Finite Group Databasebeta

← #26316483

Subgroups — #26316483

Order 12032.

784 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#20249209C211yesmaximal, characteristic, complemented
2#20244956C211yesmaximal, characteristic, complemented
3#20244909C211yesmaximal, characteristic, complemented
4#20244721C211yesmaximal, characteristic, complemented
5#20244524C211yesmaximal, characteristic, complemented
6#20243966C211yesmaximal, characteristic, complemented
7#20243959C211yesmaximal, characteristic, complemented
8#20243827C211yesmaximal, characteristic
9#20243817C211yesmaximal, characteristic
10#20243649C211yesmaximal, characteristic, complemented
11#20243649C211yesmaximal, characteristic, complemented
12#20243419C211yesmaximal, characteristic, complemented
13#20243411C211yesmaximal, characteristic, complemented
14#20243085C211yesmaximal, characteristic, complemented
15C47×#2324C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13094024C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27#15689062—21no—
28#15689059—21no—
29#15689043—21no—
30#15689035—21no—
31#15689021C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15688989C2211yescharacteristic
34#15688983C2211yescharacteristic
35#15688983C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#121×C47C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×#117×C47C2211yescharacteristic
51C47×#390—21no—
52C47×#390C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#388—21no—
55C47×#386—21no—
56C47×#385—21no—
57C47×#380C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C22×Dic94—21no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—41no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8—41no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094024—41no—
71#13094024—41no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81#13093957—22no—
82#13093955—22no—
83#13093955—41no—
84#13093955—41no—
85#13093945—22no—
86#13093945—22no—
87#13093945—41no—
88#13093945—41no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×D4×C47—21no—
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99#121×C47—22no—
100#121×C47Q814yescomplemented
101C4×D4×C47—41no—
102C4×D4×C47—41no—
103#117×C47—41no—
104#117×C47—41no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47D411yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—21no—
116C2×C22⋊C4×C47D411yescharacteristic
117C2×C22⋊C4×C47D411yescharacteristic
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124C24×C47—21no—
125C24×C47C2×D411yescharacteristic
126C2×D4×C47—24no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—81no—
133C2×D4×C47—81no—
134C2×D4×C47C2×Q812yes—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—42no—
143C22×C4×C47—81no—
144C22×C4×C47—81no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×Q812yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C2×Dic94—42no—
158C2×Dic94—81no—
159C2×Dic94—81no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—41no—
166C4⋊C4×C47—41no—
167C4⋊C4×C47—81no—
168C4⋊C4×C47C2×Q812yes—
169C22⋊C4×C47—24no—
170C22⋊C4×C47—41no—
171C22⋊C4×C47—41no—
172C22⋊C4×C47—42no—
173C22⋊C4×C47—81no—
174C22⋊C4×C47—81no—
175C22⋊C4×C47—81no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—81no—
181C22⋊Dic47—81no—
182C22⋊Dic47—81no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—42no—
192Dic47⋊C4—81no—
193Dic47⋊C4—81no—
194#10625329—44no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—44no—
206C2×C47⋊C8#4112yes—
207C2×C47⋊C8#4112yes—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—22no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47C22⋊Q812yes—
222C23×C47C22×D411yescharacteristic
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—41no—
226D4×C47—41no—
227D4×C47—84no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
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—84no—
256C2×C4×C47C22⋊Q812yes—
257C2×C4×C47C22×Q811yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47H(5,𝔽2)11yescharacteristic
262C2×C4×C47H(5,𝔽2)11yescharacteristic
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275Dic94—84no—
276C47⋊C8—22no—
277C47⋊C8—22no—
278C47⋊C8—41no—
279C47⋊C8—41no—
280#29174—471nomaximal, Sylow
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—81no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47—81no—
295C22×C47C2×C22⋊Q811yescharacteristic
296C22×C47#53211yescharacteristic
297C22×C47#53411yescharacteristic
298C22×C47#57611yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57711yescharacteristic
301C22×C47#57711yescharacteristic
302C4×C47—22no—
303C4×C47—42no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—41no—
307C4×C47—41no—
308C4×C47—81no—
309C4×C47—81no—
310C4×C47#55111yescharacteristic
311C4×C47#55111yescharacteristic
312C4×C47#55411yescharacteristic
313C4×C47#55411yescharacteristic
314Dic47—81no—
315Dic47—81no—
316Dic47—81no—
317Dic47—81no—
318#2628—471no—
319#2324D4711yescharacteristic
320#2101—471no—
321#2092—471no—
322#2058—471no—
323#2023—471no—
324#1919—471no—
325#1918—471no—
326#1895—471no—
