Finite Group Databasebeta

← #26223533

Subgroups — #26223533

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
1C2×#15689440C211yesmaximal, characteristic
2#20246075C211yesmaximal, characteristic
3#20245742C211yesmaximal, characteristic, complemented
4#20244018C211yesmaximal, characteristic, complemented
5#20244010C211yesmaximal, characteristic, complemented
6#20243972C211yesmaximal, characteristic, complemented
7#20243964C211yesmaximal, characteristic, complemented
8#20243707C211yesmaximal, characteristic
9#20243590C211yesmaximal, characteristic, complemented
10#20243473C211yesmaximal, characteristic, complemented
11#20243285C211yesmaximal, characteristic, complemented
12C4×#13093955C211yesmaximal, characteristic, complemented
13C4×#13093945C211yesmaximal, characteristic, complemented
14#20243157C211yesmaximal, characteristic, complemented
15C47×#1762C211yesmaximal, characteristic, complemented
16C2×#13093995C2211yescharacteristic
17C2×#13093982C2211yescharacteristic
18C2×#13093981C2211yescharacteristic
19C2×C4⋊C4×D47C2211yescharacteristic
20#15689440C418yescomplemented
21C2×#13093958C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×(C2×D47)⋊C8C2211yescharacteristic
28C2×(C2×D47)⋊C8C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689026C2211yescharacteristic
31#15689023—21no—
32#15689019C2211yescharacteristic
33#15689017—21no—
34#15689002—21no—
35#15688996—21no—
36#15688980C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4×Dic47⋊C4—21no—
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42#15688974—21no—
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C22⋊D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13093995—22no—
66#13093995C2×C414yes—
67#13093982—22no—
68#13093982—22no—
69#13093981—22no—
70#13093981—22no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4D411yescharacteristic
78C2×Dic47⋊C4D411yescharacteristic
79C4⋊C4×D47—22no—
80C4⋊C4×D47C2×C414yescomplemented
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4D411yescharacteristic
87C2×(C2×D47)⋊C4D411yescharacteristic
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47C2311yescharacteristic
94#13093958—22no—
95#13093956C2×C414yescomplemented
96#13093955—22no—
97#13093955C2×C414yescomplemented
98#13093945—22no—
99#13093945C2×C414yes—
100(C2×D47)⋊C8—22no—
101(C2×D47)⋊C8C2×C414yescomplemented
102C4×C47⋊C8—22no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C4×D4×C47—22no—
114C4×D4×C47—22no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yes—
123#97×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C23×D47C2×D411yescharacteristic
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47C22×C412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—81no—
138C22×C4×C47—81no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—24no—
143C2×C22⋊D47—81no—
144C2×C22⋊D47—81no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C2×C4×D47—21no—
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—24no—
157C2×C4×D47—81no—
158C2×C4×D47—81no—
159C2×C4×D47C22×C412yes—
160C2×C4×D47#4112yes—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47C22×C412yes—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47#4112yes—
173C22⋊C4×C47—42no—
174C42×C47—81no—
175C42×C47#4112yes—
176C42×C47#4112yes—
177C22⋊Dic47—24no—
178C22⋊Dic47—81no—
179C22⋊Dic47—81no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183(C2×D47)⋊C4—81no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C412yes—
190C4⋊Dic47C22×C412yes—
191C4⋊Dic47#4112yes—
192Dic47⋊C4—22no—
193Dic47⋊C4—22no—
194Dic47⋊C4—24no—
195Dic47⋊C4—81no—
196C4×Dic47—24no—
197C4×Dic47—21no—
198C4×Dic47—21no—
199C4×Dic47—81no—
200C4×Dic47—81no—
201C4×Dic47—81no—
202C4×Dic47—81no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8C22×C412yes—
209C23×C47—22no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47C22×D411yescharacteristic
217C22×D47—21no—
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—21no—
223C22×D47C4×D412yes—
224D4×C47—22no—
225D4×C47—22no—
226D4×C47—41no—
227D4×C47—41no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—84no—
242C2×C4×C47#12312yes—
243C2×C4×C47#12312yes—
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C22⋊D47—84no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—22no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47C4×D412yes—
272C4×D47—22no—
273C4×D47—22no—
274C4×D47—22no—
275C4×D47—22no—
276C4×D47—84no—
277C8×C47—22no—
278C8×C47—41no—
279C47⋊C8—22no—
280C47⋊C8—41no—
281#21584—471nomaximal, Sylow
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—81no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47—81no—
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×#12311yescharacteristic
291C22×C47#52511yescharacteristic
292C22×C47#57611yescharacteristic
293C22×C47#57711yescharacteristic
294C22×C47#57711yescharacteristic
295C22×C47#57711yescharacteristic
296C2×D47—22no—
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—41no—
300C2×D47—41no—
301C2×D47—41no—
302C2×D47—41no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47—81no—
308C4×C47C4×#4111yescharacteristic
309C4×C47C4×#4111yescharacteristic
310C4×C47#52911yescharacteristic
311C4×C47#52911yescharacteristic
312Dic47—22no—
313Dic47—41no—
314Dic47—41no—
315Dic47—42no—
316Dic47—42no—
317Dic47—42no—
318Dic47—81no—
319C2×#481—471no—
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#1762D4711yescharacteristic, complemented
