Finite Group Databasebeta

← #26222877

Subgroups — #26222877

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×#15689362C211yesmaximal, characteristic
2C2×#15689228C211yesmaximal, characteristic, complemented
3C2×#15689079C211yesmaximal, characteristic
4#20246070C211yesmaximal, characteristic
5#20244258C211yesmaximal, characteristic, complemented
6#20244123C211yesmaximal, characteristic, complemented
7#20244067C211yesmaximal, characteristic, complemented
8#20244063C211yesmaximal, characteristic, complemented
9#20243822C211yesmaximal, characteristic
10#20243819C211yesmaximal, characteristic
11#20243411C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13C4×#13093947C211yesmaximal, characteristic, complemented
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#2323C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C23×C47⋊C8C2211yescharacteristic
19C2×#13094003C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21#15689362C418yescomplemented
22#15689228—24no—
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32#15689079—24no—
33C2×#13093935C2211yescharacteristic
34C2×#13093935C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36C2×C4×C47⋊C8C2211yescharacteristic
37#15689029C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689018—21no—
40#15689017—21no—
41#15689011—21no—
42#15689004—21no—
43#15688983C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51C2×C22⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#389—21no—
56C47×#389C2211yescharacteristic
57C47×#388—21no—
58C47×#388—21no—
59C47×#387—21no—
60C47×#380C2211yescharacteristic
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yes—
66C2×#10625329—22no—
67C2×#10625329—22no—
68C2×#10625329—22no—
69C22×C47⋊C8—22no—
70C22×C47⋊C8—22no—
71C22×C47⋊C8C2×C414yes—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75C22×C47⋊C8C2311yescharacteristic
76#13094003—22no—
77#13094003C2×C414yes—
78C4×Dic94—22no—
79C4×Dic94—22no—
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—
90#13093957C2×C414yes—
91#13093955—22no—
92#13093955—22no—
93#13093948—22no—
94#13093948—22no—
95#13093948C2×C414yes—
96#13093947—22no—
97#13093947—22no—
98#13093947C2×C414yes—
99#13093935—22no—
100#13093935D414yes—
101C4×C47⋊C8—22no—
102C4×C47⋊C8—22no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C22⋊Q8×C47—22no—
111C22⋊Q8×C47C2×C414yes—
112C4×Q8×C47—22no—
113C4×Q8×C47—22no—
114#117×C47—22no—
115#117×C47—22no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47D411yescharacteristic
120C2×C4⋊C4×C47D411yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C22⋊C4×C47—21no—
125C2×C22⋊C4×C47—21no—
126C2×C22⋊C4×C47D411yescharacteristic
127C2×C22⋊C4×C47D411yescharacteristic
128C2×C42×C47—21no—
129C2×C42×C47—21no—
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#95×C47—21no—
133#95×C47—21no—
134#95×C47—21no—
135#95×C47—21no—
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×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47#4112yes—
159C22×C4×C47C2411yescharacteristic
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94C22×C412yes—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—42no—
179C4⋊C4×C47—81no—
180C4⋊C4×C47C22×C412yes—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—81no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—42no—
188C42×C47—81no—
189C42×C47C2×D412yes—
190C42×C47#4112yes—
191C22⋊Dic47—24no—
192C22⋊Dic47—81no—
193#10625337—28no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47C22×C412yes—
200Dic47⋊C4—24no—
201Dic47⋊C4—42no—
202Dic47⋊C4—81no—
203C4×Dic47—42no—
204C4×Dic47—81no—
205#10625329—28no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—28no—
213C2×C47⋊C8C22×C412yes—
214C2×C47⋊C8C22×C412yes—
215C2×C47⋊C8C2×D412yes—
216C2×C47⋊C8C2×D412yes—
217C2×C47⋊C8#4112yes—
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—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47C4×D412yes—
260C2×C4×C47#12112yes—
261C2×C4×C47#12112yes—
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C2×Dic47—22no—
269C2×Dic47—22no—
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—41no—
279C2×Dic47—41no—
280Dic94—22no—
281Dic94—22no—
282Dic94—41no—
283Dic94—41no—
284C47⋊C8—22no—
285C47⋊C8—22no—
286C47⋊C8—22no—
287C47⋊C8C4×D414yes—
288#21530—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—22no—
305C4×C47—22no—
306C4×C47—42no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—42no—
310C4×C47—81no—
311C4×C47—81no—
312C4×C47C2×C4×D411yescharacteristic
313C4×C47C4×#4111yescharacteristic
314C4×C47#52511yescharacteristic
315C4×C47#52911yescharacteristic
316Dic47—42no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#467—471no—
321C2×#441—471no—
322C2×#405—471no—
323#2323—471no—
324#2323D4711yescharacteristic
325#1971—471no—
326#1943—471no—
327#1933—471no—
328#1933—471no—
329#1894—471no—
330#1894—471no—
331#1807—471no—
332#1776—471no—
333C4×#103—471no—
334C4×#103—471no—
335C2×C47—42no—
336C2×C47C4×#12111yescharacteristic
337C2×C47#295011yescharacteristic
338C2×C47#295211yescharacteristic
339C2×C47#306311yescharacteristic
340C2×C47#306311yescharacteristic
341C2×C47#317811yescharacteristic
