no—
261C47⋊C8—24no—
262#10941—471nomaximal, Sylow
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47C4×C22⋊C411yescharacteristic
271C22×C47C4×C22⋊C411yescharacteristic
272C22×C47C4×C22⋊C411yescharacteristic
273C22×C47#38611yescharacteristic
274C22×C47#38811yescharacteristic
275C22×C47#38811yescharacteristic
276C22×C47#39011yescharacteristic
277C2×D47—44no—
278C2×D47—42no—
279C2×D47—82no—
280C2×D47—82no—
281C4×C47—44no—
282C4×C47#38611yescharacteristic
283C4×C47#38611yescharacteristic
284C4×C47#38811yescharacteristic
285C4×C47#38811yescharacteristic
286C4×C47#38811yescharacteristic
287C4×C47#38811yescharacteristic
288C4×C47#39011yescharacteristic
289C4×C47#39011yescharacteristic
290Dic47—42no—
291Dic47—82no—
292C2×#413—471no—
293C2×#410—471no—
294#2879—471no—
295#2873—471no—
296C2×#344—471no—
297C2×#344D4711yescharacteristic, complemented
298#1742—471no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47#143711yescharacteristic
307D47—82no—
308C22×#41—471no—
309C22×M16—471no—
310C22×M16—471no—
311C22×M16C2×D4711yescharacteristic
312C22×M16C2×D4711yescharacteristic
313C2×#117—471no—
314C2×#117C2×D4711yescharacteristic
315#413—942no—
316#413—942no—
317C2×#101—472no—
318C2×#100—472no—
319#410—942no—
320#410—942no—
321C2×#99—474no—
322#408—942no—
323#408—942no—
324#407—474no—
325#405—942no—
326#405—942no—
327C4×M16—942no—
328#344—22no—
329#344—942no—
330#343—942no—
331C47#1094111yescharacteristic, complemented, Sylow
332C2×#41—941no—
333C2×#41—941no—
334C2×#41—941no—
335C2×#41—941no—
336C2×#41—941no—
337C2×#41—941no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C2×#41—941no—
344C22×Q8—471no—
345C22×D4—472no—
346C22×D4—471no—
347C23×C4—472no—
348C23×C4C22×D4711yescharacteristic
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—472no—
360C2×M16—472no—
361C2×M16—472no—
362C2×M16—472no—
363C2×M16—472no—
364C2×M16—941no—
365C2×M16—941no—
366C2×M16—941no—
367C2×M16—941no—
368C2×M16—941no—
369C2×M16—941no—
370C2×M16—941no—
371C2×M16C4×D4712yes—
372C2×M16C4×D4712yes—
373C22×C8—21no—
374C22×C8—21no—
375C22×C8—472no—
376C22×C8—941no—
377#117—21no—
378#117—21no—
379#117—941no—
380#117—941no—
381#117—941no—
382#117—941no—
383#117D18812yes—
384#117C22⋊D4712yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—941no—
387C2×C22⋊C4—472no—
388C2×C22⋊C4C4×D4712yes—
389C2×C42—21no—
390C2×C42—941no—
391#101—944no—
392#100—944no—
393#99—948no—
394C22⋊C8—948no—
395#97—944no—
396C4×C8—944no—
397C24—472no—
398C24C2×C4×D4711yescharacteristic
399#41—1881no—
400#41—1881no—
401#41—1881no—
402#41—1881no—
403#41—1881no—
404#41—1881no—
405#41—1881no—
406#41—1881no—
407#41—1881no—
408#41—1881no—
409#41—1881no—
410#41—1881no—
411#41—1881no—
412#41—1881no—
413#41—1881no—
414#41—1881no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×Q8—941no—
418C2×Q8—941no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×D4—942no—
422C2×D4—942no—
423C2×D4—942no—
424C2×D4—942no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4—941no—
428C2×D4—941no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C22×C4—22no—
434C22×C4—472no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4C2×C4×D4711yescharacteristic
443C22×C4C2×C4×D4711yescharacteristic
444C22×C4C2×D18811yescharacteristic
445C22×C4#1062535011yescharacteristic
446C22×C4#1062535011yescharacteristic
447C22×C4D4×D4711yescharacteristic
448C22×C4D4×D4711yescharacteristic
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4#1062535211yescharacteristic
452C22×C4#1062535211yescharacteristic
453C22×C4#1062535411yescharacteristic
454C22×C4#1062535411yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—474no—
