Finite Group Databasebeta

← #26218683

Subgroups — #26218683

Order 12032.

1218 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
1C2×#15689125C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic
3#20245876C211yesmaximal, characteristic
4C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
5#20244028C212yesmaximal, complemented
6#20244021C212yesmaximal, complemented
7#20244014C212yesmaximal, complemented
8#20244007C212yesmaximal, complemented
9#20243109C211yesmaximal, characteristic, complemented
10#20243081C211yesmaximal, characteristic, complemented
11C2×C47×#428C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4×Dic47—21no—
16C22×C4×Dic47C2211yescharacteristic
17#15689125C418yescomplemented
18#15689122—24no—
19C2×#13093949C2212yes—
20C2×#13093947C2212yes—
21C2×#13093946C2212yes—
22C2×#13093944C2212yes—
23#15689028C2211yescharacteristic
24#15689028C2211yescharacteristic
25#15689024C2211yescharacteristic
26#15689024C2211yescharacteristic
27#15689000—21no—
28#15689000—21no—
29C4⋊C4×Dic47—21no—
30C4⋊C4×Dic47—21no—
31C4⋊C4×Dic47C2212yes—
32C4⋊C4×Dic47C2212yes—
33C4×(C2×D47)⋊C4—21no—
34C4×(C2×D47)⋊C4—21no—
35C2×C47⋊#95—21no—
36#15688885C2212yes—
37#15688885C2212yes—
38#15688877C2212yes—
39#15688877C2212yes—
40C22×M16×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#428—24no—
44C2×#107×C47C2212yes—
45C2×#106×C47C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350C2×C414yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56C23×Dic47D411yescharacteristic
57C23×Dic47D411yescharacteristic
58C22×C4×D47—21no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C22×C47⋊C8C2312yes—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47C2×C414yes—
66C2×C4⋊Dic47C2312yes—
67C2×(C2×D47)⋊C4—21no—
68C2×(C2×D47)⋊C4—21no—
69C2×(C2×D47)⋊C4—21no—
70C2×(C2×D47)⋊C4—21no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—22no—
74C2×C4×Dic47—22no—
75C2×C4×Dic47—44no—
76C2×C4×Dic47D411yescharacteristic
77C2×C4×Dic47D411yescharacteristic
78C2×C4×Dic47C2311yescharacteristic
79C2×C4×Dic47C2311yescharacteristic
80#13093949C2×C418yescomplemented
81#13093947—24no—
82#13093946C2×C418yescomplemented
83#13093944—24no—
84C47⋊#95—22no—
85C47⋊#95—22no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C23×C4×C47—21no—
91C23×C4×C47C2311yescharacteristic
92C2×M16×C47—22no—
93C2×M16×C47—22no—
94C2×M16×C47C2×C414yes—
95C22×C8×C47C2312yes—
96#117×C47—22no—
97#117×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47D414yes—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C22⋊C4×C47—21no—
105C2×C42×C47—21no—
106#107×C47—24no—
107#106×C47—24no—
108C24×C47C2×D411yescharacteristic
109C23×D47—21no—
110C22×C4×C47—24no—
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D412yes—
119C22×C4×C47#4112yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×C4×C47C2411yescharacteristic
124C2×C22⋊D47—24no—
125C2×C22⋊D47—42no—
126C2×C22⋊D47—42no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—24no—
131C22×Dic47—22no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134C22×Dic47—41no—
135C22×Dic47—41no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142#10625350—28no—
143#10625350—28no—
144#10625350—44no—
145#10625350—44no—
146C2×D188—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188C22×C412yes—
152C2×C4×D47—24no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94C22×C412yes—
161M16×C47—28no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47C22×C414yes—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C2×D414yes—
172C22⋊C4×C47—42no—
173C42×C47—42no—
174(C2×D47)⋊C4—42no—
175(C2×D47)⋊C4—42no—
176(C2×D47)⋊C4—42no—
177(C2×D47)⋊C4—42no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47C22×C414yes—
181C4×Dic47—22no—
182C4×Dic47—22no—
183C4×Dic47—21no—
184C4×Dic47—21no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—44no—
188C4×Dic47—44no—
