Finite Group Databasebeta

← #26219360

Subgroups — #26219360

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689221C211yesmaximal, characteristic
2#20246039C211yesmaximal, characteristic, complemented
3#20245748C211yesmaximal, characteristic
4#20244032C211yesmaximal, characteristic, complemented
5#20244025C211yesmaximal, characteristic, complemented
6#20243983C211yesmaximal, characteristic, complemented
7#20243962C211yesmaximal, characteristic, complemented
8#20243719C211yesmaximal, characteristic
9#20243719C211yesmaximal, characteristic
10#20243650C211yesmaximal, characteristic
11#20243636C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13C4×#13093946C211yesmaximal, characteristic
14#20243139C211yesmaximal, characteristic, complemented
15C4×#103×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×#13093986C2211yescharacteristic
19C2×#13093986C2211yescharacteristic
20#15689221C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093946C2211yescharacteristic
26C2×#13093946C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29#15689030C2211yescharacteristic
30#15689030C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689022C2211yescharacteristic
33#15689009—21no—
34#15689009—21no—
35#15688991—21no—
36#15688989—21no—
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C4×(C2×D47)⋊C4—21no—
41C4×(C2×D47)⋊C4—21no—
42#15688970—21no—
43#15688969C2211yescharacteristic
44#95×D47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×C4×C8×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C2×#10625336—22no—
65C2×#10625336—22no—
66C2×#10625336C2×C414yes—
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C4×D188—22no—
70C4×D188C2×C414yes—
71#13093986—22no—
72#13093986—22no—
73#13093986—22no—
74#13093986—22no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093957—22no—
89#13093957C2×C414yescomplemented
90#13093955—22no—
91#13093948C2×C414yescomplemented
92#13093946—22no—
93#13093946—22no—
94#13093944C2×C414yes—
95#13093935C2×C414yescomplemented
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C22×Q8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C4×Q8×C47—22no—
113C4×Q8×C47C2×C414yescomplemented
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C42×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#103×C47—22no—
122#103×C47—22no—
123C4×C8×C47—22no—
124#95×C47—21no—
125C23×D47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C2×D188—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188C22×C412yes—
150C2×D188#4112yes—
151C2×D188#4112yes—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—24no—
155C2×C4×D47—81no—
156C2×C4×D47—81no—
157C2×C4×D47—81no—
158C2×C4×D47—81no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47#4112yes—
173C4⋊C4×C47#4112yes—
174C42×C47—24no—
175C42×C47—81no—
176C42×C47C22×C412yes—
177C42×C47#4112yes—
178#10625336—28no—
179(C2×D47)⋊C4—24no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—24no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—81no—
188C4⋊Dic47—81no—
189C4⋊Dic47C22×C412yes—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—81no—
195C4×Dic47—81no—
196C4×Dic47—81no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8C22×C412yes—
202C2×C47⋊C8C22×C412yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211Q8×C47—22no—
212Q8×C47—22no—
213Q8×C47—41no—
214Q8×C47—41no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47C4×D412yes—
232C2×C4×C47C4×D412yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260D188—22no—
261D188—22no—
262D188—22no—
263D188#11714yes—
264C4×D47—84no—
265C4×D47—84no—
266C4×D47—84no—
267C8×C47—41no—
268C8×C47—41no—
269C47⋊C8—22no—
270C47⋊C8—22no—
271#21125—471nomaximal, Sylow
272C22×C47C2×C4×D411yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47#52511yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57811yescharacteristic
279C2×D47—24no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—42no—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—42no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47C2×#11711yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#52911yescharacteristic
295C4×C47#52911yescharacteristic
296Dic47—41no—
297Dic47—41no—
298Dic47—41no—
299Dic47—41no—
300Dic47—42no—
301Dic47—81no—
302Dic47—81no—
303C2×#440—471no—
304#2320—471no—
305#2275—471no—
306#1928—471no—
307#1927—471no—
308#1921—471no—
309#1918—471no—
310#1877—471no—
311#1877—471no—
312#1864—471no—
313#1862—471no—
314#1776—471no—
315C4×#103D4711yescharacteristic, complemented
316C4×#102—471no—
317#1746—471no—
318C2×C47#229011yescharacteristic
319C2×C47#294611yescharacteristic
320C2×C47#294711yescharacteristic
321C2×C47#311811yescharacteristic
322C2×C47#312011yescharacteristic
323C2×C47#312311yescharacteristic
324C2×C47#312411yescharacteristic
325D47—44no—
326C22×SD16—471no—
327C2×C4×Q8C2×D4711yescharacteristic
328C2×C4×D4—471no—
329C2×C4×D4—471no—
330C2×C4×D4—471no—
331#440—478no—
332C2×#107—471no—
333C2×#107—471no—
334C2×#106—471no—
