Finite Group Databasebeta

← #26183483

Subgroups — #26183483

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257437C211yesmaximal, characteristic
2#20252423C211yesmaximal, characteristic, complemented
3#20241911C211yesmaximal, characteristic, complemented
4#20241907C212yesmaximal, complemented
5#20241899C211yesmaximal, characteristic, complemented
6#20241890C212yesmaximal, complemented
7#20241886C211yesmaximal, characteristic, complemented
8#20241877C212yesmaximal, complemented
9#20241872C211yesmaximal, characteristic, complemented
10C2×#15688859C212yesmaximal, complemented
11C47×#2275C211yesmaximal, characteristic, complemented
12#15689843C411yescharacteristic, complemented
13#15689828—21no—
14#15689813C412yescomplemented
15#15689799—22no—
16D4×Dic94C411yescharacteristic, complemented
17#15689782C412yescomplemented
18#15689778C411yescharacteristic, complemented
19#15689777C411yescharacteristic, complemented
20#15689767—21no—
21#15689738—21no—
22#15689715—22no—
23#15689680C2211yescharacteristic
24#15689657C2211yescharacteristic
25#15689634C2212yes—
26C2×#13094003C2212yes—
27#15689602—21no—
28#15689201—21no—
29#15689195—22no—
30D4×C47⋊C8—21no—
31#15689180C2211yescharacteristic
32#15689176C2211yescharacteristic
33#15689087C2212yes—
34C2×C22⋊(C47⋊C8)C2212yes—
35#15688859C2211yescharacteristic
36#15688859C2212yes—
37#15688859C2212yes—
38#15688859C2212yes—
39#15688859C2211yescharacteristic
40#15688859C2212yes—
41#15688859C2212yes—
42#15688856C2211yescharacteristic
43#15688856C2211yescharacteristic
44#15688856C2212yes—
45C2×C4×D4×C47—21no—
46C2×C4×D4×C47C2211yescharacteristic
47C4×C4⋊C4×C47—21no—
48C4×C4⋊C4×C47C2211yescharacteristic
49C4×C22⋊C4×C47—22no—
50C4×C22⋊C4×C47C2212yes—
51C47×#376C2211yescharacteristic
52C2×#95×C47C2212yes—
53C2×#10625350C2×C412yescomplemented
54C2×#10625352—21no—
55C22×Dic94C2×C412yescomplemented
56C2×#10625329—22no—
57C22×C47⋊C8—22no—
58#13094024C2×C411yescharacteristic, complemented
59#13094024C2×C411yescharacteristic, complemented
60#13094023—21no—
61#13094018—21no—
62#13094017—22no—
63#13094017—22no—
64#13094016—21no—
65#13094016C2×C411yescharacteristic, complemented
66#13094014—22no—
67#13094012C2×C411yescharacteristic, complemented
68#13094011—21no—
69#13094010—22no—
70#13094008—22no—
71#13094007—22no—
72#13094007—21no—
73#13094006—22no—
74#13094005—22no—
75#13094004—21no—
76#13094003—22no—
77#13094003—21no—
78#13094003—22no—
79#13094003C2×C412yescomplemented
80#13094003C2×C412yescomplemented
81#13094003C2×C412yescomplemented
82#13094003C2311yescharacteristic
83#13094003C2312yes—
84#13094002—22no—
85#13094001—22no—
86#13093998C2×C412yescomplemented
87#13093998C2311yescharacteristic
88#13093996—22no—
89#13093995—21no—
90#13093993—22no—
91C4×Dic94—21no—
92C4×Dic94C2×C412yescomplemented
93C4×Dic94C2×C411yescharacteristic, complemented
94#13093989—21no—
95C4×D188C2×C411yescharacteristic, complemented
96D4×Dic47—21no—
97C4×C22⋊D47—21no—
98C4×C22⋊D47—22no—