465M16—942no—
466M16—942no—
467M16—942no—
468M16—942no—
469M16—942no—
470M16—942no—
471M16—942no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—474no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8—942no—
482C2×C8—942no—
483C2×C8—942no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—1881no—
487C4⋊C4—1881no—
488C22⋊C4—22no—
489C22⋊C4—942no—
490C22⋊C4—942no—
491C22⋊C4(C2×D47)⋊C414yes—
492C42—22no—
493C42—22no—
494C42—1881no—
495C42—1881no—
496C23—81no—
497C23—942no—
498C23—942no—
499C23—942no—
500C23—3762no—
501C23C42×D4711yescharacteristic
502C23C2×(C2×D47)⋊C411yescharacteristic
503C23#1309398111yescharacteristic
504C23(C22⋊D47)⋊C411yescharacteristic
505C23(C22⋊D47)⋊C411yescharacteristic
506C23#1309398611yescharacteristic
507C23#1309398611yescharacteristic
508Q8—1881no—
509Q8—1881no—
510Q8—1881no—
511Q8—1881no—
512D4—1882no—
513D4—1882no—
514D4—1882no—
515D4—1882no—
516D4—1881no—
517D4—1881no—
518D4—1881no—
519D4—1881no—
520C2×C4—24no—
521C2×C4—82no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—3762no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4C22⋊C4×D4711yescharacteristic
532C2×C4C22⋊C4×D4711yescharacteristic
533C2×C4#1309398211yescharacteristic
534C2×C4#1309398211yescharacteristic
535C2×C4(C22⋊D47)⋊C411yescharacteristic
536C2×C4(C22⋊D47)⋊C411yescharacteristic
537C2×C4C4×C22⋊D4711yescharacteristic
538C2×C4C4×C22⋊D4711yescharacteristic
539C2×C4#1309398611yescharacteristic
540C2×C4#1309398611yescharacteristic
541C2×C4C4×D18811yescharacteristic
542C2×C4C4×D18811yescharacteristic
543C2×C4#1309399411yescharacteristic
544C2×C4#1309399611yescharacteristic
545C2×C4#1309399611yescharacteristic
546C2×C4#1309399711yescharacteristic
547C2×C4#1309399711yescharacteristic
548C2×C4#1309399911yescharacteristic
549C2×C4#1309399911yescharacteristic
550C2×C4#1309400811yescharacteristic
551C2×C4#1309400911yescharacteristic
552C2×C4#1309400911yescharacteristic
553C2×C4#1309401011yescharacteristic
554C2×C4#1309401011yescharacteristic
555C2×C4#1309401311yescharacteristic
556C2×C4#1309401911yescharacteristic
557C2×C4#1309401911yescharacteristic
558C2×C4#1309402011yescharacteristic
559C8—42no—
560C8—42no—
561C8—944no—
562C8—944no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—3762no—
571C22—3762no—
572C22—1884no—
573C22—1882no—
574C22#1568897011yescharacteristic
575C22#1568897011yescharacteristic
576C22C4×(C2×D47)⋊C411yescharacteristic
577C22#1568900011yescharacteristic
578C22#1568901211yescharacteristic
579C22#1568901211yescharacteristic
580C22#1568902411yescharacteristic
581C4—44no—
582C4—3762no—
583C4—1882no—
584C4#1568900111yescharacteristic
585C4#1568900111yescharacteristic
586C4#1568901311yescharacteristic
587C4#1568901311yescharacteristic
588C4#1568901511yescharacteristic
589C4#1568901511yescharacteristic
590C4#1568902311yescharacteristic
591C4#1568902311yescharacteristic
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—3762no—
600C2#2024162411yescharacteristic
subgroupG/H
centerC4#15689001
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×#344C2
socleC2×C47#1437
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×#344D47
hypocenterC47#10941
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. #20251807C2semidirect
5. C2×#15688895C2semidirect
6. #20243126C2semidirect
7. C2×C47×#344C2semidirect
14. C2×#13093942C4semidirect
15. C2×#13093940C4semidirect
21. #15689086C4semidirect
297. C2×#344D47semidirect
331. C47#10941semidirect

Maximal subgroups

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

subgroupindex
C2×#156891052
C2×#156890962
#202518322
#202518072
C2×#156888952
#202431262
C2×C47×#3442
#1094147