Finite Group Databasebeta

← #26219294

Subgroups — #26219294

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689223C211yesmaximal, characteristic
2#20245865C211yesmaximal, characteristic, complemented
3#20244029C211yesmaximal, characteristic, complemented
4#20244022C211yesmaximal, characteristic, complemented
5#20243980C211yesmaximal, characteristic, complemented
6#20243959C211yesmaximal, characteristic, complemented
7#20243722C211yesmaximal, characteristic
8#20243722C211yesmaximal, characteristic
9#20243649C211yesmaximal, characteristic
10#20243635C211yesmaximal, characteristic
11#20243286C211yesmaximal, characteristic, complemented
12#20243221C211yesmaximal, characteristic, complemented
13#20243213C211yesmaximal, characteristic
14#20243137C211yesmaximal, characteristic, complemented
15C42×D4×C47C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18#15689223C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093947C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30#15689029C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689007—21no—
35#15689007—21no—
36C4×Dic47⋊C4—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688969—21no—
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C22×C42×C47—21no—
53C4×C4⋊C4×C47C2211yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C22⋊C4×C47—21no—
57C4×C22⋊C4×C47—21no—
58C4×C22⋊C4×C47—21no—
59C43×C47C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C2×#10625335—22no—
62C2×#10625335—22no—
63C2×#10625335C2×C414yes—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C4×Dic94—22no—
69C4×Dic94C2×C414yescomplemented
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78#13093957—22no—
79#13093957C2×C414yescomplemented
80#13093955—22no—
81#13093947—22no—
82#13093947—22no—
83#13093947C2×C414yescomplemented
84#13093945—22no—
85#13093945—22no—
86#13093945C2×C414yes—
87#13093935—22no—
88#13093935C2×C414yescomplemented
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47—21no—
99C23×C4×C47—21no—
100C4×D4×C47—22no—
101C4×D4×C47—22no—
102C4×D4×C47—22no—
103C4×D4×C47—22no—
104C4×D4×C47—22no—
105C4×D4×C47C2×C414yes—
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C42×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47—44no—
116C2×C42×C47D411yescharacteristic
117C2×C42×C47D411yescharacteristic
118C2×C42×C47D411yescharacteristic
119C2×C42×C47D411yescharacteristic
120C2×C42×C47C2311yescharacteristic
121C2×C42×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126C24×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47C22×C412yes—
131C2×D4×C47#4112yes—
132C2×D4×C47#4112yes—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—24no—
139C22×C4×C47—42no—
140C22×C4×C47—42no—
141C22×C4×C47—42no—
142C22×C4×C47—42no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94C22×C412yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47C22×C412yes—
166C4⋊C4×C47#4112yes—
167C4⋊C4×C47#4112yes—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—24no—
170C22⋊C4×C47—24no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—41no—
178C42×C47—41no—
179C42×C47—42no—
180C42×C47—42no—
181C42×C47—42no—
182C42×C47C22×C412yes—
183C42×C47#4112yes—
184#10625335—28no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47C22×C412yes—
191Dic47⋊C4—24no—
192Dic47⋊C4—42no—
193Dic47⋊C4—42no—
194C4×Dic47—24no—
195C4×Dic47—42no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8C22×C412yes—
207C2×C47⋊C8C22×C412yes—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C22×D411yescharacteristic
216D4×C47—22no—
217D4×C47—22no—
218D4×C47—22no—
219D4×C47#11714yes—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—48no—
238C2×C4×C47—48no—
239C2×C4×C47—48no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47C4×D412yes—
247C2×C4×C47C4×D412yes—
248C2×C4×C47AGL(1,ℤ/8)12yes—
249C2×C4×C47#13712yes—
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268Dic94—22no—
269Dic94—22no—
270Dic94—41no—
271Dic94—41no—
272C47⋊C8—22no—
273C47⋊C8—22no—
274C47⋊C8—41no—
275C47⋊C8—41no—
276#21121—471nomaximal, Sylow
277C22×C47—24no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47C2×C4×D411yescharacteristic
285C22×C47#52511yescharacteristic
286C22×C47#52511yescharacteristic
287C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C47C2×#13711yescharacteristic
289C22×C47#57511yescharacteristic
290C22×C47#57511yescharacteristic
291C4×C47—22no—
292C4×C47—22no—
293C4×C47—22no—
294C4×C47—22no—
295C4×C47—42no—
296C4×C47—41no—
297C4×C47—42no—
298C4×C47—41no—
299C4×C47—42no—
300C4×C47C2×#11711yescharacteristic
301C4×C47C4×#4111yescharacteristic
302C4×C47#52911yescharacteristic
303C4×C47#52911yescharacteristic
304Dic47—42no—
305Dic47—42no—
306Dic47—81no—
307Dic47—81no—
308C2×#440—471no—
309#2295—471no—
310C42×D4D4711yescharacteristic
311#1928—471no—
312#1927—471no—
313#1921—471no—
314#1918—471no—
315#1877—471no—
316#1877—471no—
317#1864—471no—
318#1862—471no—
319#1776—471no—
320#1763—471no—
321#1762—471no—
322#1745—471no—
323C2×C47—44no—
324C2×C47#229011yescharacteristic
