Finite Group Databasebeta

← #26223536

Subgroups — #26223536

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246083C211yesmaximal, characteristic
2#20245745C211yesmaximal, characteristic, complemented
3#20244017C211yesmaximal, characteristic, complemented
4#20244010C211yesmaximal, characteristic, complemented
5#20243975C211yesmaximal, characteristic, complemented
6#20243968C211yesmaximal, characteristic, complemented
7#20243711C211yesmaximal, characteristic
8#20243589C211yesmaximal, characteristic, complemented
9#20243474C211yesmaximal, characteristic, complemented
10#20243288C211yesmaximal, characteristic, complemented
11#106×Dic47C211yesmaximal, characteristic, complemented
12#20243215C211yesmaximal, characteristic, complemented
13#102×Dic47C211yesmaximal, characteristic, complemented
14#20243156C211yesmaximal, characteristic, complemented
15C2×C47×#481C211yesmaximal, characteristic
16C2×D4×Dic47C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093956C2211yescharacteristic
21C2×#13093945C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093934C2211yescharacteristic
25C2×C8×Dic47C2211yescharacteristic
26#15689026—21no—
27#15689026C2211yescharacteristic
28#15689022C2211yescharacteristic
29#15689016—21no—
30#15689003—21no—
31#15688998—21no—
32#15688983C2211yescharacteristic
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47—21no—
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4×C22⋊Dic47—21no—
41C4×C22⋊Dic47—21no—
42#15688886C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×#121×C47C2211yescharacteristic
53C22×C4⋊C4×C47C2211yescharacteristic
54C47×#481C418yescomplemented
55C2×#106×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×#102×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C2×C22⋊C8×C47C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63D4×Dic47—22no—
64D4×Dic47—22no—
65#13093983—22no—
66#13093983—22no—
67#13093983—22no—
68#13093983—22no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47D411yescharacteristic
71C2×C4⋊Dic47D411yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47D411yescharacteristic
82C2×C22⋊Dic47D411yescharacteristic
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093958—22no—
92#13093956—22no—
93#13093945—22no—
94#13093945—22no—
95C22⋊(C47⋊C8)—22no—
96#13093934—22no—
97C8×Dic47—22no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C22×D4×C47—21no—
106C22×D4×C47C2311yescharacteristic
107C23×C4×C47—21no—
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111#121×C47—22no—
112#121×C47C2×C414yes—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47C2×C414yescomplemented
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120#106×C47C2×C414yescomplemented
121#106×C47C2×C414yescomplemented
122#102×C47C2×C414yes—
123#102×C47C2×C414yes—
124C22⋊C8×C47C2×C414yescomplemented
125C24×C47—21no—
126C24×C47C2×D411yescharacteristic
127C2×D4×C47—24no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—81no—
134C2×D4×C47—81no—
135C2×D4×C47C22×C412yes—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—24no—
144C22×C4×C47—81no—
145C22×C4×C47—81no—
146C22×C4×C47C22×C412yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47#4112yes—
149C22×C4×C47C2411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—42no—
155C22×Dic47—81no—
156C22×Dic47—81no—
157C22×Dic47C2×D411yescharacteristic
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47C22×C412yes—
166C2×C8×C47C22×C412yes—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47C22×C412yes—
173C4⋊C4×C47C22×C412yes—
174C22⋊C4×C47—24no—
175C22⋊Dic47—24no—
176C22⋊Dic47—22no—
