Finite Group Databasebeta

← #26214793

Subgroups — #26214793

Order 12032.

1130 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689223C211yesmaximal, characteristic
2C2×#15689220C211yesmaximal, characteristic
3C2×#15689214C211yesmaximal, characteristic
4#20245752C211yesmaximal, characteristic, complemented
5#20245749C211yesmaximal, characteristic
6#20243286C211yesmaximal, characteristic, complemented
7#20243284C211yesmaximal, characteristic, complemented
8#20243280C211yesmaximal, characteristic, complemented
9#20243217C211yesmaximal, characteristic, complemented
10#20243215C211yesmaximal, characteristic
11C4×#13093949C211yesmaximal, characteristic, complemented
12#20243207C211yesmaximal, characteristic, complemented
13#102×Dic47C211yesmaximal, characteristic
14#20243074C211yesmaximal, characteristic, complemented
15C4×#102×C47C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×#13093989C2211yescharacteristic
19C2×#13093989C2211yescharacteristic
20C2×D4×Dic47C2211yescharacteristic
21C2×D4×Dic47C2211yescharacteristic
22#15689223C418yescomplemented
23#15689220C418yescomplemented
24#15689214C418yescomplemented
25C2×#13093957C2211yescharacteristic
26C2×#13093957C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×#13093949C2211yescharacteristic
29C2×#13093949C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33C2×#13093934C2211yescharacteristic
34C2×#13093934C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36C2×C8×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×Dic47⋊C4—21no—
42C4×Dic47⋊C4—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44#15688974—21no—
45C22⋊C4×Dic47—21no—
46C22⋊C4×Dic47—21no—
47C4×C22⋊Dic47—21no—
48#15688969C2211yescharacteristic
49#15688968—21no—
50C2×C47⋊#95—21no—
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4×D4×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C2×#102×C47C2211yescharacteristic
58C2×C4×C8×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C23×Dic47—21no—
64C23×Dic47—21no—
65C2×#10625335C2×C414yes—
66C2×#10625335C2×C414yes—
67C2×#10625335C2×C414yes—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C4×Dic94—22no—
71C4×Dic94C2×C414yescomplemented
72#13093989—22no—
73#13093989—22no—
74#13093989C2×C414yescomplemented
75#13093989C2×C414yescomplemented
76D4×Dic47—22no—
77D4×Dic47—22no—
78D4×Dic47C2×C414yes—
79D4×Dic47C2×C414yes—
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×C22⋊Dic47—21no—
87C2×C22⋊Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98#13093957C2×C414yescomplemented
99#13093957C2×C414yescomplemented
100#13093955C2×C414yescomplemented
101#13093949C2×C414yescomplemented
102#13093949C2×C414yescomplemented
103#13093947C2×C414yescomplemented
104#13093945C2×C414yes—
105#13093935C2×C414yescomplemented
106#13093934—22no—
107#13093934C2×C414yescomplemented
108C8×Dic47—22no—
109C8×Dic47C2×C414yescomplemented
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95D411yescharacteristic
116C47⋊#95D411yescharacteristic
117C47⋊#95D411yescharacteristic
118C47⋊#95D411yescharacteristic
119C22×D4×C47C2311yescharacteristic
120C23×C4×C47—21no—
121C22×C8×C47C2311yescharacteristic
122C22×C8×C47C2311yescharacteristic
123C4×D4×C47—22no—
124C4×D4×C47C2×C414yes—
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C22⋊C4×C47—21no—
129C2×C42×C47—21no—
130C2×C42×C47C2311yescharacteristic
131#102×C47C2×C414yes—
132#102×C47C2×C414yes—
133C4×C8×C47—22no—
134#95×C47—21no—
135C24×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47C22×C412yes—
140C2×D4×C47C22×C412yes—
141C2×D4×C47C22×C412yes—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—24no—
146C22×C4×C47—81no—
147C22×C4×C47—81no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—81no—
158C22×Dic47—81no—
159C22×Dic47—81no—
160C22×Dic47—81no—
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D411yescharacteristic
164C22×Dic47C2×D411yescharacteristic
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94C22×C412yes—
169C2×Dic94C22×C412yes—
170C2×Dic94C22×C412yes—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47C22×C412yes—
176C2×C8×C47C22×C412yes—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47C22×C412yes—
181C4⋊C4×C47C22×C412yes—
182C4⋊C4×C47C22×C412yes—
183C22⋊C4×C47—24no—
184C42×C47—81no—
185C42×C47C22×C412yes—
