Finite Group Databasebeta

← #26224840

Subgroups — #26224840

Order 12032.

1262 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689509C211yesmaximal, characteristic
2C2×#15689236C211yesmaximal, characteristic, complemented
3C2×#15689233C211yesmaximal, characteristic, complemented
4C2×#15689221C211yesmaximal, characteristic, complemented
5C4×#13094020C211yesmaximal, characteristic, complemented
6C4×#13094013C211yesmaximal, characteristic, complemented
7#20244252C211yesmaximal, characteristic, complemented
8#20244251C211yesmaximal, characteristic, complemented
9#20244244C211yesmaximal, characteristic, complemented
10#20244239C211yesmaximal, characteristic, complemented
11#20244210C211yesmaximal, characteristic, complemented
12#20244201C211yesmaximal, characteristic, complemented
13#20243555C211yesmaximal, characteristic, complemented
14#97×Dic47C211yesmaximal, characteristic, complemented
15C2×C47×#440C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C22×D376C2211yescharacteristic
18C22×#10625334C2211yescharacteristic
19C2×#13094020C2211yescharacteristic
20C2×#13094013C2211yescharacteristic
21C2×C4×D188C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4×C22⋊D47—22no—
24#15689509C418yescomplemented
25#15689236—24no—
26#15689233—24no—
27#15689221—24no—
28C2×#13093957C2211yescharacteristic
29C2×#13093956C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31C2×#13093948C2211yescharacteristic
32C2×#13093946C2211yescharacteristic
33C2×#13093946C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093944C2211yescharacteristic
37C2×#13093935C2211yescharacteristic
38C2×#13093934C2211yescharacteristic
39C2×#13093934C2211yescharacteristic
40C2×C8×Dic47C2211yescharacteristic
41C2×C8×Dic47C2211yescharacteristic
42#15689030C2211yescharacteristic
43#15689026C2211yescharacteristic
44#15689024C2211yescharacteristic
45#15689024C2211yescharacteristic
46#15689023—22no—
47#15689013—22no—
48#15688992C2211yescharacteristic
49#15688988C2211yescharacteristic
50C4×(C2×D47)⋊C4—22no—
51C42×Dic47C2211yescharacteristic
52C22×SD16×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C47×#440—24no—
56C2×#107×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C22⋊D47—22no—
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C2×#10625336—22no—
66C2×#10625336—22no—
67C2×#10625336C2×C414yescomplemented
68C2×D376—22no—
69C2×D376—22no—
70C2×D376C2×C414yescomplemented
71C2×#10625334—22no—
72C2×#10625334—22no—
73C2×#10625334C2×C414yescomplemented
74C22×C47⋊C8C2311yescharacteristic
75C22×C47⋊C8C2311yescharacteristic
76#13094020—22no—
77#13094020C2×C414yescomplemented
78#13094013—22no—
79#13094013C2×C414yescomplemented
80C4×D188—22no—
81C4×D188—22no—
82C4×D188—22no—
83C4×D188—22no—
84C4×C22⋊D47—48no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×Dic47⋊C4—22no—
88C2×C22⋊Dic47—22no—
89C2×(C2×D47)⋊C4—22no—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—22no—
93C2×(C2×D47)⋊C4—22no—
94C2×C4×Dic47D412yes—
95C2×C4×Dic47D412yes—
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98#13093957—22no—
99#13093956—22no—
100#13093955—22no—
101#13093948—22no—
102#13093946—22no—
103#13093946—22no—
104#13093945—22no—
105#13093944—22no—
106#13093944—22no—
107#13093935D414yescomplemented
108#13093934—22no—
109#13093934C2×C414yescomplemented
110C8×Dic47—22no—
111C8×Dic47—22no—
112C47⋊#95—22no—
113C47⋊#95—22no—
114C22×Q8×C47C2311yescharacteristic
115C22×D4×C47C2311yescharacteristic
116C23×C4×C47—21no—
117C2×SD16×C47—22no—
118C2×SD16×C47—22no—
119C2×SD16×C47C2×C414yescomplemented
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C4×Q8×C47—22no—
123C4×Q8×C47—22no—
124C4×D4×C47—22no—