99C4×C22⋊D47—22no—
100(C22⋊D47)⋊C4—22no—
101#13093983—21no—
102#13093983C2×C412yescomplemented
103#13093981—21no—
104#13093980—22no—
105C2×C4⋊Dic47C2×C412yescomplemented
106C2×Dic47⋊C4—22no—
107C4⋊C4×D47—21no—
108C2×C22⋊Dic47—22no—
109C4⋊(C47⋊C8)—21no—
110C4⋊(C47⋊C8)—21no—
111C22⋊(C47⋊C8)—22no—
112C22⋊(C47⋊C8)—22no—
113C22⋊(C47⋊C8)C2311yescharacteristic
114C22⋊(C47⋊C8)C2312yes—
115C22⋊(C47⋊C8)C2311yescharacteristic
116#13093935—21no—
117C4×C47⋊C8—21no—
118C22×D4×C47D411yescharacteristic
119C23×C4×C47—22no—
120C23×C4×C47C2312yes—
121C4×D4×C47—28no—
122C4×D4×C47C2×C411yescharacteristic
123C4×D4×C47C2×C411yescharacteristic
124C4×D4×C47C2×C412yes—
125C4×D4×C47D411yescharacteristic
126C4×D4×C47D411yescharacteristic
127C4×D4×C47D412yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47D411yescharacteristic
130C2×C22⋊C4×C47—22no—
131C2×C22⋊C4×C47D412yes—
132C2×C42×C47—21no—
133C2×C42×C47—21no—
134C2×C42×C47—22no—
135C2×C42×C47C2311yescharacteristic
136#95×C47—22no—
137#95×C47—21no—
138#95×C47C2311yescharacteristic
139#95×C47C2312yes—
140#95×C47C2311yescharacteristic
141C24×C47C2×D412yes—
142C2×D4×C47—24no—
143C2×D4×C47C22⋊C411yescharacteristic
144C2×D4×C47C22⋊C412yes—
145C2×D4×C47C22⋊C411yescharacteristic
146C22×C4×C47—21no—
147C22×C4×C47—22no—
148C22×C4×C47—21no—
149C22×C4×C47—28no—
150C22×C4×C47—82no—
151C22×C4×C47—82no—
152C22×C4×C47—82no—
153C22×C4×C47C22×C412yes—
154C22×C4×C47C22×C412yes—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2411yescharacteristic
161C2×C22⋊D47—22no—
162C2×C22⋊D47—21no—
163C2×C22⋊D47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166#10625352—44no—
167#10625350—82no—
168C2×D188C22×C411yescharacteristic
169C2×C4×D47—21no—
170C2×C4×D47—22no—
171C2×Dic94—21no—
172C2×Dic94—22no—
173C2×Dic94—82no—
174C2×Dic94C22×C411yescharacteristic
175C2×Dic94C22×C412yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47C22⋊C411yescharacteristic
178C4⋊C4×C47C22⋊C412yes—
179C4⋊C4×C47C22⋊C411yescharacteristic
180C22⋊C4×C47—28no—
181C22⋊C4×C47C22⋊C412yes—
182C22⋊C4×C47C22⋊C412yes—
183C22⋊C4×C47C22⋊C412yes—
184C22⋊C4×C47C22⋊C412yes—
185C42×C47—24no—
186C42×C47—81no—
187C42×C47—82no—
188C42×C47C22×C411yescharacteristic
189C42×C47C22×C411yescharacteristic
190C42×C47C2×D411yescharacteristic
191C42×C47C2×D411yescharacteristic
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194C22⋊Dic47—22no—
195C22⋊Dic47—22no—
196C22⋊Dic47—21no—
197C22⋊Dic47—22no—
198(C2×D47)⋊C4—21no—
199(C2×D47)⋊C4—21no—
200(C2×D47)⋊C4—22no—
201(C2×D47)⋊C4—21no—
202(C2×D47)⋊C4—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—22no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47C22×C411yescharacteristic
208C4⋊Dic47C22×C412yes—
209C4⋊Dic47C22×C411yescharacteristic
210Dic47⋊C4—22no—
211Dic47⋊C4—22no—
212Dic47⋊C4—21no—
213Dic47⋊C4—22no—
214Dic47⋊C4—21no—
215Dic47⋊C4—22no—
216Dic47⋊C4—21no—
217Dic47⋊C4—21no—
218Dic47⋊C4—22no—
219C4×Dic47—22no—
220C4×Dic47—22no—
221C4×Dic47—21no—