189#10625329—44no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47C4×D412yes—
198C23×C47#12112yes—
199C23×C47#12312yes—
200C23×C47C22×D411yescharacteristic
201C22×D47—22no—
202C22×D47—41no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—41no—
208C2×C4×C47—24no—
209C2×C4×C47—48no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47#11714yes—
219C2×C4×C47C4×D414yes—
220C2×C4×C47C4×D412yes—
221C2×C4×C47#12112yes—
222C2×C4×C47#12312yes—
223C2×C4×C47#12414yes—
224C2×C4×C47#12614yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C22⋊D47—48no—
232C2×Dic47—28no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—44no—
246C2×Dic47—48no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47AGL(1,ℤ/8)12yes—
252C2×Dic47#13712yes—
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—48no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C8×C47—24no—
263C47⋊C8—42no—
264#21059—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47#38814yes—
272C22×C47C2×C4×D411yescharacteristic
273C22×C47C2×#12111yescharacteristic
274C22×C47C2×#12311yescharacteristic
275C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C47C2×#13711yescharacteristic
277C22×C47#57511yescharacteristic
278C22×C47#57511yescharacteristic
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—81no—
283C2×D47—81no—
284C2×D47—81no—
285C2×D47—81no—
286C4×C47—44no—
287C4×C47—81no—
288C4×C47—81no—
289C4×C47#38814yes—
290C4×C47C2×#11711yescharacteristic
291C4×C47C2×C4×D411yescharacteristic
292C4×C47C2×#12411yescharacteristic
293C4×C47C2×#12611yescharacteristic
294Dic47—22no—
295Dic47—22no—
296Dic47—41no—
297Dic47—41no—
298Dic47—44no—
299Dic47—42no—
300C2×#428D4711yescharacteristic, complemented
301C2×#417—471no—
302C2×#417—471no—
303#2297—471no—
304C2×C4×C4⋊C4—471no—
305#1928—472no—
306#1927—472no—
307#1926—472no—
308#1925—472no—
309#1736—471no—
310#1728—471no—
311C2×C47—24no—
312C2×C47C2×#38811yescharacteristic
313C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C47C4×#13711yescharacteristic
315C2×C47#303611yescharacteristic
316C2×C47#303711yescharacteristic
317C2×C47#309211yescharacteristic
318C2×C47#309311yescharacteristic
319D47—82no—
320C22×#41—471no—
321C22×M16—471no—
322C22×M16C2×D4711yescharacteristic
323C2×#117C2×D4711yescharacteristic
324C22×C4⋊C4—471no—
325C22×C4⋊C4C2×D4711yescharacteristic
326C22×C42—471no—
327C22×C42—941no—
328#428—24no—
329C2×#107C2×D4712yes—
330C2×#106C2×D4712yes—
331#417—478no—
332#417—944no—
333C2×#103—472no—
334C2×#103—472no—
335C2×#102—472no—
336C2×#102—472no—
337#391—471no—
338#391—471no—
339#390—471no—
340#390—471no—
341#386—941no—
342#386—941no—
343C4×C4⋊C4—472no—
344C4×C4⋊C4—472no—
345C4×C4⋊C4—941no—
346C4×C4⋊C4—941no—
347C4×C22⋊C4—941no—
348C4×C22⋊C4—941no—
349C2×#95—941no—
350#340—472no—
351#340—472no—
352#336—472no—
353#336—472no—
354C47#2105911yescharacteristic, complemented, Sylow
355C2×#41—474no—
356C2×#41—942no—
357C2×#41—942no—
358C2×#41—942no—
359C2×#41—942no—
360C2×#41—942no—
361C22×Q8—471no—
362C22×D4—471no—
363C22×D4—941no—
364C23×C4—21no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—941no—
368C23×C4—941no—
369C23×C4C22×D4711yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—942no—
373C2×M16—942no—
374C2×M16—942no—
375C2×M16C4×D4714yes—
376C22×C8—472no—
377C22×C8C22×D4712yes—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—472no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4C22⋊D4714yes—
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4712yes—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—21no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—942no—
401C2×C42—942no—
402C2×C42—942no—
403C2×C42—942no—
404C2×C42—1884no—
405#107—24no—
406#106—24no—
407#103—478no—
408#103—944no—
409#102—478no—
410#102—944no—
411#95—942no—
412#95—942no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417C24—941no—
418C24D4×D4711yescharacteristic
419#41—948no—
