Finite Group Databasebeta

← #26222967

Subgroups — #26222967

Order 12032.

954 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×#15689379C211yesmaximal, characteristic
2C2×#15689215C211yesmaximal, characteristic, complemented
3C2×#15689078C211yesmaximal, characteristic
4#20246072C211yesmaximal, characteristic
5#20246066C211yesmaximal, characteristic
6#20244256C211yesmaximal, characteristic, complemented
7#20244130C211yesmaximal, characteristic, complemented
8#20244065C211yesmaximal, characteristic, complemented
9#20244061C211yesmaximal, characteristic, complemented
10#20243811C211yesmaximal, characteristic
11#20243419C211yesmaximal, characteristic
12#20243276C211yesmaximal, characteristic, complemented
13#20243220C211yesmaximal, characteristic, complemented
14#20243217C211yesmaximal, characteristic, complemented
15C47×#1892C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093989C2211yescharacteristic
21C2×#13093983C2211yescharacteristic
22#15689379C418yescomplemented
23#15689215—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×#13093949C2211yescharacteristic
28C2×#13093949C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31#15689078—24no—
32C2×#13093934C2211yescharacteristic
33C2×#13093934C2211yescharacteristic
34C2×C8×Dic47C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36#15689030C2211yescharacteristic
37#15689028C2211yescharacteristic
38#15689022—21no—
39#15689022C2211yescharacteristic
40#15689020—21no—
41#15689019C2211yescharacteristic
42#15689017—21no—
43#15689016—21no—
44#15689016—21no—
45#15689011—21no—
46#15689010—21no—
47#15689009—21no—
48#15688983C2211yescharacteristic
49C4⋊C4×Dic47C2211yescharacteristic
50C4⋊C4×Dic47C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C23×C8×C47C2211yescharacteristic
56C2×C22⋊Q8×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#103×C47C2211yescharacteristic
59C47×#380C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yes—
66C2×#10625329—22no—
67C2×#10625329—22no—
68C2×#10625329C2×C414yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71#13094003—22no—
72#13094003C2×C414yes—
73Q8×Dic47—22no—
74Q8×Dic47—22no—
75#13093989—22no—
76#13093989—22no—
77#13093983—22no—
78#13093983—22no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47D411yescharacteristic
82C2×C4⋊Dic47D411yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×C22⋊Dic47—21no—
90C2×C22⋊Dic47—21no—
91C2×C22⋊Dic47D411yescharacteristic
92C2×C22⋊Dic47D411yescharacteristic
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47—21no—
97C2×C4×Dic47C2311yescharacteristic
98C2×C4×Dic47C2311yescharacteristic
99#13093957—22no—
100#13093955—22no—
101#13093955C2×C414yes—
102#13093949—22no—
103#13093949—22no—
104#13093948C2×C414yes—
105#13093947C2×C414yes—
106#13093934—22no—
107#13093934—22no—
108C8×Dic47—22no—
109C8×Dic47D414yes—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C22×Q8×C47C2311yescharacteristic
120C23×C4×C47C2311yescharacteristic
121C22×C8×C47—22no—
122C22×C8×C47—22no—
123C22×C8×C47—22no—
124C22×C8×C47C2311yescharacteristic
125C22×C8×C47C2311yescharacteristic
126C22⋊Q8×C47—22no—
127C22⋊Q8×C47C2×C414yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C22⋊C4×C47—21no—
133C2×C22⋊C4×C47—21no—
134#103×C47—22no—
135#103×C47—22no—
136#95×C47—21no—
137C24×C47C2×D411yescharacteristic
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47C22×C412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47#4112yes—
155C22×C4×C47C2411yescharacteristic
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D411yescharacteristic
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94C22×C412yes—
170C2×C8×C47—28no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47C2×D412yes—
176C2×C8×C47#4112yes—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47C22×C412yes—