325C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C47C4×#13711yescharacteristic
327C2×C47#311011yescharacteristic
328C2×C47#311111yescharacteristic
329C2×C47#311411yescharacteristic
330C2×C47#311611yescharacteristic
331C22×SD16—471no—
332C2×C4×Q8—471no—
333C2×C4×D4C2×D4711yescharacteristic
334C2×C4×D4C2×D4711yescharacteristic
335C2×C4×D4C2×D4711yescharacteristic
336C22×C42—21no—
337#440—478no—
338C2×#107—471no—
339C2×#107—471no—
340C2×#106—471no—
341C2×#103—471no—
342C2×#103—471no—
343C2×#103—471no—
344C2×#102—471no—
345C2×#102—471no—
346C2×#102—471no—
347C2×#97—471no—
348C2×#97—471no—
349#391—471no—
350#391—471no—
351#389—471no—
352#389—471no—
353#387—941no—
354#387—941no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—941no—
357C4×C4⋊C4C2×D4711yescharacteristic
358C4×C4⋊C4C2×D4711yescharacteristic
359C4×C4⋊C4C2×D4711yescharacteristic
360C4×C22⋊C4—21no—
361C4×C22⋊C4—21no—
362C4×C22⋊C4—21no—
363#376—941no—
364C43C2×D4711yescharacteristic
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371#336—471no—
372#336—471no—
373#336—471no—
374#336—471no—
375C47#2112111yescharacteristic, complemented, Sylow
376C22×Q8—471no—
377C22×D4C22×D4711yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4—21no—
381C2×SD16—474no—
382C2×SD16—942no—
383C2×SD16—942no—
384C22×C8—471no—
385C22×C8—471no—
386C22×C8—471no—
387C22×C8—471no—
388C4×Q8—474no—
389C4×Q8—942no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4C4×D4714yes—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C4⋊C4C22×D4711yescharacteristic
404C2×C4⋊C4C22×D4711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—44no—
412C2×C42—941no—
413C2×C42—941no—
414C2×C42C22⋊D4711yescharacteristic
415C2×C42C22⋊D4711yescharacteristic
416C2×C42C22⋊D4711yescharacteristic
417C2×C42C22⋊D4711yescharacteristic
418C2×C42C22×D4711yescharacteristic
419C2×C42C22×D4711yescharacteristic
420C2×C42C22×D4711yescharacteristic
421#107—474no—
422#107—942no—
423#106—942no—
424#103—474no—
425#103—942no—
426#103—942no—
427#102—474no—
428#102—942no—
429#102—942no—
430#97—474no—
431#97—942no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—941no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442C24—21no—
443C2×Q8—472no—
444C2×Q8—941no—
445C2×Q8—941no—
446C2×Q8—941no—
447C2×Q8—941no—
448C2×Q8—941no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D4712yes—
453C2×D4#1062535012yes—
454C2×D4#1062535012yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478SD16—948no—
479C2×C8—472no—
480C2×C8—472no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—472no—
495C4⋊C4—944no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—1882no—
502C4⋊C4—1882no—
503C4⋊C4C2×C4×D4712yes—
504C4⋊C4#1062535012yes—
505C4⋊C4#1062535012yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—944no—
521C42—1882no—
522C42C2×C4×D4712yes—
523C42#1062535012yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C22⋊D4711yescharacteristic
532Q8—942no—
533Q8—942no—
534Q8—1881no—
535Q8—1881no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1309398114yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4C4×C22⋊D4712yes—
579C2×C4C4×C22⋊D4712yes—
580C2×C4#1309405012yes—
581C2×C4#1309405412yes—
582C2×C4C22×C4×D4711yescharacteristic
583C2×C4C2×#1062535011yescharacteristic
584C2×C4C2×#1062535011yescharacteristic
585C2×C4C2×#1062535011yescharacteristic
586C2×C4C2×#1062535011yescharacteristic
587C2×C4C2×#1062535011yescharacteristic
588C8—942no—
589C8—942no—
590C8—1881no—
591C8—1881no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×C4×C22⋊D4711yescharacteristic
600C22#1568963411yescharacteristic
601C22#1568963411yescharacteristic
602C22C2×#1309405011yescharacteristic
603C22C2×#1309405411yescharacteristic
604C22#1568992111yescharacteristic
605C22#1568992111yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—3761no—
616C4—3761no—
617C4—1882no—
618C4—1882no—
619C4C2×#1309398111yescharacteristic
620C4C4×#1062535011yescharacteristic
621C4#1568965711yescharacteristic
622C4#1568965711yescharacteristic
623C2—44no—
624C2#2024583111yescharacteristic
625C2C4×#1309405011yescharacteristic
626C2C4×#1309405411yescharacteristic
627C2#2025348711yescharacteristic
628C2#2025349711yescharacteristic
629C2#2025352611yescharacteristic
630C2#2025355011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC42×D4×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC42×D4D47
hypocenterC47#21121
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #20245865C2semidirect
3. #20244029C2semidirect
4. #20244022C2semidirect
5. #20243980C2semidirect
6. #20243959C2semidirect
11. #20243286C2semidirect
12. #20243221C2semidirect
14. #20243137C2semidirect
18. #15689223C4semidirect
69. C4×Dic94C2×C4semidirect
79. #13093957C2×C4semidirect
83. #13093947C2×C4semidirect
88. #13093935C2×C4semidirect
375. C47#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#156892232
#202458652
#202440292
#202440222
#202439802
#202439592
#202437222
#202436492
#202436352
#202432862
#202432212
#202432132
#202431372
C42×D4×C472
#2112147