332C4×#102—471no—
333#1750—471no—
334C2×C47—82no—
335C2×C47C4×#12311yescharacteristic
336C2×C47#294611yescharacteristic
337C2×C47#294911yescharacteristic
338C2×C47#309611yescharacteristic
339C2×C47#309911yescharacteristic
340C2×C47#311911yescharacteristic
341C2×C47#312511yescharacteristic
342D47—42no—
343C2×#121—471no—
344C2×C4×D4C2×D4711yescharacteristic
345C2×#117—471no—
346C2×#117—471no—
347C22×C4⋊C4—471no—
348#481—478no—
349C2×#107—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#106—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102C2×D4711yescharacteristic
356C2×#102C2×D4711yescharacteristic
357C2×C22⋊C8—471no—
358C2×C22⋊C8—471no—
359C2×#97C2×D4711yescharacteristic
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—941no—
373C4×C4⋊C4—941no—
374C4×C4⋊C4C2×D4711yescharacteristic
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#340C2×D4711yescharacteristic
384#340C2×D4711yescharacteristic
385#336—471no—
386#336—471no—
387C47#2158411yescharacteristic, complemented, Sylow
388C22×D4—941no—
389C22×D4C22×D4711yescharacteristic
390C23×C4—21no—
391C23×C4—471no—
392C23×C4—941no—
393C22×C8—471no—
394C22×C8—471no—
395C22×C8C22×D4711yescharacteristic
396C22×C8C22×D4711yescharacteristic
397#121—474no—
398#121—942no—
399C4×D4—22no—
400C4×D4—22no—
401#117—942no—
402#117—942no—
403#117—942no—
404#117—942no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
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—21no—
427C2×C42—471no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42—941no—
433C2×C42C22×D4711yescharacteristic
434#107—942no—
435#106—474no—
436#106—474no—
437#106—942no—
438#102—22no—
439#102—474no—
440#102—942no—
441#102C4×D4714yes—
442C22⋊C8—474no—
443C22⋊C8—942no—
444#97—22no—
445C4×C8—942no—
446#95—21no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453C24—21no—
454C24—471no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—944no—
461C2×D4—3761no—
462C2×D4—3761no—
463C2×D4C2×C4×D4712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—471no—
471C22×C4—472no—
472C22×C4—472no—
473C22×C4—471no—
474C22×C4—471no—
475C22×C4—944no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—941no—
484C22×C4—3761no—
485C22×C4—3761no—
486C22×C4D4×D4711yescharacteristic
487C22×C4D4×D4711yescharacteristic
488C22×C4C23×D4711yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—472no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8C2×C4×D4712yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—472no—
506C4⋊C4—472no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—3761no—
517C4⋊C4C2×C4×D4712yes—
518C4⋊C4#1062535012yes—
519C22⋊C4—42no—
520C22⋊C4—944no—
521C22⋊C4—944no—
522C22⋊C4—942no—
523C22⋊C4—942no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—3761no—
527C42—81no—
528C42—944no—
529C42—941no—
530C42—941no—
531C42—3761no—
532C42—3761no—
533C42—3761no—
534C42—3761no—
535C42#1062535012yes—
536C42#1062535212yes—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—472no—
545C23—941no—
546C23—941no—
547C23—941no—
548C23—941no—
549C23—941no—
550C23—941no—
551C23C2×D4×D4711yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—3764no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—472no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—941no—
581C2×C4—941no—
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#1309400612yes—
598C2×C4#1309400612yes—
599C2×C4C22×C4×D4711yescharacteristic
600C2×C4#41×D4711yescharacteristic
601C2×C4#41×D4711yescharacteristic
602C2×C4C2×#1062535211yescharacteristic
603C2×C4C2×#1062535011yescharacteristic
604C2×C4C2×#1062535011yescharacteristic
605C8—22no—
606C8—41no—
607C8—942no—
608C8—1881no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—942no—
617C22—942no—
618C22—942no—
619C22—1881no—
620C22—1881no—
621C22—1881no—
622C22—1881no—
623C22C4×D4×D4711yescharacteristic
624C22C2×#1309400611yescharacteristic
625C22#1568963211yescharacteristic
626C22#1568992711yescharacteristic
627C22#1568993211yescharacteristic
628C22#1568993311yescharacteristic
629C22#1568993311yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—942no—
636C4—3761no—
637C4—1882no—
638C4—1882no—
639C4—1882no—
640C4—1881no—
641C4—1881no—
642C4C4×#1062535211yescharacteristic
643C4C4×#1062535011yescharacteristic
644C4#1568965111yescharacteristic
645C4#1568965511yescharacteristic
646C2—82no—
647C2—1882no—
648C2C4×#1309400611yescharacteristic
649C2#2025221711yescharacteristic
650C2#2025224411yescharacteristic
651C2#2025334411yescharacteristic
652C2#2025337611yescharacteristic
653C2#2025358511yescharacteristic
654C2#2025366911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21584
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. #20245742C2semidirect
4. #20244018C2semidirect
5. #20244010C2semidirect
6. #20243972C2semidirect
7. #20243964C2semidirect
9. #20243590C2semidirect
10. #20243473C2semidirect
11. #20243285C2semidirect
12. C4×#13093955C2semidirect
13. C4×#13093945C2semidirect
14. #20243157C2semidirect
15. C47×#1762C2semidirect
20. #15689440C4semidirect
80. C4⋊C4×D47C2×C4semidirect
95. #13093956C2×C4semidirect
97. #13093955C2×C4semidirect
101. (C2×D47)⋊C8C2×C4semidirect
331. #1762D47semidirect
387. C47#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#156894402
#202460752
#202457422
#202440182
#202440102
#202439722
#202439642
#202437072
#202435902
#202434732
#202432852
C4×#130939552
C4×#130939452
#202431572
C47×#17622
#2158447