Finite Group Databasebeta

← #26236135

Subgroups — #26236135

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689223C211yesmaximal, characteristic
2#20246361C211yesmaximal, characteristic
3#20246347C211yesmaximal, characteristic, complemented
4#20244270C211yesmaximal, characteristic
5#20244243C211yesmaximal, characteristic
6#20244027C211yesmaximal, characteristic, complemented
7#20244020C211yesmaximal, characteristic, complemented
8#20244018C211yesmaximal, characteristic
9#20244011C211yesmaximal, characteristic
10#20243554C211yesmaximal, characteristic, complemented
11#20243404C211yesmaximal, characteristic, complemented
12#20243284C211yesmaximal, characteristic, complemented
13#20243280C211yesmaximal, characteristic, complemented
14#20243265C211yesmaximal, characteristic, complemented
15C47×#1802C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18#15689223C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093949C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15689028C2211yescharacteristic
32#15689028C2211yescharacteristic
33#15689026C2211yescharacteristic
34#15689026C2211yescharacteristic
35#15689016—21no—
36#15689016—21no—
37#15689004—21no—
38#15689004—21no—
39#15688989C2211yescharacteristic
40#15688988—21no—
41#15688986—21no—
42C4⋊C4×Dic47—21no—
43C4⋊C4×Dic47C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46C4⋊C4×Dic47C2211yescharacteristic
47C4×C4⋊Dic47C2211yescharacteristic
48C4×C22⋊Dic47—21no—
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×C4⋊C8×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#384C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C2×#10625335—22no—
62C2×#10625335—22no—
63C2×#10625335C2×C414yes—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C4×Dic94—22no—
67C4×Dic94C2×C414yes—
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47D411yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85#13093957—22no—
86#13093957—41no—
87#13093957C2×C414yes—
88#13093955—22no—
89#13093955—41no—
90#13093949—41no—
91#13093949—41no—
92#13093947C2×C414yes—
93#13093945C2×C414yes—
94#13093935C2×C414yes—
95#13093934—22no—
96C8×Dic47—22no—
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⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C4×D4×C47—22no—
114C4×D4×C47C2×C414yes—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C22⋊C4×C47—21no—
121C2×C42×C47C2311yescharacteristic
122C4⋊C8×C47—22no—
123#102×C47—41no—
124#102×C47—41no—
125C24×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—24no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—81no—
137C22×C4×C47—81no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94C22×C412yes—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—42no—
167C4⋊C4×C47—81no—
168C4⋊C4×C47C22×C412yes—
169C22⋊C4×C47—24no—
170C42×C47C22×C412yes—
171C42×C47#4112yes—
172C22⋊Dic47—42no—
173C22⋊Dic47—42no—
174C22⋊Dic47—81no—
175C22⋊Dic47—81no—
176#10625335—28no—
177C4⋊Dic47—21no—
178C4⋊Dic47—22no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—41no—
184C4⋊Dic47C22×C412yes—
185Dic47⋊C4—24no—
186C4×Dic47—24no—
187C4×Dic47—21no—
188C4×Dic47—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—81no—
192C4×Dic47—81no—
193C4×Dic47—81no—
194C4×Dic47—81no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8C22×C412yes—
198C2×C47⋊C8C22×C412yes—
199C2×C47⋊C8#4112yes—
200C2×C47⋊C8#4112yes—
201C23×C47—22no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47C22×D411yescharacteristic
209D4×C47—22no—
210D4×C47—22no—
211D4×C47—41no—
212D4×C47—41no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—84no—
227C2×C4×C47C4×D412yes—
228C2×C4×C47C4×D412yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47H(5,𝔽2)11yescharacteristic
234C2×C4×C472−1+411yescharacteristic
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—21no—
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—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47#13712yes—
259Dic94—22no—
260Dic94—22no—
261Dic94—41no—
262Dic94—41no—
263C8×C47—41no—
264C8×C47—41no—
265C47⋊C8—22no—
266C47⋊C8#11714yes—
267#22819—471nomaximal, Sylow
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—81no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47C2×C4×D411yescharacteristic
276C22×C47#54711yescharacteristic
277C22×C47#55011yescharacteristic
278C22×C47C2×#13711yescharacteristic
279C22×C47#57511yescharacteristic
280C22×C47#57611yescharacteristic
281C22×C47#57711yescharacteristic
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—42no—
285C4×C47—81no—
286C4×C47—81no—
287C4×C47C2×#11711yescharacteristic
288C4×C47#52011yescharacteristic
289C4×C47#55511yescharacteristic
290C4×C47#55611yescharacteristic
291Dic47—22no—
292Dic47—41no—
293Dic47—41no—
294Dic47—41no—
295Dic47—42no—
296Dic47—42no—
297Dic47—81no—
298Dic47—81no—
299C2×#440—471no—
300#2357—471no—
301#2355—471no—
302#1973—471no—
303#1968—471no—
304#1928—471no—
