Finite Group Databasebeta

← #26218986

Subgroups — #26218986

Order 12032.

1198 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×#15689125C211yesmaximal, characteristic
3#20245878C211yesmaximal, characteristic
4C2×#15689021C211yesmaximal, characteristic, complemented
5#20244085C212yesmaximal, complemented
6#20244078C212yesmaximal, complemented
7#20244071C212yesmaximal, complemented
8#20244064C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10#20243108C211yesmaximal, characteristic, complemented
11C2×C4×M16×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4—21no—
17#15689125—24no—
18#15689125C418yescomplemented
19C2×#13093949C2212yes—
20C2×#13093949C2212yes—
21C2×#13093946C2212yes—
22C2×#13093946C2212yes—
23#15689029C2211yescharacteristic
24#15689029C2211yescharacteristic
25#15689025C2211yescharacteristic
26#15689025C2211yescharacteristic
27#15689021—21no—
28#15689021—21no—
29#15689021C2212yes—
30#15689021C2212yes—
31#15689002—21no—
32#15689002—21no—
33C4×(C2×D47)⋊C4—21no—
34C4×(C2×D47)⋊C4—21no—
35C2×C47⋊#95—21no—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688886C2212yes—
40C22×M16×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C22×C42×C47C2211yescharacteristic
43C4×M16×C47—24no—
44C2×#97×C47C2212yes—
45C2×C4×C8×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×Dic47—21no—
57C22×C4×D47—21no—
58#13093983—22no—
59#13093983—22no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2×C414yes—
66C2×C4⋊Dic47C2312yes—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2312yes—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4—22no—
72C2×Dic47⋊C4—44no—
73C2×C22⋊Dic47—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×C4×Dic47—21no—
79#13093949—24no—
80#13093949C2×C418yescomplemented
81#13093946—24no—
82#13093946C2×C418yescomplemented
83C47⋊#95—22no—
84C47⋊#95—22no—
85C47⋊#95—21no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C23×C4×C47D411yescharacteristic
90C23×C4×C47D411yescharacteristic
91C23×C4×C47C2311yescharacteristic
92C2×M16×C47—22no—
93C2×M16×C47—22no—
94C2×M16×C47—22no—
95C2×M16×C47—22no—
96C2×M16×C47C2×C414yes—
97C2×M16×C47D414yes—
98C22×C8×C47C2312yes—
99C22×C8×C47C2312yes—
100C2×C42×C47—22no—
101C2×C42×C47—22no—
102C2×C42×C47D411yescharacteristic
103C2×C42×C47D411yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C2×C42×C47C2311yescharacteristic
106#97×C47—24no—
107C4×C8×C47—24no—
108C24×C47C2×D411yescharacteristic
109C23×D47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—41no—
115C22×C4×C47—41no—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D412yes—
123C22×C4×C47#4112yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×C4×C47#4112yes—
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—42no—
130C2×C22⋊D47—42no—
131C22×Dic47—24no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47—42no—
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—
162M16×C47—28no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47C22×C414yes—
168C2×C8×C47C2×D414yes—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C22⋊Dic47—42no—
176(C2×D47)⋊C4—42no—
177(C2×D47)⋊C4—42no—
178(C2×D47)⋊C4—42no—
179(C2×D47)⋊C4—42no—
180C4⋊Dic47—24no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—42no—
189C4⋊Dic47C22×C414yes—
190Dic47⋊C4—44no—
191Dic47⋊C4—44no—
192C4×Dic47—42no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47C4×D412yes—
198C23×C47#12112yes—
199C23×C47#12412yes—
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—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—44no—
219C2×C4×C47—44no—
220C2×C4×C47#11714yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47C4×D412yes—
223C2×C4×C47#12112yes—
224C2×C4×C47#12314yes—
225C2×C4×C47#12412yes—
226C2×C4×C47#12614yes—
227C2×C4×C47AGL(1,ℤ/8)12yes—
228C2×C4×C47#13712yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C22⋊D47—48no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—48no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—42no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—42no—
252C2×Dic47—84no—
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—48no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C8×C47—24no—
263C8×C47—24no—
264#21093—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×#12411yescharacteristic
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—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47#38814yes—
291C4×C47C2×#11711yescharacteristic
292C4×C47C2×C4×D411yescharacteristic
293C4×C47C2×#12311yescharacteristic
294C4×C47C2×#12611yescharacteristic
295Dic47—44no—
296Dic47—42no—
297Dic47—81no—
298Dic47—81no—
299Dic47—82no—
300C2×#417—471no—
301C2×#417—471no—
302C2×C4×M16D4711yescharacteristic, complemented
303#2297—471no—
304C2×#389—471no—
305#1936—472no—
306#1935—472no—
307#1934—472no—
308#1933—472no—
309#1736—471no—
310#1736—471no—
311C2×C47—24no—
312C2×C47C2×#38811yescharacteristic
313C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C47C4×#13711yescharacteristic
315C2×C47#305511yescharacteristic
316C2×C47#305611yescharacteristic
317C2×C47#314111yescharacteristic
318C2×C47#314211yescharacteristic
319D47—82no—
320C22×#41—471no—
321C22×M16C2×D4711yescharacteristic
322C22×M16C2×D4711yescharacteristic