335C2×#103—471no—
336C2×#103C2×D4711yescharacteristic
337C2×#103C2×D4711yescharacteristic
338C2×#102—471no—
339C2×#102—471no—
340C2×#102—471no—
341C2×#97—471no—
342C2×C4×C8C2×D4711yescharacteristic
343#391—471no—
344#391—471no—
345#389—471no—
346#389—471no—
347#387—941no—
348#387—941no—
349#383—941no—
350#382—941no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4C2×D4711yescharacteristic
355C4×C22⋊C4—941no—
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358#376—471no—
359C2×#95—941no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340C2×D4711yescharacteristic
365#340C2×D4711yescharacteristic
366#336—471no—
367#336—471no—
368#336—471no—
369#336—471no—
370C47#2112511yescharacteristic, complemented, Sylow
371C22×Q8C22×D4711yescharacteristic
372C22×D4—471no—
373C23×C4—941no—
374C23×C4—941no—
375C23×C4—941no—
376C2×SD16—474no—
377C2×SD16—942no—
378C2×SD16—942no—
379C22×C8—471no—
380C22×C8—471no—
381C22×C8C22×D4711yescharacteristic
382C22×C8C22×D4711yescharacteristic
383C4×Q8—22no—
384C4×Q8C4×D4714yescomplemented
385C4×D4—474no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4—942no—
390C4×D4—942no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—471no—
406C2×C42—471no—
407C2×C42—941no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42C22×D4711yescharacteristic
411#107—474no—
412#107—942no—
413#106—942no—
414#103—22no—
415#103—22no—
416#103—474no—
417#102—474no—
418#102—942no—
419#102—942no—
420#97—474no—
421C4×C8—22no—
422#95—21no—
423#95—471no—
424#95—471no—
425#95—471no—
426#95—471no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436C24—941no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D4712yes—
443C2×D4—472no—
444C2×D4—472no—
445C2×D4—472no—
446C2×D4—941no—
447C2×D4—941no—
448C2×D4—941no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—941no—
460C22×C4—944no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—3761no—
464C22×C4—3761no—
465C22×C4—3761no—
466C22×C4—3761no—
467C22×C4—3761no—
468C22×C4—3761no—
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4C23×D4711yescharacteristic
472SD16—948no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—472no—
480C2×C8—472no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—472no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—3761no—
496C4⋊C4—3761no—
497C4⋊C4C2×C4×D4712yes—
498C4⋊C4#1062535012yes—
499C4⋊C4#1062535012yes—
500C22⋊C4—944no—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C42—24no—
504C42—81no—
505C42—941no—
506C42—941no—
507C42—941no—
508C42—941no—
509C42—3761no—
510C42—3761no—
511C42—3761no—
512C42C2×C4×D4712yes—
513C42#1062535212yes—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23C2×D4×D4711yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—474no—
527D4—942no—
528D4—942no—
529D4—942no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—942no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—3764no—
560C2×C4—3764no—
561C2×C4—3764no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4(C22⋊D47)⋊C412yes—
571C2×C4(C22⋊D47)⋊C412yes—
572C2×C4C22×C4×D4711yescharacteristic
573C2×C4#41×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C2×C4C2×#1062535011yescharacteristic
577C2×C4C2×#1062535011yescharacteristic
578C8—41no—
579C8—41no—
580C8—942no—
581C8—942no—
582C22—944no—
583C22—1882no—
584C22—1882no—
585C22—1882no—
586C22—1882no—
587C22—1882no—
588C22—1882no—
589C22C2×(C22⋊D47)⋊C411yescharacteristic
590C22#1568963211yescharacteristic
591C22#1568963211yescharacteristic
592C22#1568992511yescharacteristic
593C22#1568992911yescharacteristic
594C22#1568993011yescharacteristic
595C22#1568993611yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—3761no—
603C4—3761no—
604C4—1882no—
605C4—1881no—
606C4—1881no—
607C4—1881no—
608C4—1881no—
609C4#117×D4711yescharacteristic
610C4C4×#1062535011yescharacteristic
611C4#1568965511yescharacteristic
612C4#1568965511yescharacteristic
613C2—1884no—
614C2#2024583411yescharacteristic
615C2#2025222311yescharacteristic
616C2#2025222511yescharacteristic
617C2#2025357811yescharacteristic
618C2#2025360811yescharacteristic
619C2#2025363711yescharacteristic
620C2#2025365611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#103×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterC4×#103D47
hypocenterC47#21125
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. #20246039C2semidirect
4. #20244032C2semidirect
5. #20244025C2semidirect
6. #20243983C2semidirect
7. #20243962C2semidirect
12. #20243286C2semidirect
14. #20243139C2semidirect
15. C4×#103×C47C2semidirect
20. #15689221C4semidirect
89. #13093957C2×C4semidirect
91. #13093948C2×C4semidirect
95. #13093935C2×C4semidirect
113. C4×Q8×C47C2×C4semidirect
315. C4×#103D47semidirect
370. C47#21125semidirect
384. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892212
#202460392
#202457482
#202440322
#202440252
#202439832
#202439622
#202437192
#202436502
#202436362
#202432862
C4×#130939462
#202431392
C4×#103×C472
#2112547