177C22⋊Dic47—22no—
178C22⋊Dic47—42no—
179C22⋊Dic47—81no—
180C22⋊Dic47—81no—
181C22⋊Dic47—81no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—81no—
191C4⋊Dic47#4112yes—
192C4⋊Dic47#4112yes—
193C4×Dic47—24no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—81no—
197C4×Dic47—81no—
198C4×Dic47—81no—
199C4×Dic47—81no—
200C4×Dic47—81no—
201C4×Dic47#4112yes—
202C4×Dic47#4112yes—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—22no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47C4×D412yes—
223C23×C47C22×D411yescharacteristic
224D4×C47—22no—
225D4×C47—22no—
226D4×C47—41no—
227D4×C47—41no—
228D4×C47—84no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—84no—
245C2×C4×C47C4×D412yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—84no—
275C2×Dic47#12312yes—
276C2×Dic47#12312yes—
277C8×C47—22no—
278C8×C47—22no—
279C47⋊C8—41no—
280C47⋊C8—41no—
281#21584—471nomaximal, Sylow
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47—81no—
295C22×C47—81no—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47C2×#12311yescharacteristic
298C22×C47#52511yescharacteristic
299C22×C47#57611yescharacteristic
300C22×C47#57711yescharacteristic
301C22×C47#57711yescharacteristic
302C22×C47#57711yescharacteristic
303C4×C47—22no—
304C4×C47—42no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47C4×#4111yescharacteristic
308C4×C47C4×#4111yescharacteristic
309C4×C47#52911yescharacteristic
310C4×C47#52911yescharacteristic
311Dic47—22no—
312Dic47—22no—
313Dic47—41no—
314Dic47—41no—
315Dic47—42no—
316Dic47—42no—
317Dic47—81no—
318Dic47—81no—
319C2×#481D4711yescharacteristic
320#2324—471no—
321#2274—471no—
322#1926—471no—
323#1925—471no—
324#1920—471no—
325#1919—471no—
326#1875—471no—
327#1854—471no—
328#1823—471no—
329#1776—471no—
330C4×#106—471no—
331#1762—471no—
332C4×#102—471no—
333#1750—471no—
334C2×C47—42no—
335C2×C47—82no—
336C2×C47C4×#12311yescharacteristic
337C2×C47#294611yescharacteristic
338C2×C47#294911yescharacteristic
339C2×C47#309611yescharacteristic
340C2×C47#309911yescharacteristic
341C2×C47#311911yescharacteristic
342C2×C47#312511yescharacteristic
343C2×#121C2×D4711yescharacteristic
344C2×C4×D4—471no—
345C2×#117—471no—
346C2×#117—471no—
347C22×C4⋊C4C2×D4711yescharacteristic
348#481Dic4718yescomplemented
349C2×#107—471no—
350C2×#106—471no—
351C2×#106C2×D4711yescharacteristic
352C2×#106C2×D4711yescharacteristic
353C2×#102—471no—
354C2×#102—471no—
355C2×#102C2×D4711yescharacteristic
356C2×#102C2×D4711yescharacteristic
357C2×C22⋊C8—471no—
358C2×C22⋊C8C2×D4711yescharacteristic
359C2×#97—471no—
360C2×C4×C8—471no—
361#390—471no—
362#390—941no—
363#389—471no—
364#388—941no—
365#386—941no—
366#385—941no—
367#380—471no—
368C4×C4⋊C4—471no—
369C4×C4⋊C4—471no—
370C4×C4⋊C4—471no—
371C4×C4⋊C4—471no—
372C4×C4⋊C4—471no—
373C4×C4⋊C4—941no—
374C4×C4⋊C4—941no—
375C4×C22⋊C4—941no—
376C4×C22⋊C4—941no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#336—471no—
386#336—471no—
387C47#2158411yescharacteristic, complemented, Sylow
388C22×D4—21no—
389C22×D4C22×D4711yescharacteristic
390C23×C4—21no—
391C23×C4—941no—
392C23×C4C22×D4711yescharacteristic
393C22×C8—471no—
394C22×C8—471no—
395C22×C8C22×D4711yescharacteristic
396C22×C8C22×D4711yescharacteristic
397#121—22no—
398#121C2×Dic4714yes—
399C4×D4—942no—
400C4×D4—942no—
401#117—942no—
402#117—942no—
403#117—942no—
404#117—942no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C2×Dic4714yescomplemented
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—471no—
420C2×C22⋊C4—471no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42—941no—
433C2×C42—941no—