420#41—948no—
421#41—1884no—
422#41—1884no—
423C2×Q8—472no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×Q8—941no—
429C2×D4—472no—
430C2×D4—944no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—1882no—
437C2×D4—1882no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—941no—
454C22×C4—942no—
455C22×C4—944no—
456C22×C4—1881no—
457C22×C4—1881no—
458C22×C4—1882no—
459C22×C4—1882no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4C2×C4×D4712yes—
463C22×C4#1062535012yes—
464C22×C4D4×D4711yescharacteristic
465C22×C4#1062535212yes—
466C22×C4#1062535412yes—
467C22×C4#1062535412yes—
468C22×C4C2×C22⋊D4712yes—
469C22×C4C23×D4711yescharacteristic
470M16—28no—
471M16—1884no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—942no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8C2×C4×D4714yes—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—474no—
485C4⋊C4—944no—
486C4⋊C4—942no—
487C4⋊C4C2×C22⋊D4714yes—
488C22⋊C4—42no—
489C22⋊C4—1882no—
490C22⋊C4—1882no—
491C22⋊C4—1882no—
492C22⋊C4—1882no—
493C42—42no—
494C42—942no—
495C42—942no—
496C42—941no—
497C42—941no—
498C42—941no—
499C42—941no—
500C42—1884no—
501C42—1884no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—942no—
507C23—1881no—
508C23—1881no—
509C23—1881no—
510C23—1881no—
511C23—1881no—
512C23—1881no—
513C23#1309398612yes—
514C23#1309399612yes—
515C23#1309400912yes—
516C23C2×D4×D4711yescharacteristic
517Q8—942no—
518Q8—942no—
519Q8—1881no—
520Q8—1881no—
521D4—942no—
522D4—942no—
523D4—1888no—
524D4—1881no—
525D4—1881no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—472no—
537C2×C4—472no—
538C2×C4—948no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—944no—
544C2×C4—942no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—1888no—
552C2×C4—1884no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1888no—
558C2×C4#1309398214yes—
559C2×C4C4×C22⋊D4714yes—
560C2×C4#1309398612yes—
561C2×C4#1309399612yes—
562C2×C4#1309400912yes—
563C2×C4#1309401314yes—
564C2×C4#1309401914yes—
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C22×C22⋊D4711yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4C2×#1062535011yescharacteristic
569C2×C4C2×#1062535411yescharacteristic
570C2×C4C2×#1062535411yescharacteristic
571C8—24no—
572C8—1882no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—3761no—
580C22—3761no—
581C22—3761no—
582C22—3761no—
583C22—1882no—
584C22—1882no—
585C22—1882no—
586C22#1568901314yes—
587C22C2×#1309398611yescharacteristic
588C22C2×#1309399611yescharacteristic
589C22C2×#1309400911yescharacteristic
590C22AGL(1,ℤ/8)×D4711yescharacteristic
591C22#137×D4711yescharacteristic
592C22#1568991911yescharacteristic
593C22#1568992211yescharacteristic
594C4—44no—
595C4—81no—
596C4—81no—
597C4—942no—
598C4—942no—
599C4—1884no—
600C4—1882no—
601C4—1881no—
602C4—1881no—
603C4#1568901314yes—
604C4C2×#1309398211yescharacteristic
605C4C2×C4×C22⋊D4711yescharacteristic
606C4C2×#1309401311yescharacteristic
607C4C2×#1309401911yescharacteristic
608C2—24no—
609C2—3762no—
610C2C2×#1568901311yescharacteristic
611C2#2025219511yescharacteristic
612C2#2025220611yescharacteristic
613C2#2025284911yescharacteristic
614C2#2025285711yescharacteristic
615C2#2025331411yescharacteristic
616C2#2025332211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#21059
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊C4×Dic47C2semidirect
5. #20244028C2semidirect
6. #20244021C2semidirect
7. #20244014C2semidirect
8. #20244007C2semidirect
9. #20243109C2semidirect
10. #20243081C2semidirect
11. C2×C47×#428C2semidirect
17. #15689125C4semidirect
80. #13093949C2×C4semidirect
82. #13093946C2×C4semidirect
300. C2×#428D47semidirect
354. C47#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#156891252
C2×#156891222
#202458762
C2×C4⋊C4×Dic472
#202440282
#202440212
#202440142
#202440072
#202431092
#202430812
C2×C47×#4282
#2105947