305#1927—471no—
306#1926—471no—
307#1925—471no—
308#1847—471no—
309#1805—471no—
310#1802D4711yescharacteristic
311#1776—471no—
312C4×#107—471no—
313#1771—471no—
314C2×C47—82no—
315C2×C47#235411yescharacteristic
316C2×C47C4×#13711yescharacteristic
317C2×C47#294611yescharacteristic
318C2×C47#334611yescharacteristic
319C2×C47#335011yescharacteristic
320C2×C47#340011yescharacteristic
321C2×C47#340311yescharacteristic
322C22×SD16—471no—
323C2×C4×Q8—471no—
324C2×C4×D4C2×D4711yescharacteristic
325#440—478no—
326C2×#107—471no—
327C2×#107—471no—
328C2×#107—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×C4⋊C8C2×D4711yescharacteristic
332C2×#103—471no—
333C2×#103—471no—
334C2×#103—471no—
335C2×#102—471no—
336C2×#102C2×D4711yescharacteristic
337C2×#102C2×D4711yescharacteristic
338C2×#97—471no—
339C2×#97—471no—
340C2×C4×C8—471no—
341#391—471no—
342#391—471no—
343#390—471no—
344#390—471no—
345#388—941no—
346#388—941no—
347#387—941no—
348#387—941no—
349#384C2×D4711yescharacteristic
350#382—471no—
351#382—941no—
352#381—941no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—941no—
359C4×C22⋊C4—941no—
360#340—471no—
361#340—471no—
362#340C2×D4711yescharacteristic
363#340C2×D4711yescharacteristic
364#336—471no—
365#336—471no—
366C47#2281911yescharacteristic, complemented, Sylow
367C22×Q8—471no—
368C22×D4C22×D4711yescharacteristic
369C23×C4—21no—
370C2×SD16—474no—
371C2×SD16—942no—
372C2×SD16—942no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8C22×D4711yescharacteristic
376C22×C8C22×D4711yescharacteristic
377C4×Q8—474no—
378C4×Q8—942no—
379C4×D4—22no—
380C4×D4C4×D4714yes—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402C2×C42—941no—
403C2×C42—941no—
404C2×C42C22×D4711yescharacteristic
405#107—474no—
406#107—942no—
407#107—1881no—
408#106—942no—
409#106—1881no—
410C4⋊C8—22no—
411#103—474no—
412#103—1881no—
413#103—1881no—
414#102—41no—
415#102—41no—
416#102—474no—
417#97—474no—
418#97—942no—
419C4×C8—942no—
420#95—471no—
421#95—471no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432C24—21no—
433C2×Q8—472no—
434C2×Q8—941no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D4712yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4D4×D4711yescharacteristic
460C22×C4D4×D4711yescharacteristic
461C22×C4C23×D4711yescharacteristic
462SD16—948no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—472no—
470C2×C8—472no—
471C2×C8—472no—
472C2×C8—472no—
473C2×C8—941no—
474C2×C8—941no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—472no—
483C4⋊C4—942no—
484C4⋊C4—944no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—1881no—
491C4⋊C4C2×C4×D4712yes—
492C22⋊C4—24no—
493C22⋊C4—3761no—
494C22⋊C4—3761no—
495C22⋊C4—1882no—
496C22⋊C4—1882no—
497C42—944no—
498C42—941no—
499C42—941no—
500C42—941no—
501C42—941no—
502C42—3761no—
503C42—3761no—
504C42—3761no—
505C42—3761no—
506C42C2×C4×D4712yes—
507C42#1062535212yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D4711yescharacteristic
516Q8—942no—
517Q8—942no—
518Q8—1881no—
519Q8—1881no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—472no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4(C22⋊D47)⋊C412yes—
563C2×C4(C22⋊D47)⋊C412yes—
564C2×C4C22×C4×D4711yescharacteristic
565C2×C4#41×D4711yescharacteristic
566C2×C4#41×D4711yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4#1309406911yescharacteristic
569C2×C4#1309407211yescharacteristic
570C8—41no—
571C8—41no—
572C8—474no—
573C8—942no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×(C22⋊D47)⋊C411yescharacteristic
582C22#1568977511yescharacteristic
583C22#1568979111yescharacteristic
584C22#137×D4711yescharacteristic
585C22#1568991911yescharacteristic
586C22#1568992411yescharacteristic
587C22#1568992911yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—942no—
594C4—3761no—
595C4—3761no—
596C4—1882no—
597C4—1882no—
598C4—1881no—
599C4—1881no—
600C4—1881no—
601C4#117×D4711yescharacteristic
602C4#1568960411yescharacteristic
603C4#1568983711yescharacteristic
604C4#1568984411yescharacteristic
605C2—82no—
606C2#2024633211yescharacteristic
607C2#2025221211yescharacteristic
608C2#2025221411yescharacteristic
609C2#2025587011yescharacteristic
610C2#2025594011yescharacteristic
611C2#2025643611yescharacteristic
612C2#2025647111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22819
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #20246347C2semidirect
6. #20244027C2semidirect
7. #20244020C2semidirect
10. #20243554C2semidirect
11. #20243404C2semidirect
12. #20243284C2semidirect
13. #20243280C2semidirect
14. #20243265C2semidirect
18. #15689223C4semidirect
366. C47#22819semidirect

Maximal subgroups

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

subgroupindex
C2×#156892232
#202463612
#202463472
#202442702
#202442432
#202440272
#202440202
#202440182
#202440112
#202435542
#202434042
#202432842
#202432802
#202432652
C47×#18022
#2281947