434#107—942no—
435#106—942no—
436#106C2×Dic4714yescomplemented
437#106C2×Dic4714yescomplemented
438#102—942no—
439#102—942no—
440#102C2×Dic4714yes—
441#102C2×Dic4714yes—
442C22⋊C8—942no—
443C22⋊C8C2×Dic4714yescomplemented
444#97—942no—
445C4×C8—942no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453C24—21no—
454C24D4×D4711yescharacteristic
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C22×Dic4712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—3761no—
483C22×C4—3761no—
484C22×C4—1882no—
485C22×C4D4×D4711yescharacteristic
486C22×C4#1062535212yes—
487C22×C4C22×Dic4712yes—
488C22×C4C23×D4711yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8C22×Dic4712yes—
500C2×C8C22×Dic4712yes—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—472no—
506C4⋊C4—472no—
507C4⋊C4—942no—
508C4⋊C4—942no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—3761no—
516C4⋊C4C22×Dic4712yes—
517C4⋊C4C22×Dic4712yes—
518C4⋊C4C22×Dic4712yes—
519C22⋊C4—24no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—944no—
523C22⋊C4—3761no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—1882no—
527C42—472no—
528C42—472no—
529C42—944no—
530C42—941no—
531C42—941no—
532C42—3761no—
533C42—3761no—
534C42—3761no—
535C42—3761no—
536C42—3761no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23D4×Dic4712yes—
551C23C2×D4×D4711yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—3764no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4D4×Dic4712yes—
599C2×C4C23×Dic4711yescharacteristic
600C2×C4#41×D4711yescharacteristic
601C2×C4#41×D4711yescharacteristic
602C2×C4#41×D4711yescharacteristic
603C2×C4#41×D4711yescharacteristic
604C2×C4C2×#1062535211yescharacteristic
605C8—22no—
606C8—22no—
607C8—1881no—
608C8—1881no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×D4×Dic4711yescharacteristic
624C22#123×D4711yescharacteristic
625C22#1568963111yescharacteristic
626C22#1568992711yescharacteristic
627C22#1568992911yescharacteristic
628C22#1568992911yescharacteristic
629C22#1568993311yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—942no—
635C4—942no—
636C4—3761no—
637C4—3761no—
638C4—1882no—
639C4—1881no—
640C4—1881no—
641C4—1882no—
642C4#41×Dic4711yescharacteristic
643C4#41×Dic4711yescharacteristic
644C4#1568965011yescharacteristic
645C4#1568965011yescharacteristic
646C2—42no—
647C2—82no—
648C2#123×Dic4711yescharacteristic
649C2#2025222111yescharacteristic
650C2#2025224811yescharacteristic
651C2#2025334511yescharacteristic
652C2#2025337811yescharacteristic
653C2#2025358211yescharacteristic
654C2#2025366811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#481C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#481D47
hypocenterC47#21584
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. #20245745C2semidirect
3. #20244017C2semidirect
4. #20244010C2semidirect
5. #20243975C2semidirect
6. #20243968C2semidirect
8. #20243589C2semidirect
9. #20243474C2semidirect
10. #20243288C2semidirect
11. #106×Dic47C2semidirect
12. #20243215C2semidirect
13. #102×Dic47C2semidirect
14. #20243156C2semidirect
54. C47×#481C4semidirect
115. C2×C4⋊C4×C47C2×C4semidirect
120. #106×C47C2×C4semidirect
121. #106×C47C2×C4semidirect
124. C22⋊C8×C47C2×C4semidirect
348. #481Dic47semidirect
387. C47#21584semidirect
414. C2×C4⋊C4C2×Dic47semidirect
436. #106C2×Dic47semidirect
437. #106C2×Dic47semidirect
443. C22⋊C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202460832
#202457452
#202440172
#202440102
#202439752
#202439682
#202437112
#202435892
#202434742
#202432882
#106×Dic472
#202432152
#102×Dic472
#202431562
C2×C47×#4812
#2158447