Finite Group Databasebeta

← #26224266

Subgroups — #26224266

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689476C211yesmaximal, characteristic
2#20245947C211yesmaximal, characteristic, complemented
3#20245935C211yesmaximal, characteristic
4#20244351C211yesmaximal, characteristic, complemented
5#20244198C211yesmaximal, characteristic
6#20244074C211yesmaximal, characteristic
7#20244064C211yesmaximal, characteristic, complemented
8#20244063C211yesmaximal, characteristic, complemented
9#20243517C211yesmaximal, characteristic, complemented
10#20243207C211yesmaximal, characteristic, complemented
11#20243207C211yesmaximal, characteristic, complemented
12#102×Dic47C211yesmaximal, characteristic
13#102×Dic47C211yesmaximal, characteristic
14C4×C8×Dic47C211yesmaximal, characteristic, complemented
15C47×#1934C211yesmaximal, characteristic
16C2×#13094015C2211yescharacteristic
17C2×#13094011C2211yescharacteristic
18C2×#13093989C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20#15689476C418yescomplemented
21C2×#13093949C2211yescharacteristic
22C2×#13093949C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15689029C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689011—21no—
35#15689011—21no—
36#15689008—21no—
37#15689004—21no—
38#15688988C2211yescharacteristic
39#15688988C2211yescharacteristic
40C4⋊C4×Dic47—21no—
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43#15688974—21no—
44C22⋊C4×Dic47—21no—
45C22⋊C4×Dic47—21no—
46C42×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#102×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C47×#390C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094015—22no—
65#13094015C2×C414yescomplemented
66#13094011—22no—
67#13094011C2×C414yes—
68#13093989—22no—
69#13093989—22no—
70D4×Dic47—22no—
71D4×Dic47—22no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093949—22no—
92#13093949C2×C414yescomplemented
93#13093947—22no—
94#13093947C2×C414yescomplemented
95#13093945—22no—
96#13093945C2×C414yes—
97C4×C47⋊C8—22no—
98C8×Dic47—22no—
99C8×Dic47—22no—
100C8×Dic47—22no—
101C8×Dic47C2×C414yescomplemented
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C22×D4×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yes—
123C4×C8×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2411yescharacteristic
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94C22×C412yes—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47C22×C412yes—
160C2×C8×C47#4112yes—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—42no—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—42no—
169C22⋊C4×C47—42no—
170C42×C47—21no—
171C42×C47—21no—
172C22⋊Dic47—24no—
173C22⋊Dic47—24no—
174C22⋊Dic47—42no—
175C4⋊Dic47—21no—
176C4⋊Dic47—24no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47C22×C412yes—
182Dic47⋊C4—24no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—42no—
190C4×Dic47—42no—
191C4×Dic47—42no—
192C4×Dic47C22×C412yes—
193C4×Dic47#4112yes—
194C4×Dic47#4112yes—
195C4×Dic47#4112yes—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8#4112yes—
202C23×C47—22no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47C22×D411yescharacteristic
210D4×C47—22no—
211D4×C47—22no—
212D4×C47—41no—
213D4×C47—41no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—22no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47#13512yes—
228C2×C4×C47#13512yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—22no—
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—84no—
254C2×Dic47C4×D412yes—
255C2×Dic47C4×D412yes—
256C2×Dic47#12412yes—
257C2×Dic47#12412yes—
258C2×Dic47#13512yes—
259C2×Dic47#13512yes—
260Dic94—22no—
261Dic94—22no—
262Dic94—41no—
263Dic94—41no—
264C8×C47—22no—
265C8×C47—22no—
266C47⋊C8—22no—
267C47⋊C8—22no—
268C4×#488—471nomaximal, Sylow
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12411yescharacteristic
278C22×C47#53211yescharacteristic
279C22×C47C2×#13511yescharacteristic
280C22×C47C2×#13511yescharacteristic