222#10625329—82no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—22no—
226C2×C47⋊C8—82no—
227C23×C47—22no—
228C23×C47—22no—
229C23×C47—24no—
230C23×C47—42no—
231C23×C47C2×C22⋊C412yes—
232C23×C47C2×C22⋊C412yes—
233C23×C47C22×D411yescharacteristic
234C22×D47—21no—
235D4×C47—24no—
236D4×C47—22no—
237D4×C47#10414yes—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—24no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—42no—
252C2×C4×C47—41no—
253C2×C4×C47—42no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—88no—
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C2×C22⋊C411yescharacteristic
260C2×C4×C47C2×C22⋊C411yescharacteristic
261C2×C4×C47C2×C22⋊C412yes—
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C23×C411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C22⋊D47—41no—
266C22⋊D47—41no—
267C22⋊D47—44no—
268C22⋊D47—82no—
269C2×Dic47—21no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—21no—
273C2×Dic47—21no—
274C2×Dic47—44no—
275C2×Dic47—82no—
276D188—41no—
277C4×D47—42no—
278C4×D47—82no—
279Dic94—42no—
280Dic94—41no—
281Dic94—42no—
282Dic94—82no—
283C47⋊C8—41no—
284C47⋊C8—42no—
285C47⋊C8—41no—
286#17548—471nomaximal, Sylow
287C22×C47—22no—
288C22×C47—21no—
289C22×C47—22no—
290C22×C47—21no—
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—44no—
295C22×C47—44no—
296C22×C47C2×#10414yes—
297C22×C47C22×C22⋊C411yescharacteristic
298C22×C47#57711yescharacteristic
299C22×C47#57811yescharacteristic
300C2×D47—41no—
301C2×D47—41no—
302C2×D47—41no—
303C4×C47—21no—
304C4×C47—21no—
305C4×C47—22no—
306C4×C47—22no—
307C4×C47—21no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—42no—
311C4×C47—44no—
312C4×C47—42no—
313C4×C47C2×#10412yes—
314Dic47—41no—
315Dic47—42no—
316Dic47—41no—
317Dic47—42no—
318Dic47—41no—
319#3520—471no—
320#2974—471no—
321#2275D4711yescharacteristic, complemented
322#1509—471no—
323#1508—472no—
324#1506—471no—
325#1504—472no—
326#1503—471no—
327#1501—472no—
328#1500—471no—
329C2×#328—472no—
330C2×C47—21no—
331C2×C47—21no—
332C2×C47—24no—
333C2×C47—42no—
334C2×C47#288311yescharacteristic
335C2×C47#289811yescharacteristic
336C2×C47C22×#10411yescharacteristic
337D47—81no—
338#556—471no—
339#554—941no—
340#552—472no—
341#551—942no—
342D4×Q8—471no—
343#548—472no—
344#547—471no—
345#547—471no—
346#546—941no—
347#542—941no—
348#539—942no—
349#533—471no—
350#529—471no—
351#525—472no—
352C2×C22⋊Q8—472no—
353#520—941no—
354C2×C4×D4—21no—
355C2×C4×D4C2×D4711yescharacteristic
356#436—941no—
357#435—942no—
358C8×D4—941no—
359#432—471no—
360#431—471no—
361#407—472no—
362C2×C22⋊C8—472no—
363C4×C4⋊C4—21no—
364C4×C4⋊C4C2×D4711yescharacteristic
365C4×C22⋊C4—22no—
366C4×C22⋊C4C2×D4712yes—
367#376C2×D4711yescharacteristic
368C2×#95C2×D4712yes—
369#328—471no—
370#328—472no—
371#328—472no—
372#328—472no—
373#328—471no—
374#328—472no—
375#328—472no—
376#327—471no—
377#327—471no—
378#327—472no—
379C47#1754811yescharacteristic, complemented, Sylow
380C2×#41—472no—