184C22⋊C4×C47—24no—
185C22⋊C4×C47—81no—
186C22⋊Dic47—24no—
187C22⋊Dic47—22no—
188C22⋊Dic47—22no—
189C22⋊Dic47—81no—
190#10625337—28no—
191C4⋊Dic47—21no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—42no—
199C4⋊Dic47—81no—
200C4⋊Dic47C22×C412yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—42no—
203Dic47⋊C4—81no—
204C4×Dic47—21no—
205C4×Dic47—21no—
206C4×Dic47—42no—
207C4×Dic47—42no—
208C4×Dic47—81no—
209C4×Dic47—81no—
210C4×Dic47C2×D412yes—
211C4×Dic47#4112yes—
212#10625329—28no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8C22×C412yes—
216C2×C47⋊C8C22×C412yes—
217C2×C47⋊C8C2×D412yes—
218C2×C47⋊C8#4112yes—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47C4×D412yes—
226C23×C47C22×D411yescharacteristic
227Q8×C47—22no—
228Q8×C47—22no—
229Q8×C47—41no—
230Q8×C47—41no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47C4×D412yes—
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47#12112yes—
279C2×Dic47#12112yes—
280Dic94—22no—
281Dic94—22no—
282Dic94—41no—
283Dic94—41no—
284C8×C47—22no—
285C8×C47—22no—
286C47⋊C8—22no—
287C47⋊C8C4×D414yes—
288#21537—471nomaximal, Sylow
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—41no—
293C22×C47—41no—
294C22×C47—41no—
295C22×C47—41no—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47C2×#12111yescharacteristic
298C22×C47#53211yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57711yescharacteristic
301C22×C47#57811yescharacteristic
302C22×C47#57811yescharacteristic
303C4×C47—22no—
304C4×C47—42no—
305C4×C47—42no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C2×C4×D411yescharacteristic
309C4×C47C4×#4111yescharacteristic
310C4×C47#52511yescharacteristic
311C4×C47#52911yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—42no—
315Dic47—41no—
316Dic47—41no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#470—471no—
321C2×C4×Q16—471no—
322C2×#405—471no—
323#2323—471no—
324#2323—471no—
325#1971—471no—
326#1944—471no—
327#1933—471no—
328#1933—471no—
329#1892—471no—
330#1892D4711yescharacteristic
331#1809—471no—
332C4×#106—471no—
333#1763—471no—
334#1763—471no—
335C2×C47—42no—
336C2×C47C4×#12111yescharacteristic
337C2×C47#294711yescharacteristic
338C2×C47#294711yescharacteristic
339C2×C47#305911yescharacteristic
340C2×C47#306011yescharacteristic
341C2×C47#317811yescharacteristic
342C2×C47#317811yescharacteristic
343C22×Q16—471no—
344C22×M16—471no—
345C23×C8C2×D4711yescharacteristic
346C2×C22⋊Q8—471no—
347C2×C22⋊Q8C2×D4711yescharacteristic
348C2×C4×Q8—471no—
349C2×C4×Q8—471no—
350C2×#117—471no—
351#470—478no—
352C4×Q16—944no—
353C2×#107—471no—
354C2×#106—471no—
355C2×#106—471no—
356C2×#103—471no—
357C2×#103—471no—
358C2×#103—471no—
359C2×#103—471no—
360C2×#103C2×D4711yescharacteristic
361C2×#103C2×D4711yescharacteristic
362#405—944no—
363C2×#97—471no—
364C2×#97—471no—
365C2×C4×C8—471no—
366C2×C4×C8—471no—
367#391—471no—
368#391—471no—
369#389—471no—
370#389—471no—
371#389—941no—
372#389—941no—
373#388—941no—
374#388—941no—
375#388—941no—
376#388—941no—
377#387—941no—
378#387—941no—
379#380—471no—
380#380C2×D4711yescharacteristic
381C4×C4⋊C4—471no—
382C4×C4⋊C4—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340C2×D4711yescharacteristic
388#340C2×D4711yescharacteristic
389C47#2153711yescharacteristic, complemented, Sylow
390C22×Q8—471no—
391C22×Q8C22×D4711yescharacteristic
392C23×C4C22×D4711yescharacteristic
393C2×Q16—474no—
394C2×Q16—942no—
395C2×Q16—942no—
396C2×M16—474no—
397C2×M16—942no—
398C2×M16—942no—
399C22×C8—22no—
400C22×C8—22no—
401C22×C8—22no—
402C22×C8—471no—
403C22×C8—471no—
404C22×C8C22×D4711yescharacteristic
405C22×C8C22×D4711yescharacteristic
406C22⋊Q8—22no—
407C22⋊Q8—474no—
408C22⋊Q8—942no—
409C22⋊Q8C4×D4714yes—
410C4×Q8—942no—
411C4×Q8—942no—
