Finite Group Databasebeta

← #26316273

Subgroups — #26316273

Order 12032.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251698C211yesmaximal, characteristic
2C4×#13094003C211yesmaximal, characteristic
3#20244957C211yesmaximal, characteristic, complemented
4#20244935C211yesmaximal, characteristic, complemented
5#20244727C211yesmaximal, characteristic, complemented
6#20244540C211yesmaximal, characteristic, complemented
7#20243974C211yesmaximal, characteristic, complemented
8#20243960C211yesmaximal, characteristic, complemented
9#20243811C211yesmaximal, characteristic
10#20243811C211yesmaximal, characteristic
11#20243659C211yesmaximal, characteristic, complemented
12#20243645C211yesmaximal, characteristic, complemented
13#20243417C211yesmaximal, characteristic
14#20243413C211yesmaximal, characteristic
15C2×C47×#336C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093949C2211yescharacteristic
24C2×#13093949C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2211yescharacteristic
27#15689029C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689021C2211yescharacteristic
31#15689007—21no—
32#15689007—21no—
33#15689007—21no—
34#15688996—21no—
35#15688996—21no—
36#15688996—21no—
37#15688989C2211yescharacteristic
38#15688983C2211yescharacteristic
39#15688983C2211yescharacteristic
40#15688983C2211yescharacteristic
41C4×Dic47⋊C4—21no—
42C4×C4⋊Dic47C2211yescharacteristic
43C4×C22⋊Dic47—21no—
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52C23×C8×C47C2211yescharacteristic
53C23×C8×C47C2211yescharacteristic
54C22×C42×C47C2211yescharacteristic
55C47×#336C2211yescharacteristic
56C47×#336C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C22×Dic94C2311yescharacteristic
60#13094016—41no—
61#13094016—41no—
62#13094003—22no—
63#13094003Q814yes—
64C4×Dic94—41no—
65C4×Dic94—41no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86#13093958—22no—
87#13093958—41no—
88#13093956—22no—
89#13093956—41no—
90#13093949—22no—
91#13093949—22no—
92#13093949—41no—
93#13093949—41no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C23×C4×C47D411yescharacteristic
106C23×C4×C47D411yescharacteristic
107C23×C4×C47C2311yescharacteristic
108C22×C8×C47—22no—
109C22×C8×C47—22no—
110C22×C8×C47—22no—
111C22×C8×C47—22no—
112C22×C8×C47—41no—
113C22×C8×C47—41no—
114C22×C8×C47C2311yescharacteristic
115C22×C8×C47C2311yescharacteristic
116C22×C8×C47C2311yescharacteristic
117C22×C8×C47C2311yescharacteristic
118C2×C42×C47—41no—
119C2×C42×C47—41no—
120C2×C42×C47D411yescharacteristic
121C2×C42×C47D411yescharacteristic
122C2×C42×C47C2311yescharacteristic
123C2×C42×C47C2311yescharacteristic
124C24×C47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—41no—
133C22×C4×C47—41no—
134C22×C4×C47—41no—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×Q812yes—
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94C2×Q812yes—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—44no—
168C2×C8×C47—44no—
169C2×C8×C47#4112yes—
170C2×C8×C47#4112yes—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—41no—
176C42×C47—41no—
177C22⋊Dic47—24no—
178C22⋊Dic47—42no—
179C22⋊Dic47—42no—
180C22⋊Dic47—42no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
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—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47C2×Q812yes—
195Dic47⋊C4—24no—
196Dic47⋊C4—42no—
197Dic47⋊C4—42no—
198Dic47⋊C4—42no—
199Dic47⋊C4—42no—
200Dic47⋊C4—42no—
201Dic47⋊C4—42no—
202C4×Dic47—42no—
203C4×Dic47—42no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47C22⋊Q812yes—
209C23×C47#13512yes—
210C23×C47#13712yes—
211C23×C47C22×D411yescharacteristic
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—44no—
230C2×C4×C47—44no—
231C2×C4×C47C22⋊Q812yes—
232C2×C4×C47#13512yes—
233C2×C4×C47C22×Q811yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47H(5,𝔽2)11yescharacteristic
238C2×C4×C47H(5,𝔽2)11yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263Dic94—22no—
264Dic94—22no—
265Dic94—41no—
266Dic94—41no—
267C8×C47—22no—
268C8×C47—22no—
269C8×C47—41no—
270C8×C47—41no—
271#29158—471nomaximal, Sylow
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47#47914yes—
279C22×C47C2×C22⋊Q811yescharacteristic
280C22×C47#53211yescharacteristic
281C22×C47#53411yescharacteristic
282C22×C47C2×#13511yescharacteristic
283C22×C47C2×#13511yescharacteristic
284C22×C47C2×#13711yescharacteristic
285C22×C47#57511yescharacteristic
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47#55111yescharacteristic
292C4×C47#55111yescharacteristic
293C4×C47#55411yescharacteristic
294C4×C47#55411yescharacteristic
295Dic47—42no—
296Dic47—42no—
297Dic47—81no—
298Dic47—81no—
299Dic47—81no—
300Dic47—81no—
301Dic47—81no—
302Dic47—81no—
303#2851—471no—
304C4×C22⋊Q8—471no—
305#2101—471no—
306#2097—471no—
307#2059—471no—
308#2026—471no—
309#1920—471no—
310#1918—471no—