327#1893—471no—
328#1864—471no—
329#1864—471no—
330#1809—471no—
331#1807—471no—
332#1729—471no—
333C2×C47—42no—
334C2×C47—82no—
335C2×C47#256411yescharacteristic
336C2×C47#308111yescharacteristic
337C2×C47#308411yescharacteristic
338C2×C47#317311yescharacteristic
339C2×C47#317911yescharacteristic
340C2×C47#319811yescharacteristic
341C2×C47#320011yescharacteristic
342C22×M16—471no—
343C23×C8—471no—
344C2×C4⋊Q8—471no—
345C2×#121C2×D4711yescharacteristic
346C2×C4×D4C2×D4711yescharacteristic
347C2×#117C2×D4711yescharacteristic
348C2×#107—471no—
349C2×#106—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356#400—941no—
357#399—941no—
358#395—941no—
359#393—941no—
360#390—21no—
361#390C2×D4711yescharacteristic
362#389—471no—
363#389—471no—
364#389C2×D4711yescharacteristic
365#388—21no—
366#386—21no—
367#385—21no—
368#382—471no—
369#380—471no—
370#380—471no—
371#380C2×D4711yescharacteristic
372C4×C4⋊C4C2×D4711yescharacteristic
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384#336—471no—
385C47#2917411yescharacteristic, complemented, Sylow
386C22×Q8—941no—
387C22×D4—21no—
388C22×D4C22×D4711yescharacteristic
389C23×C4—21no—
390C23×C4C22×D4711yescharacteristic
391C2×M16—942no—
392C2×M16—942no—
393C2×M16—1881no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—942no—
399C22×C8—942no—
400C22×C8—1881no—
401C4⋊Q8—1881no—
402C4⋊Q8—1881no—
403#121—22no—
404#121Dic9414yescomplemented
405C4×D4—41no—
406C4×D4—41no—
407#117—41no—
408#117—41no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—471no—
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×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—941no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4C22⋊D4711yescharacteristic
430C2×C22⋊C4C22⋊D4711yescharacteristic
431C2×C42—21no—
432C2×C42C22×D4711yescharacteristic
433C2×C42C22×D4711yescharacteristic
434#107—942no—
435#106—942no—
436#106—1881no—
437#106—1881no—
438#102—942no—
439#102—942no—
440#102—1881no—
441#102—1881no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—941no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451C24—21no—
452C24C2×C22⋊D4711yescharacteristic
453C2×Q8—3761no—
454C2×Q8—3761no—
455C2×Q8—1882no—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—81no—
463C2×D4—81no—
464C2×D4C2×Dic9412yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4C2×Dic9412yes—
480C22×C4#1062535012yes—
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4711yescharacteristic
486C22×C4C23×D4711yescharacteristic
487M16—1884no—
488C2×C8—472no—
489C2×C8—472no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—1884no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—81no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—3761no—
518C4⋊C4—3761no—
519C4⋊C4—1882no—
520C4⋊C4C2×Dic9412yes—
521C22⋊C4—24no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C22⋊C4—3761no—
529C22⋊C4—3761no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—81no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#1309400312yes—
549C23C22×C22⋊D4711yescharacteristic
550Q8—3764no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4#1309400312yes—
597C2×C4C22×Dic9411yescharacteristic
598C2×C4#41×D4711yescharacteristic
599C2×C4#41×D4711yescharacteristic
600C2×C4C2×#1062535011yescharacteristic
601C2×C4#1309406911yescharacteristic
602C2×C4#1309406911yescharacteristic
603C8—942no—
604C8—942no—
605C8—1881no—
606C8—1881no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#1309400311yescharacteristic
622C22#1568966911yescharacteristic
623C22#1568968211yescharacteristic
624C22#1568992611yescharacteristic
625C22#1568993111yescharacteristic
626C22#1568993411yescharacteristic
627C22#1568993411yescharacteristic
628C4—22no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—3761no—
637C4—3761no—
638C4—3761no—
639C4—3761no—
640C4#1568980211yescharacteristic
641C4#1568980211yescharacteristic
642C4#1568983211yescharacteristic
643C4#1568983211yescharacteristic
644C2—42no—
645C2—82no—
646C2#2024856411yescharacteristic
647C2#2025321911yescharacteristic
648C2#2025325011yescharacteristic
649C2#2025399911yescharacteristic
650C2#2025407311yescharacteristic
651C2#2025423411yescharacteristic
652C2#2025426011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2324C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenter#2324D47
hypocenterC47#29174
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. #20249209C2semidirect
2. #20244956C2semidirect
3. #20244909C2semidirect
4. #20244721C2semidirect
5. #20244524C2semidirect
6. #20243966C2semidirect
7. #20243959C2semidirect
10. #20243649C2semidirect
11. #20243649C2semidirect
12. #20243419C2semidirect
13. #20243411C2semidirect
14. #20243085C2semidirect
100. #121×C47Q8semidirect
385. C47#29174semidirect
404. #121Dic94semidirect

Maximal subgroups

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

subgroupindex
#202492092
#202449562
#202449092
#202447212
#202445242
#202439662
#202439592
#202438272
#202438172
#202436492
#202434192
#202434112
#202430852
C47×#23242
#2917447