Finite Group Databasebeta

← #26316416

Subgroups — #26316416

Order 12032.

776 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246081C211yesmaximal, characteristic
2#20244950C211yesmaximal, characteristic, complemented
3#20244917C211yesmaximal, characteristic, complemented
4#20244723C211yesmaximal, characteristic, complemented
5#20244531C211yesmaximal, characteristic, complemented
6#20243977C211yesmaximal, characteristic, complemented
7#20243956C211yesmaximal, characteristic, complemented
8#20243824C211yesmaximal, characteristic
9#20243814C211yesmaximal, characteristic
10#20243648C211yesmaximal, characteristic, complemented
11#20243634C211yesmaximal, characteristic, complemented
12#20243419C211yesmaximal, characteristic, complemented
13#20243415C211yesmaximal, characteristic, complemented
14#20243085C211yesmaximal, characteristic, complemented
15C47×#2629C211yesmaximal, characteristic, complemented
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13093998C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28#15689025C2211yescharacteristic
29#15689023—21no—
30#15689021C2211yescharacteristic
31#15689013—21no—
32#15688999—21no—
33#15688994—21no—
34#15688983C2211yescharacteristic
35C4×C4⋊Dic47C2211yescharacteristic
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×#125×C47C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C47×#398—21no—
51C47×#398—21no—
52C47×#396—21no—
53C47×#396—21no—
54C47×#389C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#384C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#380C2211yescharacteristic
59C22×D188C2311yescharacteristic
60C22×C22⋊D47—21no—
61C22×C4×D47—21no—
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329—41no—
65C22×C47⋊C8—22no—
66C22×C47⋊C8—22no—
67C22×C47⋊C8—41no—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72#13093998—22no—
73#13093998Q814yes—
74C4×D188—41no—
75C4×D188—41no—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×C4×Dic47—21no—
86#13093957—22no—
87#13093957—22no—
88#13093957—41no—
89#13093955—41no—
90#13093944—22no—
91#13093944—22no—
92#13093944—41no—
93#13093944—41no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C23×C4×C47C2311yescharacteristic
98#125×C47—41no—
99#125×C47—41no—
100#117×C47—41no—
101#117×C47—41no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47D411yescharacteristic
109C2×C4⋊C4×C47D411yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—21no—
116C2×C22⋊C4×C47D411yescharacteristic
117C2×C22⋊C4×C47D411yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124C24×C47C2×D411yescharacteristic
125C23×D47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×Q812yes—
140C22×C4×C47#4112yes—
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—24no—
143C2×C22⋊D47—81no—
144C2×C22⋊D47—81no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188C2×Q812yes—
154C2×C4×D47—42no—
155C2×C4×D47—81no—
156C2×C4×D47—81no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
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—42no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47—81no—
170C4⋊C4×C47—81no—
171C4⋊C4×C47—81no—
172C4⋊C4×C47—81no—
173C22⋊C4×C47—41no—
174C22⋊C4×C47—41no—
175C22⋊C4×C47—81no—
176C22⋊C4×C47—81no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—21no—
181C22⋊Dic47—81no—
182(C2×D47)⋊C4—24no—
183(C2×D47)⋊C4—42no—
184(C2×D47)⋊C4—81no—
185(C2×D47)⋊C4—81no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47C2×Q812yes—
192Dic47⋊C4—81no—
193C4×Dic47—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×C47C22⋊Q812yes—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
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—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
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×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47C22⋊Q812yes—
254C2×C4×C47C22×Q811yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C47H(5,𝔽2)11yescharacteristic
260C22⋊D47—84no—
261C2×Dic47—22no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270D188—22no—
271D188—22no—
272D188—41no—
273D188—41no—
274C4×D47—84no—
275C47⋊C8—22no—
276C47⋊C8—22no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279#29169—471nomaximal, Sylow
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47C2×C22⋊Q811yescharacteristic
288C22×C47#53211yescharacteristic
289C22×C47#53411yescharacteristic
290C22×C47#57611yescharacteristic
291C22×C47#57611yescharacteristic
292C22×C47#57711yescharacteristic
293C22×C47#57811yescharacteristic
294C2×D47—42no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—81no—
298C2×D47—81no—
299C2×D47—81no—
300C2×D47—81no—
301C4×C47—22no—
302C4×C47—41no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47—81no—
309C4×C47—81no—
310C4×C47#55111yescharacteristic
311C4×C47#55111yescharacteristic
312C4×C47#55411yescharacteristic
313C4×C47#55411yescharacteristic
314Dic47—42no—
315Dic47—81no—
316Dic47—81no—
317#2629D4711yescharacteristic, complemented
318#2324—471no—
319#2100—471no—
320#2093—471no—
321#2059—471no—
322#2025—471no—
323#1921—471no—
324#1918—471no—
325#1895—471no—
326#1893—471no—
327#1864—471no—
328#1862—471no—
329#1809—471no—
330#1808—471no—
331#1729—471no—
332C2×C47—42no—
333C2×C47#256411yescharacteristic
334C2×C47#308111yescharacteristic
335C2×C47#308311yescharacteristic
336C2×C47#317611yescharacteristic