381C2×#41—941no—
382C22×Q8—472no—
383C22×D4C22⋊D4711yescharacteristic
384C23×C4—22no—
385C23×C4C22×D4712yes—
386C2×M16—942no—
387C22×C8—942no—
388C4⋊Q8—471no—
389C4⋊Q8—471no—
390C4⋊Q8—941no—
391#126—942no—
392#126—942no—
393#126—941no—
394#125—471no—
395#125—942no—
396#125—941no—
397#124—471no—
398#124—941no—
399#124—942no—
400#123—942no—
401#123—942no—
402#123—941no—
403#123—942no—
404#123—942no—
405C22⋊Q8—472no—
406C22⋊Q8—472no—
407C22⋊Q8—472no—
408C22⋊Q8—471no—
409C22⋊Q8—472no—
410C22⋊Q8—942no—
411C22⋊Q8—942no—
412C22⋊Q8—941no—
413C22⋊Q8—942no—
414C22⋊Q8—941no—
415C22⋊Q8—942no—
416#121—472no—
417#121—471no—
418#121—942no—
419#121—941no—
420#120—942no—
421C4×Q8—471no—
422C4×Q8—472no—
423C4×Q8—941no—
424C4×Q8—941no—
425C4×D4—28no—
426C4×D4—471no—
427C4×D4—941no—
428C4×D4—942no—
429C4×D4—942no—
430C4×D4—941no—
431C4×D4—942no—
432C4×D4C4×D4711yescharacteristic
433C4×D4C4×D4711yescharacteristic
434C4×D4C4×D4712yes—
435C4×D4D18811yescharacteristic
436C4×D4D18812yes—
437C4×D4D18811yescharacteristic
438#117—472no—
439#117—941no—
440#117—942no—
441#117—941no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—472no—
444C2×C4⋊C4—942no—
445C2×C4⋊C4—941no—
446C2×C4⋊C4C22⋊D4711yescharacteristic
447C2×C22⋊C4—22no—
448C2×C22⋊C4—942no—
449C2×C22⋊C4C22⋊D4712yes—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—22no—
453C2×C42C22×D4711yescharacteristic
454C4⋊C8—941no—
455C4⋊C8—941no—
456C22⋊C8—471no—
457C22⋊C8—472no—
458C22⋊C8—471no—
459C22⋊C8—942no—
460C22⋊C8—942no—
461#97—941no—
462C4×C8—941no—
463#95—22no—
464#95—21no—
465#95C22×D4711yescharacteristic
466#95C22×D4712yes—
467#95C22×D4711yescharacteristic
468C24C2×C22⋊D4712yes—
469#41—3762no—
470#41—1884no—
471C2×Q8—471no—
472C2×Q8—472no—
473C2×Q8—942no—
474C2×Q8—941no—
475C2×Q8—3762no—
476C2×D4—24no—
477C2×D4—471no—
478C2×D4—942no—
479C2×D4—942no—
480C2×D4—941no—
481C2×D4(C2×D47)⋊C412yes—
482C2×D4(C2×D47)⋊C411yescharacteristic
483C2×D4(C2×D47)⋊C411yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—941no—
495C22×C4C2×C4×D4712yes—
496C22×C4C2×C4×D4712yes—
497C22×C4C2×D18812yes—
498C22×C4C2×D18812yes—
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4712yes—
502C22×C4C23×D4711yescharacteristic
503M16—3762no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—942no—
507C2×C8—3762no—
508C4⋊C4—24no—
509C4⋊C4—471no—
510C4⋊C4—472no—
511C4⋊C4—471no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—941no—
515C4⋊C4—942no—
516C4⋊C4—941no—
517C4⋊C4—942no—
518C4⋊C4—941no—
519C4⋊C4—942no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—941no—
524C4⋊C4—942no—
525C4⋊C4(C2×D47)⋊C411yescharacteristic
526C4⋊C4(C2×D47)⋊C412yes—
527C4⋊C4(C2×D47)⋊C411yescharacteristic
528C22⋊C4—28no—
529C22⋊C4—942no—
530C22⋊C4—942no—
531C22⋊C4—942no—
532C22⋊C4—942no—
533C22⋊C4—941no—
534C22⋊C4—942no—
535C22⋊C4—941no—
536C22⋊C4—941no—
537C22⋊C4—942no—
538C22⋊C4—941no—
539C22⋊C4—941no—
540C22⋊C4(C2×D47)⋊C412yes—
541C22⋊C4(C2×D47)⋊C412yes—
542C22⋊C4(C2×D47)⋊C412yes—