311#1892—471no—
312#1892—471no—
313#1865—471no—
314#1863—471no—
315#1808—471no—
316#1807—471no—
317C2×#336D4711yescharacteristic
318C2×C47—24no—
319C2×C47#256411yescharacteristic
320C2×C47C2×#47911yescharacteristic
321C2×C47#307611yescharacteristic
322C2×C47#316811yescharacteristic
323C2×C47#318111yescharacteristic
324C2×C47#319011yescharacteristic
325C2×C47#319111yescharacteristic
326C23×C8C2×D4711yescharacteristic
327C23×C8C2×D4711yescharacteristic
328C2×#125—471no—
329C2×C22⋊Q8—471no—
330C2×C4×Q8—471no—
331C22×C42C2×D4711yescharacteristic
332C2×#107—471no—
333C2×#107—471no—
334C2×#106—471no—
335C2×#106—471no—
336C2×#103—471no—
337C2×#103—471no—
338C2×#103—471no—
339C2×#103—471no—
340#391—471no—
341#389—471no—
342#389—471no—
343#389—471no—
344#387—941no—
345#387—941no—
346#387—941no—
347#385—941no—
348#385—941no—
349#385—941no—
350#382—471no—
351#380—471no—
352#380—471no—
353#380—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—941no—
356C4×C22⋊C4—941no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#336C2×D4711yescharacteristic
366#336C2×D4711yescharacteristic
367#336C2×D4711yescharacteristic
368#336C2×D4711yescharacteristic
369C47#2915811yescharacteristic, complemented, Sylow
370C22×Q8—471no—
371C23×C4D18811yescharacteristic
372C23×C4D18811yescharacteristic
373C23×C4C22×D4711yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—41no—
379C22×C8—41no—
380C22×C8C22×D4711yescharacteristic
381C22×C8C22×D4711yescharacteristic
382C22×C8C22×D4711yescharacteristic
383C22×C8C22×D4711yescharacteristic
384#125—1881no—
385#125—1881no—
386C22⋊Q8—474no—
387C22⋊Q8—942no—
388C4×Q8—1881no—
389C4×Q8—1881no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C42—41no—
409C2×C42—41no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42D18811yescharacteristic
413C2×C42D18811yescharacteristic
414C2×C42C22×D4711yescharacteristic
415C2×C42C22×D4711yescharacteristic
416#107—942no—
417#107—1881no—
418#106—942no—
419#106—1881no—
420#103—942no—
421#103—942no—
422#103—1881no—
423#103—1881no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435C24C2×D18811yescharacteristic
436C2×Q8—472no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4C2×D18811yescharacteristic
461C22×C4C2×D18811yescharacteristic
462C22×C4C2×D18811yescharacteristic
463C22×C4C2×D18811yescharacteristic
464C22×C4C2×D18811yescharacteristic
465C22×C4#1062535212yes—
466C22×C4Q8×D4712yes—
467C22×C4C23×D4711yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8#1062535012yes—
481C2×C8#1062535012yes—
482C4⋊C4—472no—
483C4⋊C4—944no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4—1882no—
500C4⋊C4—1882no—
501C4⋊C4—1882no—
502C4⋊C4—1882no—
503C22⋊C4—944no—
504C22⋊C4—1882no—
505C22⋊C4—1882no—
506C22⋊C4—1882no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—1882no—
514C42—1882no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1309400412yes—
520C23#1309404612yes—
521C23#1309405712yes—
522C23C22×D18811yescharacteristic
523Q8—942no—
524Q8—942no—
525Q8—1881no—
526Q8—1881no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4#1309400412yes—
571C2×C4#1309404612yes—
572C2×C4C2×Q8×D4711yescharacteristic
573C2×C4C2×#1062535211yescharacteristic
574C2×C4C2×#1062535011yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C2×C4#1309406911yescharacteristic
577C2×C4#1309406911yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1568943114yes—
589C22C2×#1309400411yescharacteristic
590C22#1568967311yescharacteristic
591C22#1568968511yescharacteristic
592C22C2×#1309404611yescharacteristic
593C22C2×#1309404611yescharacteristic
594C22C2×#1309405711yescharacteristic
595C22#1568992311yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—3761no—
602C4—3761no—
603C4—3761no—
604C4—3761no—
605C4—3761no—
606C4—3761no—
607C4—1882no—
608C4—1882no—
609C4#1568979911yescharacteristic
610C4#1568979911yescharacteristic
611C4#1568982811yescharacteristic
612C4#1568982811yescharacteristic
613C2—24no—
614C2#2024856811yescharacteristic
615C2C2×#1568943111yescharacteristic
616C2#2025317811yescharacteristic
617C2#2025395811yescharacteristic
618C2#2025409711yescharacteristic
619C2#2025417111yescharacteristic
620C2#2025418111yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#336C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC2×#336D47
hypocenterC47#29158
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
3. #20244957C2semidirect
4. #20244935C2semidirect
5. #20244727C2semidirect
6. #20244540C2semidirect
7. #20243974C2semidirect
8. #20243960C2semidirect
11. #20243659C2semidirect
12. #20243645C2semidirect
369. C47#29158semidirect

Maximal subgroups

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

subgroupindex
#202516982
C4×#130940032
#202449572
#202449352
#202447272
#202445402
#202439742
#202439602
#202438112
#202436592
#202436452
#202434172
#202434132
C2×C47×#3362
#2915847