186C42×C47#4112yes—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189#10625335C42116yescomplemented
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47C22×C412yes—
194C4⋊Dic47C22×C412yes—
195C4⋊Dic47C22×C412yes—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199C4×Dic47—24no—
200C4×Dic47—24no—
201C4×Dic47—24no—
202C4×Dic47—81no—
203C4×Dic47—81no—
204C4×Dic47—81no—
205C4×Dic47—81no—
206C4×Dic47—81no—
207C4×Dic47C22×C412yes—
208C4×Dic47C22×C412yes—
209C4×Dic47#4112yes—
210C4×Dic47#4112yes—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8C22×C412yes—
214C2×C47⋊C8C22×C412yes—
215C2×C47⋊C8C22×C412yes—
216C2×C47⋊C8C22×C412yes—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47C22×D411yescharacteristic
225D4×C47—22no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47C2×C4214yes—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—84no—
243C2×C4×C47C4×D412yes—
244C2×C4×C47C4×D412yes—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C23×C411yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—84no—
272C2×Dic47—84no—
273C2×Dic47C4×D412yes—
274C2×Dic47C4×D412yes—
275C2×Dic47C4×D412yes—
276C2×Dic47C4×D412yes—
277Dic94—22no—
278Dic94—22no—
279Dic94—22no—
280Dic94C2×C4214yes—
281C8×C47—22no—
282C8×C47—22no—
283C47⋊C8—22no—
284C47⋊C8C2×C4214yes—
285#20504—471nomaximal, Sylow
286C22×C47—24no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47C2×C4×D411yescharacteristic
294C22×C47C2×C4×D411yescharacteristic
295C22×C47C2×C4×D411yescharacteristic
296C22×C47#57611yescharacteristic
297C22×C47#57711yescharacteristic
298C22×C47#57711yescharacteristic
299C22×C47#57811yescharacteristic
300C4×C47—22no—
301C4×C47—22no—
302C4×C47—42no—
303C4×C47—42no—
304C4×C47—42no—
305C4×C47C22×C4211yescharacteristic
306C4×C47C4×#4111yescharacteristic
307C4×C47C4×#4111yescharacteristic
308C4×C47C4×#4111yescharacteristic
309Dic47—22no—
310Dic47—22no—
311Dic47—22no—
312Dic47—22no—
313Dic47—24no—
314Dic47—42no—
315Dic47—42no—
316Dic47—42no—
317C2×#440—471no—
318C2×#440—471no—
319C2×C4×SD16—471no—
320#2276—471no—
321#2275—471no—
322#1776—471no—
323#1776—471no—
324C4×#107—471no—
325#1763—471no—
326#1762—471no—
327C4×#103—471no—
328C4×#103—471no—
329C4×#102—471no—
330C4×#102D4711yescharacteristic
331#1726—471no—
332C2×C47—44no—
333C2×C47C42×D411yescharacteristic
334C2×C47#294611yescharacteristic
335C2×C47#294611yescharacteristic
336C2×C47#294711yescharacteristic
337C2×C47#294711yescharacteristic
338C2×C47#294911yescharacteristic
339C2×C47#295111yescharacteristic
340C22×SD16—471no—
341C2×C4×Q8—471no—
342C2×C4×Q8—471no—
343C2×C4×Q8—471no—
344C2×C4×D4—471no—
345C2×C4×D4—471no—
346C2×C4×D4C2×D4711yescharacteristic
347#440—478no—
348#440—478no—
349C4×SD16—478no—
350C2×#107—471no—
351C2×#107—471no—
352C2×#106—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×#103—471no—
356C2×#102—471no—
357C2×#102C2×D4711yescharacteristic
358C2×#102C2×D4711yescharacteristic
359C2×#97—471no—
360C2×#97—471no—
361C2×#97—471no—
362C2×C4×C8—471no—
363C2×C4×C8—471no—
364C2×C4×C8C2×D4711yescharacteristic
365C4×C4⋊C4—471no—
366C4×C4⋊C4—471no—
367C4×C4⋊C4—471no—
368C4×C4⋊C4—471no—
369C4×C4⋊C4—471no—
370C4×C4⋊C4—941no—
371C4×C4⋊C4—941no—
372C4×C4⋊C4—941no—
373C4×C4⋊C4C2×D4711yescharacteristic
374C4×C22⋊C4—941no—
375C4×C22⋊C4—941no—
376C4×C22⋊C4—941no—
377#376—471no—
378#376—941no—
379C2×#95—941no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340C2×D4711yescharacteristic
385#340C2×D4711yescharacteristic
386C47#2050411yescharacteristic, complemented, Sylow
387C22×Q8—471no—
388C22×D4C22×D4711yescharacteristic
389C23×C4—21no—
390C23×C4—941no—
391C23×C4—941no—
392C2×SD16—474no—
393C2×SD16—474no—
394C2×SD16—474no—
395C22×C8—471no—
396C22×C8—471no—
397C22×C8C22×D4711yescharacteristic
398C22×C8C22×D4711yescharacteristic
399C4×Q8—474no—
400C4×Q8—474no—
401C4×Q8—474no—
402C4×Q8—942no—
403C4×Q8—942no—
404C4×Q8—942no—
405C4×D4—22no—
406C4×D4—474no—
407C4×D4—474no—
408C4×D4—942no—
409C4×D4—942no—
410C4×D4C4×D4714yes—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C42—21no—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—941no—
427C2×C42—941no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42—941no—
433C2×C42—941no—
434C2×C42C22×D4711yescharacteristic
435#107—474no—
436#107—474no—