543C22⋊C4(C2×D47)⋊C412yes—
544C42—24no—
545C42—81no—
546C42—82no—
547C42—942no—
548C42—942no—
549C42—941no—
550C42C2×C4×D4711yescharacteristic
551C42C2×C4×D4711yescharacteristic
552C42C2×D18811yescharacteristic
553C42C2×D18811yescharacteristic
554C23—24no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—941no—
559C23C2×(C2×D47)⋊C412yes—
560C23C2×(C2×D47)⋊C412yes—
561C23C22×C22⋊D4711yescharacteristic
562Q8—3762no—
563Q8—1881no—
564Q8—1882no—
565Q8—1882no—
566D4—24no—
567D4—22no—
568D4—3762no—
569D4—1881no—
570D4—1884no—
571D4—1881no—
572D4—1881no—
573D4#1309395214yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—941no—
594C2×C4—942no—
595C2×C4—941no—
596C2×C4—942no—
597C2×C4—941no—
598C2×C4—3762no—
599C2×C4—3762no—
600C2×C4—1884no—
601C2×C4—1882no—
602C2×C4C2×(C2×D47)⋊C411yescharacteristic
603C2×C4C2×(C2×D47)⋊C412yes—
604C2×C4C2×(C2×D47)⋊C411yescharacteristic
605C2×C4C2×(C2×D47)⋊C411yescharacteristic
606C2×C4C2×(C2×D47)⋊C412yes—
607C2×C4C2×(C2×D47)⋊C411yescharacteristic
608C2×C4C22×C4×D4711yescharacteristic
609C2×C4C22×D18811yescharacteristic
610C8—1881no—
611C8—1882no—
612C8—1881no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—1881no—
623C22—1881no—
624C22—1881no—
625C22C2×#1309395214yes—
626C22C22×(C2×D47)⋊C411yescharacteristic
627C22#1568993411yescharacteristic
628C22#1568993811yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—1881no—
640C4—1881no—
641C4—1882no—
642C4—1882no—
643C4—1881no—
644C4C2×#1309395212yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—3761no—
650C2#2025184811yescharacteristic
651C2#2025191811yescharacteristic
652C2C22×#1309395211yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2275C2
socleC22×C47C22×C22⋊C4
prime radical#20257437C2
cosocleC22×C4×C47C24
hypercenter#2275D47
hypocenterC47#17548
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #20252423C2semidirect
3. #20241911C2semidirect
4. #20241907C2semidirect
5. #20241899C2semidirect
6. #20241890C2semidirect
7. #20241886C2semidirect
8. #20241877C2semidirect
9. #20241872C2semidirect
10. C2×#15688859C2semidirect
11. C47×#2275C2semidirect
12. #15689843C4semidirect
14. #15689813C4semidirect
16. D4×Dic94C4semidirect
17. #15689782C4semidirect
18. #15689778C4semidirect
19. #15689777C4semidirect
53. C2×#10625350C2×C4semidirect
55. C22×Dic94C2×C4semidirect
58. #13094024C2×C4semidirect
59. #13094024C2×C4semidirect
65. #13094016C2×C4semidirect
67. #13094012C2×C4semidirect
79. #13094003C2×C4semidirect
80. #13094003C2×C4semidirect
81. #13094003C2×C4semidirect
86. #13093998C2×C4semidirect
92. C4×Dic94C2×C4semidirect
93. C4×Dic94C2×C4semidirect
95. C4×D188C2×C4semidirect
102. #13093983C2×C4semidirect
105. C2×C4⋊Dic47C2×C4semidirect
321. #2275D47semidirect
379. C47#17548semidirect

Maximal subgroups

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

subgroupindex
#202574372
#202524232
#202419112
#202419072
#202418992
#202418902
#202418862
#202418772
#202418722
C2×#156888592
C47×#22752
#1754847