323C2×#117—471no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4—941no—
327C22×C42C2×D4711yescharacteristic
328#417—478no—
329#417—944no—
330C2×#103—472no—
331C2×#103—472no—
332C2×#102—472no—
333C2×#102—472no—
334C4×M16—24no—
335C2×#97C2×D4712yes—
336C2×C4×C8C2×D4712yes—
337#391—471no—
338#391—471no—
339#390—471no—
340#390—471no—
341#389—472no—
342#389—472no—
343#389—941no—
344#389—941no—
345#386—941no—
346#386—941no—
347C4×C22⋊C4—941no—
348C4×C22⋊C4—941no—
349C2×#95—941no—
350#340—472no—
351#340—472no—
352#340—472no—
353#340—472no—
354C47#2109311yescharacteristic, 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—941no—
365C23×C4—941no—
366C23×C4—941no—
367C23×C4D18811yescharacteristic
368C23×C4D18811yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16C4×D4714yes—
375C2×M16C22⋊D4714yes—
376C22×C8C22×D4712yes—
377C22×C8C22×D4712yes—
378#117—942no—
379#117—942no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—472no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—1884no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—941no—
401C2×C42D18811yescharacteristic
402C2×C42D18811yescharacteristic
403C2×C42C22×D4711yescharacteristic
404C2×C42C22×D4711yescharacteristic
405#103—478no—
406#103—944no—
407#102—478no—
408#102—944no—
409#97—24no—
410C4×C8—24no—
411#95—942no—
412#95—942no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417C24—941no—
418C24C2×D18811yescharacteristic
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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—944no—
445C22×C4—941no—
446C22×C4—942no—
447C22×C4—944no—
448C22×C4—941no—
449C22×C4—941no—
450C22×C4—942no—
451C22×C4—1882no—
452C22×C4—1882no—
453C22×C4—1882no—
454C22×C4—1882no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4C2×C4×D4712yes—
459C22×C4C2×D18811yescharacteristic
460C22×C4C2×D18811yescharacteristic
461C22×C4C2×D18811yescharacteristic
462C22×C4C2×D18811yescharacteristic
463C22×C4C2×D18811yescharacteristic
464C22×C4#1062535012yes—
465C22×C4#1062535012yes—
466C22×C4#1062535012yes—
467C22×C4#1062535012yes—
468C22×C4C2×C22⋊D4712yes—
469C22×C4C23×D4711yescharacteristic
470M16—28no—
471M16—28no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C2×C4×D4714yes—
477C2×C8C2×C22⋊D4714yes—
478C4⋊C4—474no—
479C4⋊C4—944no—
480C4⋊C4—942no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—1884no—
488C4⋊C4—1882no—
489C4⋊C4—1884no—
490C22⋊C4—1882no—
491C22⋊C4—1882no—
492C22⋊C4—1882no—
493C22⋊C4—1882no—
494C22⋊C4—1882no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—1882no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—942no—
507C23—1881no—
508C23—1881no—
509C23—1881no—
510C23—1881no—
511C23—1881no—
512C23—1881no—
513C23C4×D18812yes—
514C23#1309399812yes—
515C23#1309401212yes—
516C23C22×D18811yescharacteristic
517Q8—942no—
518Q8—942no—
519Q8—1881no—
520Q8—1881no—
521D4—942no—
522D4—942no—
523D4—1888no—
524D4—1881no—
525D4—1881no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—944no—
539C2×C4—942no—
540C2×C4—3764no—
541C2×C4—1882no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1882no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1882no—
550C2×C4—1888no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1888no—
556C2×C4#1309398114yes—
557C2×C4C4×C22⋊D4714yes—
558C2×C4C4×D18812yes—
559C2×C4#1309399812yes—
560C2×C4#1309400714yes—
561C2×C4#1309401212yes—
562C2×C4#1309401814yes—
563C2×C4#1309405212yes—
564C2×C4#1309405712yes—
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C22×C22⋊D4711yescharacteristic
567C2×C4C2×#1062535011yescharacteristic
568C2×C4C2×#1062535011yescharacteristic
569C2×C4C2×#1062535011yescharacteristic
570C2×C4C2×#1062535011yescharacteristic
571C8—24no—
572C8—24no—
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#1568901414yes—
587C22C2×C4×D18811yescharacteristic
588C22C2×#1309399811yescharacteristic
589C22C2×#1309401211yescharacteristic
590C22C2×#1309405211yescharacteristic
591C22C2×#1309405711yescharacteristic
592C22#1568992311yescharacteristic
593C22#1568992311yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—3762no—
599C4—3761no—
600C4—3761no—
601C4—1884no—
602C4—1882no—
603C4#1568901414yes—
604C4C2×#1309398111yescharacteristic
605C4C2×C4×C22⋊D4711yescharacteristic
606C4C2×#1309400711yescharacteristic
607C4C2×#1309401811yescharacteristic
608C2—24no—
609C2—3762no—
610C2C2×#1568901411yescharacteristic
611C2C4×#1309405211yescharacteristic
612C2C4×#1309405711yescharacteristic
613C2#2025300711yescharacteristic
614C2#2025301811yescharacteristic
615C2#2025378211yescharacteristic
616C2#2025378911yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C4×M16×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×C4×M16D47
hypocenterC47#21093
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
4. C2×#15689021C2semidirect
5. #20244085C2semidirect
6. #20244078C2semidirect
7. #20244071C2semidirect
8. #20244064C2semidirect
9. #20243108C2semidirect
10. #20243108C2semidirect
11. C2×C4×M16×C47C2semidirect
18. #15689125C4semidirect
80. #13093949C2×C4semidirect
82. #13093946C2×C4semidirect
302. C2×C4×M16D47semidirect
354. C47#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#156891252
#202458782
C2×#156890212
#202440852
#202440782
#202440712
#202440642
#202431082
C2×C4×M16×C472
#2109347