125C4×D4×C47—22no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C22⋊C4×C47—21no—
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132#107×C47—22no—
133#103×C47—22no—
134#102×C47—22no—
135#97×C47D414yescomplemented
136C24×C47—21no—
137C23×D47—21no—
138C23×D47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47C22×C412yes—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47—21no—
150C2×D4×C47C22×C412yes—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—44no—
155C22×C4×C47—42no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2411yescharacteristic
159C2×C22⋊D47—28no—
160C2×C22⋊D47—44no—
161C2×C22⋊D47—44no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D412yes—
166C2×D188—21no—
167C2×D188—21no—
168C2×D188—21no—
169C2×D188—21no—
170C2×D188—21no—
171C2×D188—21no—
172C2×D188—21no—
173C2×D188—21no—
174C2×D188—21no—
175C2×D188—21no—
176C2×D188C22×C412yes—
177C2×D188C22×C412yes—
178C2×C4×D47—44no—
179C2×C4×D47—44no—
180C2×C4×D47—42no—
181C2×C4×D47—42no—
182SD16×C47—28no—
183C2×C8×C47—21no—
184C2×C8×C47—21no—
185C2×C8×C47C22×C412yes—
186C2×C8×C47C2×D412yes—
187C2×C8×C47C2×D412yes—
188C2×C8×C47#4112yes—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47—21no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47—42no—
194C22⋊C4×C47—42no—
195C42×C47—42no—
196C42×C47C2×D412yes—
197C42×C47C2×D412yes—
198C22⋊Dic47—44no—
199#10625336—28no—
200#10625334—28no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—42no—
203(C2×D47)⋊C4—42no—
204(C2×D47)⋊C4—44no—
205(C2×D47)⋊C4—44no—
206C4⋊Dic47—21no—
207C4⋊Dic47—21no—
208C4⋊Dic47—21no—
209C4⋊Dic47—21no—
210Dic47⋊C4—44no—
211C4×Dic47—21no—
212C4×Dic47—21no—
213C4×Dic47—24no—
214C4×Dic47—42no—
215C4×Dic47C22×C412yes—
216C4×Dic47#4112yes—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8C22×C412yes—
220C2×C47⋊C8C2×D412yes—
221C2×C47⋊C8C2×D412yes—
222C2×C47⋊C8#4112yes—
223D376—28no—
224C23×C47—22no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—42no—
228C23×C47—42no—
229C23×C47C22×D411yescharacteristic
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—41no—
233C22×D47—41no—
234C22×D47—41no—
235C22×D47—41no—
236C22×D47—42no—
237C22×D47—42no—
238C22×D47—42no—
239C22×D47—42no—
240Q8×C47—22no—
241Q8×C47—22no—
242Q8×C47—41no—
243Q8×C47—41no—
244D4×C47—22no—
245D4×C47—22no—
246D4×C47—41no—
247D4×C47—41no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47—84no—
258C2×C4×C47#12712yes—
259C2×C4×C47#12712yes—
260C2×C4×C47C23×C411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C22⋊D47—416no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47C4×D414yes—
276C2×Dic47AGL(1,ℤ/8)14yes—
277D188—22no—
278D188—22no—
279D188—22no—
280D188—22no—
281D188—41no—
282D188—41no—
283D188—41no—
284D188—41no—
285C4×D47—84no—
286C4×D47—84no—
287C8×C47—22no—
288C8×C47C4×D414yescomplemented
289C47⋊C8—22no—
290C47⋊C8C4×D414yescomplemented
291#21701—471nomaximal, Sylow
292C22×C47—42no—
293C22×C47—44no—
294C22×C47—81no—
295C22×C47—81no—
296C22×C47—82no—
297C22×C47C2×C4×D411yescharacteristic
298C22×C47C2×#12711yescharacteristic
299C22×C47#53211yescharacteristic
300C22×C47C2×AGL(1,ℤ/8)12yes—
301C22×C47#57512yes—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—44no—
305C2×D47—44no—
306C2×D47—81no—
307C2×D47—81no—
308C2×D47—81no—
309C2×D47—81no—
310C2×D47—82no—
311C2×D47—82no—
312C4×C47—22no—
313C4×C47—22no—
314C4×C47—42no—
315C4×C47—81no—
316C4×C47—81no—
317C4×C47C2×C4×D411yescharacteristic
318C4×C47C2×C4×D411yescharacteristic
319C4×C47#53211yescharacteristic
320C4×C47#53211yescharacteristic
321Dic47—24no—
322Dic47—24no—
323Dic47—81no—
324Dic47—81no—
325C2×#496—471no—
326C2×#442—471no—
327C2×#442—471no—
328C2×#440—471no—