337C2×C47#317811yescharacteristic
338C2×C47#319811yescharacteristic
339C2×C47#320311yescharacteristic
340D47—82no—
341C22×M16—471no—
342C23×C8—471no—
343C2×#125C2×D4711yescharacteristic
344C2×#121—471no—
345C2×C4×D4—471no—
346C2×#117C2×D4711yescharacteristic
347C2×#107—471no—
348C2×#107—471no—
349C2×#107—471no—
350C2×#106—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102—471no—
355#398—21no—
356#398—21no—
357#396—21no—
358#396—21no—
359#390—471no—
360#390—941no—
361#389—471no—
362#389C2×D4711yescharacteristic
363#389C2×D4711yescharacteristic
364#388—941no—
365#386—941no—
366#385—941no—
367#384C2×D4711yescharacteristic
368#380—471no—
369#380C2×D4711yescharacteristic
370#380C2×D4711yescharacteristic
371C4×C4⋊C4—471no—
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#336—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384C47#2916911yescharacteristic, complemented, Sylow
385C22×D4—471no—
386C22×D4—941no—
387C23×C4—941no—
388C23×C4C22×D4711yescharacteristic
389C2×M16—942no—
390C2×M16—942no—
391C2×M16—1881no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8—942no—
397C22×C8—942no—
398C22×C8—1881no—
399#125—41no—
400#125—41no—
401#121—474no—
402#121—942no—
403C4×D4—1881no—
404C4×D4—1881no—
405#117—41no—
406#117—41no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
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×C4⋊C4C22×D4711yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C22⋊C4—941no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4—941no—
430C2×C22⋊C4C22⋊D4711yescharacteristic
431C2×C22⋊C4C22⋊D4711yescharacteristic
432C2×C42—941no—
433C2×C42C22×D4711yescharacteristic
434C2×C42C22×D4711yescharacteristic
435#107—942no—
436#107—942no—
437#107—1881no—
438#106—1881no—
439#102—942no—
440#102—942no—
441#102—1881no—
442#102—1881no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—941no—
448#95—941no—
449#95—941no—
450C24—941no—
451C24C2×C22⋊D4711yescharacteristic
452C2×D4—472no—
453C2×D4—944no—
454C2×D4—941no—
455C2×D4—941no—
456C2×D4—941no—
457C2×D4—941no—
458C2×D4—941no—
459C2×D4—3761no—
460C2×D4—3761no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—3761no—
473C22×C4—3761no—
474C22×C4—1882no—
475C22×C4Q8×D4712yes—
476C22×C4#1062535412yes—
477C22×C4C2×C22⋊D4711yescharacteristic
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C23×D4711yescharacteristic
483M16—1884no—
484C2×C8—472no—
485C2×C8—472no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—1884no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—472no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—3761no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—944no—
525C22⋊C4—3761no—
526C22⋊C4—3761no—
527C22⋊C4—3761no—
528C22⋊C4—1882no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—3761no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—942no—
541C23—1881no—
542C23—1881no—
543C23—1881no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23#1309400012yes—
548C23C22×C22⋊D4711yescharacteristic
549D4—942no—
550D4—942no—
551D4—3764no—
552D4—1881no—
553D4—1881no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
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—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—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—942no—
585C2×C4—3764no—
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#1309400012yes—
595C2×C4C2×Q8×D4711yescharacteristic
596C2×C4#41×D4711yescharacteristic
597C2×C4#41×D4711yescharacteristic
598C2×C4C2×#1062535411yescharacteristic
599C2×C4#1309406911yescharacteristic
600C2×C4#1309406911yescharacteristic
601C8—942no—
602C8—942no—
603C8—1881no—
604C8—1881no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—3761no—
613C22—3761no—
614C22—3761no—
615C22—3761no—
616C22—1882no—
617C22—1882no—
618C22—1882no—
619C22C2×#1309400011yescharacteristic
620C22#1568966911yescharacteristic
621C22#1568968211yescharacteristic
622C22#1568992611yescharacteristic
623C22#1568992611yescharacteristic
624C22#1568993111yescharacteristic
625C22#1568993811yescharacteristic
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—3761no—
636C4—3761no—
637C4—1882no—
638C4#1568980011yescharacteristic
639C4#1568980011yescharacteristic
640C4#1568982711yescharacteristic
641C4#1568982711yescharacteristic
642C2—42no—
643C2—3762no—
644C2#2024857011yescharacteristic
645C2#2025321311yescharacteristic
646C2#2025323511yescharacteristic
647C2#2025403211yescharacteristic
648C2#2025405811yescharacteristic
649C2#2025423411yescharacteristic
650C2#2025428511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2629C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenter#2629D47
hypocenterC47#29169
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
2. #20244950C2semidirect
3. #20244917C2semidirect
4. #20244723C2semidirect
5. #20244531C2semidirect
6. #20243977C2semidirect
7. #20243956C2semidirect
10. #20243648C2semidirect
11. #20243634C2semidirect
12. #20243419C2semidirect
13. #20243415C2semidirect
14. #20243085C2semidirect
15. C47×#2629C2semidirect
317. #2629D47semidirect
384. C47#29169semidirect

Maximal subgroups

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

subgroupindex
#202460812
#202449502
#202449172
#202447232
#202445312
#202439772
#202439562
#202438242
#202438142
#202436482
#202436342
#202434192
#202434152
#202430852
C47×#26292
#2916947