412C4×Q8—942no—
413C4×Q8—942no—
414#117—942no—
415#117—942no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—471no—
433C2×C22⋊C4—471no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42—941no—
442#107—942no—
443#106—474no—
444#106—942no—
445#103—22no—
446#103—22no—
447#103—474no—
448#103—474no—
449#103—942no—
450#103—942no—
451#97—942no—
452#97—942no—
453C4×C8—474no—
454C4×C8—942no—
455#95—21no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465C24D4×D4711yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—472no—
472C2×Q8—941no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8C2×C4×D4712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—471no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4C2×C4×D4712yes—
495C22×C4D4×D4711yescharacteristic
496C22×C4#1062535212yes—
497C22×C4C23×D4711yescharacteristic
498Q16—948no—
499M16—948no—
500C2×C8—28no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—472no—
506C2×C8—472no—
507C2×C8—472no—
508C2×C8—472no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8D4×D4712yes—
512C2×C8#1062535212yes—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—472no—
520C4⋊C4—942no—
521C4⋊C4—942no—
522C4⋊C4—944no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—3761no—
529C4⋊C4—3761no—
530C4⋊C4—1882no—
531C4⋊C4—1882no—
532C4⋊C4C2×C4×D4712yes—
533C22⋊C4—24no—
534C22⋊C4—81no—
535C22⋊C4—942no—
536C22⋊C4—942no—
537C22⋊C4—944no—
538C22⋊C4—3761no—
539C42—472no—
540C42—472no—
541C42—941no—
542C42—941no—
543C42—3761no—
544C42—3761no—
545C42—1882no—
546C42—1882no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23(C22⋊D47)⋊C412yes—
554C23C2×D4×D4711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—942no—
560Q8—942no—
561Q8—1881no—
562Q8—1881no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—472no—
582C2×C4—472no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—941no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4(C22⋊D47)⋊C412yes—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C2×D4×D4711yescharacteristic
608C2×C4#41×D4711yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4C2×#1062535211yescharacteristic
611C2×C4C2×#1062535211yescharacteristic
612C8—22no—
613C8—22no—
614C8—474no—
615C8—942no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×(C22⋊D47)⋊C411yescharacteristic
624C22#121×D4711yescharacteristic
625C22#1568966811yescharacteristic
626C22#1568992911yescharacteristic
627C22#1568993011yescharacteristic
628C22#1568993611yescharacteristic
629C22#1568993711yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—942no—
636C4—942no—
637C4—3761no—
638C4—3761no—
639C4—1882no—
640C4—1881no—
641C4—1881no—
642C4—1882no—
643C4C4×D4×D4711yescharacteristic
644C4C4×#1062535211yescharacteristic
645C4#1568962911yescharacteristic
646C4#1568965011yescharacteristic
647C2—42no—
648C2#2024589811yescharacteristic
649C2#2025222511yescharacteristic
650C2#2025223411yescharacteristic
651C2#2025303811yescharacteristic
652C2#2025304511yescharacteristic
653C2#2025405111yescharacteristic
654C2#2025405911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1892C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1892D47
hypocenterC47#21537
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#15689215C2semidirect
6. #20244256C2semidirect
7. #20244130C2semidirect
8. #20244065C2semidirect
9. #20244061C2semidirect
12. #20243276C2semidirect
13. #20243220C2semidirect
14. #20243217C2semidirect
22. #15689379C4semidirect
389. C47#21537semidirect

Maximal subgroups

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

subgroupindex
C2×#156893792
C2×#156892152
C2×#156890782
#202460722
#202460662
#202442562
#202441302
#202440652
#202440612
#202438112
#202434192
#202432762
#202432202
#202432172
C47×#18922
#2153747