329C2×#440D4711yescharacteristic, complemented
330C4×#127—471no—
331C4×#124—471no—
332#1970—471no—
333#1970—471no—
334#1968—471no—
335#1968—471no—
336#1961—471no—
337#1959—471no—
338#1847—471no—
339C4×#97—471no—
340C2×C47—82no—
341C2×C47C4×#12711yescharacteristic
342C2×C47C4×AGL(1,ℤ/8)12yes—
343C2×C47#315111yescharacteristic
344C2×C47#315311yescharacteristic
345C2×C47#318012yes—
346D47—82no—
347D47—82no—
348C22×SD16—471no—
349C22×SD16C2×D4711yescharacteristic
350C22×D8—471no—
351C22×D8—471no—
352C2×#127—471no—
353C2×#124—471no—
354C2×C4×Q8C2×D4711yescharacteristic
355C2×C4×D4—471no—
356C2×C4×D4—471no—
357C2×C4×D4—942no—
358C2×C4×D4C2×D4711yescharacteristic
359#496—478no—
360#442—944no—
361#442—944no—
362#440—24no—
363#440—944no—
364C2×#107—471no—
365C2×#107C2×D4711yescharacteristic
366C2×#106—471no—
367C2×#106—471no—
368C2×#103—471no—
369C2×#103C2×D4711yescharacteristic
370C2×#102—471no—
371C2×#102—471no—
372C2×#102—471no—
373C2×#102—471no—
374C2×#102—471no—
375C2×#102C2×D4711yescharacteristic
376C2×#97—471no—
377C2×#97—471no—
378C2×#97—471no—
379C2×#97C2×D4711yescharacteristic
380C2×C4×C8—471no—
381C2×C4×C8—471no—
382#391—471no—
383#390—471no—
384#390—471no—
385#390—471no—
386#390—942no—
387#388—942no—
388#384—471no—
389#382—471no—
390C4×C22⋊C4—942no—
391C43—471no—
392C47#2170111yescharacteristic, complemented, Sylow
393C22×Q8C22×D4711yescharacteristic
394C22×D4—471no—
395C22×D4—471no—
396C22×D4—942no—
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C23×C4—941no—
400C23×C4—941no—
401C2×SD16—22no—
402C2×SD16—22no—
403C2×SD16—474no—
404C2×SD16—942no—
405C2×SD16—942no—
406C2×SD16C4×D4714yescomplemented
407C2×D8—474no—
408C2×D8—474no—
409C2×D8—942no—
410C2×D8—942no—
411C2×D8—942no—
412C2×D8—942no—
413C22×C8—471no—
414C22×C8—471no—
415C22×C8C22×D4711yescharacteristic
416C22×C8C22×D4711yescharacteristic
417#127—474no—
418#127—942no—
419#124—474no—
420#124—942no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—942no—
426C4×D4—942no—
427C4×D4—942no—
428C4×D4—942no—
429C4×D4—1888no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—942no—
434C2×C4⋊C4C22×D4711yescharacteristic
435C2×C4⋊C4C22×D4711yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—942no—
438C2×C22⋊C4—942no—
439C2×C22⋊C4—942no—
440C2×C22⋊C4—942no—
441C2×C22⋊C4—941no—
442C2×C22⋊C4—941no—
443C2×C42—21no—
444C2×C42—471no—
445C2×C42—472no—
446C2×C42—471no—
447C2×C42—472no—
448C2×C42C22×D4711yescharacteristic
449#107—22no—
450#107—942no—
451#106—942no—
452#106—942no—
453#103—22no—
454#103—942no—
455#102—22no—
456#102—942no—
457#102—942no—
458#102—942no—
459#102—942no—
460#102—942no—
461#97—474no—
462#97—474no—
463#97—942no—
464#97C22⋊D4714yescomplemented
465C4×C8—942no—
466C4×C8—942no—
467#95—942no—
468#95—942no—
469C24—21no—
470C24—941no—
471C24—941no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8C2×C4×D4712yes—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—472no—
484C2×D4—472no—
485C2×D4—948no—
486C2×D4—941no—
487C2×D4—941no—
488C2×D4—941no—
489C2×D4—941no—
490C2×D4—941no—
491C2×D4—941no—
492C2×D4—941no—
493C2×D4—941no—
494C2×D4—941no—
495C2×D4—941no—
496C2×D4—1884no—
497C2×D4—1884no—
498C2×D4C2×C4×D4712yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—472no—
505C22×C4—472no—
506C22×C4—941no—
507C22×C4—941no—
508C22×C4—1884no—
509C22×C4—1884no—
510C22×C4—1882no—
511C22×C4—1882no—
512C22×C4D4×D4711yescharacteristic
513C22×C4D4×D4711yescharacteristic
514C22×C4C23×D4711yescharacteristic
515SD16—28no—
516SD16—948no—
517D8—948no—
518D8—948no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—472no—
522C2×C8—472no—
523C2×C8—472no—
524C2×C8—472no—
525C2×C8—941no—