437#106—474no—
438#103—474no—
439#103—474no—
440#103—474no—
441#102—474no—
442#102C4×D4714yes—
443#102C4×D4714yes—
444#97—474no—
445#97—474no—
446#97—942no—
447C4×C8—22no—
448C4×C8—474no—
449C4×C8—942no—
450#95—21no—
451#95—471no—
452#95—471no—
453#95—471no—
454#95—471no—
455#95—941no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460C24—21no—
461C2×Q8—472no—
462C2×Q8—472no—
463C2×Q8—472no—
464C2×Q8—941no—
465C2×Q8—941no—
466C2×Q8—941no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C4×D4712yes—
471C2×D4C2×C4×D4712yes—
472C2×D4C2×C4×D4712yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—944no—
484C22×C4—944no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—3761no—
490C22×C4—3761no—
491C22×C4—3761no—
492C22×C4—3761no—
493C22×C4D4×D4711yescharacteristic
494C22×C4D4×D4711yescharacteristic
495C22×C4C23×D4711yescharacteristic
496SD16—4716no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—472no—
504C2×C8—472no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8C2×C4×D4712yes—
508C2×C8C2×C4×D4712yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—472no—
513C4⋊C4—472no—
514C4⋊C4—472no—
515C4⋊C4—944no—
516C4⋊C4—944no—
517C4⋊C4—944no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—941no—
521C4⋊C4C2×C4×D4712yes—
522C4⋊C4C2×C4×D4712yes—
523C4⋊C4C2×C4×D4712yes—
524C22⋊C4—24no—
525C22⋊C4—944no—
526C22⋊C4—944no—
527C42—81no—
528C42—472no—
529C42—472no—
530C42—472no—
531C42—472no—
532C42—944no—
533C42—944no—
534C42—944no—
535C42—3761no—
536C42—3761no—
537C42—3761no—
538C42—3761no—
539C42—3761no—
540C42C2×C4×D4712yes—
541C42#1062535212yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D4711yescharacteristic
550Q8—474no—
551Q8—942no—
552Q8—942no—
553Q8—942no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C42×D4714yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—472no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—472no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—3764no—
591C2×C4—3764no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4(C22⋊D47)⋊C412yes—
599C2×C4(C22⋊D47)⋊C412yes—
600C2×C4C22×C4×D4711yescharacteristic
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4C22×C4×D4711yescharacteristic
603C2×C4#41×D4711yescharacteristic
604C2×C4#41×D4711yescharacteristic
605C2×C4C2×#1062535211yescharacteristic
606C8—22no—
607C8—22no—
608C8—474no—
609C8—942no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D4711yescharacteristic
618C22C4×D4×D4711yescharacteristic
619C22C2×(C22⋊D47)⋊C411yescharacteristic
620C22#1568992411yescharacteristic
621C22#1568992911yescharacteristic
622C22#1568993011yescharacteristic
623C22#1568993711yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—942no—
630C4—942no—
631C4—942no—
632C4—942no—
633C4—944no—
634C4—1882no—
635C4—1882no—
636C4—1882no—
637C4C2×C42×D4711yescharacteristic
638C4C4×#1062535211yescharacteristic
639C4C4×#1062535211yescharacteristic
640C4#1568959111yescharacteristic
641C2—44no—
642C2C4×(C22⋊D47)⋊C411yescharacteristic
643C2#2025221311yescharacteristic
644C2#2025221411yescharacteristic
645C2#2025222411yescharacteristic
646C2#2025223411yescharacteristic
647C2#2025224211yescharacteristic
648C2#2025225811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47#20504
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
4. #20245752C2semidirect
6. #20243286C2semidirect
7. #20243284C2semidirect
8. #20243280C2semidirect
9. #20243217C2semidirect
11. C4×#13093949C2semidirect
12. #20243207C2semidirect
14. #20243074C2semidirect
22. #15689223C4semidirect
23. #15689220C4semidirect
24. #15689214C4semidirect
71. C4×Dic94C2×C4semidirect
74. #13093989C2×C4semidirect
75. #13093989C2×C4semidirect
98. #13093957C2×C4semidirect
99. #13093957C2×C4semidirect
100. #13093955C2×C4semidirect
101. #13093949C2×C4semidirect
102. #13093949C2×C4semidirect
103. #13093947C2×C4semidirect
105. #13093935C2×C4semidirect
107. #13093934C2×C4semidirect
109. C8×Dic47C2×C4semidirect
189. #10625335C42semidirect
386. C47#20504semidirect

Maximal subgroups

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

subgroupindex
C2×#156892232
C2×#156892202
C2×#156892142
#202457522
#202457492
#202432862
#202432842
#202432802
#202432172
#202432152
C4×#130939492
#202432072
#102×Dic472
#202430742
C4×#102×C472
#2050447