342C2×C47#317811yescharacteristic
343C22×Q16—471no—
344C22×M16—471no—
345C23×C8—471no—
346C2×C22⋊Q8—471no—
347C2×C22⋊Q8C2×D4711yescharacteristic
348C2×C4×Q8—471no—
349C2×C4×Q8C2×D4711yescharacteristic
350C2×#117C2×D4711yescharacteristic
351#467—478no—
352#441—944no—
353C2×#107—471no—
354C2×#106—471no—
355C2×#106—471no—
356C2×#103—471no—
357C2×#103—471no—
358C2×#103—471no—
359C2×#103—471no—
360C2×#103—471no—
361C2×#103—471no—
362#405—944no—
363C2×#97—471no—
364C2×#97—471no—
365C2×C4×C8—471no—
366C2×C4×C8—471no—
367#391—471no—
368#391C2×D4711yescharacteristic
369#389—21no—
370#389—471no—
371#389—941no—
372#389C2×D4711yescharacteristic
373#388—21no—
374#388—21no—
375#388—941no—
376#388—941no—
377#387—21no—
378#387—941no—
379#380—471no—
380#380C2×D4711yescharacteristic
381C4×C4⋊C4—471no—
382C4×C4⋊C4C2×D4711yescharacteristic
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340—471no—
388#340—471no—
389C47#2153011yescharacteristic, complemented, Sylow
390C22×Q8—471no—
391C22×Q8C22×D4711yescharacteristic
392C23×C4C22×D4711yescharacteristic
393C2×Q16—474no—
394C2×Q16—942no—
395C2×Q16—942no—
396C2×M16—942no—
397C2×M16—942no—
398C2×M16—942no—
399C22×C8—471no—
400C22×C8—474no—
401C22×C8—471no—
402C22×C8—471no—
403C22×C8—471no—
404C22×C8—942no—
405C22×C8—942no—
406C22⋊Q8—22no—
407C22⋊Q8—474no—
408C22⋊Q8—942no—
409C22⋊Q8C4×D4714yes—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—942no—
413C4×Q8—942no—
414#117—22no—
415#117—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
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⋊C4C22⋊D4711yescharacteristic
426C2×C4⋊C4C22⋊D4711yescharacteristic
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4C22⋊D4711yescharacteristic
435C2×C22⋊C4C22⋊D4711yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42C22×D4711yescharacteristic
441C2×C42C22×D4711yescharacteristic
442#107—474no—
443#106—942no—
444#106—942no—
445#103—474no—
446#103—474no—
447#103—942no—
448#103—942no—
449#103—942no—
450#103—942no—
451#97—474no—
452#97—942no—
453C4×C8—942no—
454C4×C8—942no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—941no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465C24C2×C22⋊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—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4C2×C4×D4712yes—
491C22×C4#1062535012yes—
492C22×C4C2×C22⋊D4711yescharacteristic
493C22×C4C2×C22⋊D4711yescharacteristic
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498Q16—948no—
499M16—948no—
500C2×C8—472no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—472no—
504C2×C8—472no—
505C2×C8—472no—
506C2×C8—948no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—472no—
524C4⋊C4—944no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—3761no—
531C4⋊C4—1882no—
532C4⋊C4C2×C4×D4712yes—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—81no—
537C22⋊C4—944no—
538C22⋊C4—3761no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—3761no—
544C42—1882no—
545C42D4×D4712yes—
546C42#1062535212yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23C4×C22⋊D4712yes—
554C23C22×C22⋊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—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—1881no—
594C2×C4—1881no—
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×C4C4×C22⋊D4712yes—
604C2×C4#1309399612yes—
605C2×C4#1309399612yes—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C2×D4×D4711yescharacteristic
608C2×C4#41×D4711yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4C2×#1062535211yescharacteristic
611C2×C4C2×#1062535011yescharacteristic
612C8—474no—
613C8—942no—
614C8—942no—
615C8—942no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D4711yescharacteristic
624C22C2×#1309399611yescharacteristic
625C22#1568966911yescharacteristic
626C22#1568993111yescharacteristic
627C22#1568993411yescharacteristic
628C22#1568993811yescharacteristic
629C22#1568993811yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—3761no—
640C4—3761no—
641C4—1882no—
642C4—1882no—
643C4C4×D4×D4711yescharacteristic
644C4C4×#1062535211yescharacteristic
645C4#1568963211yescharacteristic
646C4#1568965311yescharacteristic
647C2—42no—
648C2C4×#1309399611yescharacteristic
649C2#2025225411yescharacteristic
650C2#2025226911yescharacteristic
651C2#2025307511yescharacteristic
652C2#2025308111yescharacteristic
653C2#2025405011yescharacteristic
654C2#2025405811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2323C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2323D47
hypocenterC47#21530
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#15689228C2semidirect
5. #20244258C2semidirect
6. #20244123C2semidirect
7. #20244067C2semidirect
8. #20244063C2semidirect
12. #20243286C2semidirect
13. C4×#13093947C2semidirect
14. C4×#13093948C2semidirect
21. #15689362C4semidirect
389. C47#21530semidirect

Maximal subgroups

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

subgroupindex
C2×#156893622
C2×#156892282
C2×#156890792
#202460702
#202442582
#202441232
#202440672
#202440632
#202438222
#202438192
#202434112
#202432862
C4×#130939472
C4×#130939482
C47×#23232
#2153047