526C2×C8—941no—
527C2×C8C2×C4×D4712yes—
528C2×C8#1062535012yes—
529C2×C8C2×C22⋊D4712yes—
530C2×C8C2×C22⋊D4712yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—941no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C4⋊C4—1884no—
541C22⋊C4—42no—
542C22⋊C4—948no—
543C22⋊C4—1884no—
544C22⋊C4—1884no—
545C22⋊C4—1884no—
546C22⋊C4—1882no—
547C22⋊C4—1882no—
548C42—42no—
549C42—472no—
550C42—472no—
551C42—944no—
552C42—941no—
553C42—941no—
554C42—1882no—
555C42D4×D4712yes—
556C42C2×C22⋊D4712yes—
557C23—22no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—942no—
563C23—942no—
564C23—1882no—
565C23—1882no—
566C23—1881no—
567C23—1881no—
568C23—1882no—
569C23—1882no—
570C23—1881no—
571C23—1881no—
572C23C2×D4×D4711yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577D4—22no—
578D4—22no—
579D4—41no—
580D4—41no—
581D4—942no—
582D4—942no—
583D4—942no—
584D4—942no—
585D4—18816no—
586D4—1881no—
587D4—1881no—
588D4—1881no—
589D4—1881no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—474no—
601C2×C4—474no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—942no—
606C2×C4—3764no—
607C2×C4—3764no—
608C2×C4—1882no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1882no—
612C2×C4#1309402012yes—
613C2×C4#1309402012yes—
614C2×C4C22×C4×D4711yescharacteristic
615C2×C4C2×D4×D4711yescharacteristic
616C2×C4C22×C22⋊D4711yescharacteristic
617C2×C4#41×D4711yescharacteristic
618C2×C4#41×D4711yescharacteristic
619C2×C4C2×#1062535011yescharacteristic
620C8—22no—
621C8—474no—
622C8—942no—
623C8C4×C22⋊D4714yescomplemented
624C22—44no—
625C22—42no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—3762no—
630C22—3762no—
631C22—3761no—
632C22—3761no—
633C22—3761no—
634C22—3761no—
635C22—1884no—
636C22—1884no—
637C22—1882no—
638C22—1882no—
639C22C4×D4×D4711yescharacteristic
640C22C2×#1309402011yescharacteristic
641C22#1568967011yescharacteristic
642C22AGL(1,ℤ/8)×D4712yes—
643C22#1568992212yes—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—944no—
650C4—944no—
651C4—3761no—
652C4—3761no—
653C4C4×D4×D4711yescharacteristic
654C4C2×C4×C22⋊D4711yescharacteristic
655C4#1568966911yescharacteristic
656C4#1568967011yescharacteristic
657C2—82no—
658C2—3762no—
659C2—3762no—
660C2C4×#1309402011yescharacteristic
661C2#2025219312yes—
662C2#2025383611yescharacteristic
663C2#2025385511yescharacteristic
664C2#2025408212yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#440C2
socleC23×C47C22×D4
prime radicalC2×#15689509C2
cosocleC22×C4×C47C24
hypercenterC2×#440D47
hypocenterC47#21701
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×#15689236C2semidirect
3. C2×#15689233C2semidirect
4. C2×#15689221C2semidirect
5. C4×#13094020C2semidirect
6. C4×#13094013C2semidirect
7. #20244252C2semidirect
8. #20244251C2semidirect
9. #20244244C2semidirect
10. #20244239C2semidirect
11. #20244210C2semidirect
12. #20244201C2semidirect
13. #20243555C2semidirect
14. #97×Dic47C2semidirect
15. C2×C47×#440C2semidirect
24. #15689509C4semidirect
67. C2×#10625336C2×C4semidirect
70. C2×D376C2×C4semidirect
73. C2×#10625334C2×C4semidirect
77. #13094020C2×C4semidirect
79. #13094013C2×C4semidirect
107. #13093935D4semidirect
109. #13093934C2×C4semidirect
119. C2×SD16×C47C2×C4semidirect
135. #97×C47D4semidirect
288. C8×C47C4×D4semidirect
290. C47⋊C8C4×D4semidirect
329. C2×#440D47semidirect
392. C47#21701semidirect
406. C2×SD16C4×D47semidirect
464. #97C22⋊D47semidirect
623. C8C4×C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895092
C2×#156892362
C2×#156892332
C2×#156892212
C4×#130940202
C4×#130940132
#202442522
#202442512
#202442442
#202442392
#202442102
#202442012
#202435552
#97×Dic472
C2×C47×#4402
#2170147