281C22×C47C2×#13511yescharacteristic
282C22×C47C2×#13511yescharacteristic
283C4×C47—22no—
284C4×C47—22no—
285C4×C47—42no—
286C4×C47—81no—
287C4×C47—81no—
288C4×C47C4×#4111yescharacteristic
289C4×C47C4×#4111yescharacteristic
290C4×C47#52911yescharacteristic
291C4×C47#52911yescharacteristic
292Dic47—22no—
293Dic47—22no—
294Dic47—22no—
295Dic47—42no—
296Dic47—42no—
297Dic47—81no—
298Dic47—81no—
299Dic47#48814yescomplemented
300C2×#488—471no—
301C4×#125—471no—
302C4×#124—471no—
303#1991—471no—
304#1958—471no—
305#1934—471no—
306#1934D4711yescharacteristic
307#1933—471no—
308#1933—471no—
309#1835—471no—
310C4×#103—471no—
311C4×#103—471no—
312C4×#102—471no—
313C4×#102—471no—
314C42×C8—471no—
315C2×C47—82no—
316C2×C47C4×#12411yescharacteristic
317C2×C47C4×#13511yescharacteristic
318C2×C47C4×#13511yescharacteristic
319C2×C47C2×#48811yescharacteristic
320C2×C47#313111yescharacteristic
321C2×C47#316811yescharacteristic
322C2×C47#316811yescharacteristic
323C2×#125—471no—
324C2×#124—471no—
325C2×C4×Q8—471no—
326C2×C4×D4—471no—
327#488—478no—
328C2×#103—471no—
329C2×#103—471no—
330C2×#103—471no—
331C2×#103—471no—
332C2×#102—471no—
333C2×#102—471no—
334C2×#102C2×D4711yescharacteristic
335C2×#102C2×D4711yescharacteristic
336C2×C4×C8—471no—
337C2×C4×C8—471no—
338C2×C4×C8—471no—
339C2×C4×C8—471no—
340C2×C4×C8—471no—
341C2×C4×C8C2×D4711yescharacteristic
342#391—471no—
343#390C2×D4711yescharacteristic
344#389—471no—
345#389C2×D4711yescharacteristic
346#388—941no—
347#388—941no—
348#387—941no—
349#387—941no—
350#382—471no—
351#382—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—941no—
355C4×C4⋊C4—941no—
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358C43—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340C2×D4711yescharacteristic
366#340C2×D4711yescharacteristic
367C47C4×#48811yescharacteristic, complemented, Sylow
368C22×Q8—471no—
369C22×D4C22×D4711yescharacteristic
370C23×C4—941no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375#125—474no—
376#125—942no—
377#124—474no—
378#124—942no—
379C4×Q8—942no—
380C4×Q8—942no—
381C4×D4—942no—
382C4×D4—942no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42—941no—
408C2×C42C22×D4711yescharacteristic
409#103—474no—
410#103—474no—
411#103—942no—
412#103—942no—
413#102—22no—
414#102—474no—
415#102—942no—
416#102C4×D4714yes—
417C4×C8—22no—
418C4×C8—474no—
419C4×C8—942no—
420C4×C8—942no—
421C4×C8—942no—
422C4×C8—942no—
423#95—21no—
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—
433C24—21no—
434C2×Q8—472no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C2×C4×D4712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—1882no—
458C22×C4—1882no—
459C22×C4—1882no—
460C22×C4D4×D4711yescharacteristic
461C22×C4D4×D4711yescharacteristic
462C22×C4C23×D4711yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—472no—
468C2×C8—472no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8C2×C4×D4712yes—
474C2×C8#1062535212yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—472no—
482C4⋊C4—944no—
483C4⋊C4—944no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C4⋊C4C2×C4×D4712yes—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—944no—
495C22⋊C4—944no—
496C22⋊C4—1882no—
497C42—21no—
498C42—21no—
499C42—472no—
500C42—472no—
501C42—472no—
502C42—472no—
503C42—942no—
504C42—942no—
505C42—942no—
506C42—942no—
507C42—1882no—
508C42—1882no—
509C42—1882no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D4711yescharacteristic
518Q8—942no—
519Q8—942no—
520Q8—1881no—
521Q8—1881no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—472no—
540C2×C4—472no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—942no—
552C2×C4—942no—
553C2×C4—3764no—
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#1309404312yes—
565C2×C4#1309404512yes—
566C2×C4C22×C4×D4711yescharacteristic
567C2×C4#41×D4711yescharacteristic
568C2×C4#41×D4711yescharacteristic
569C2×C4#41×D4711yescharacteristic
570C2×C4#41×D4711yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C8—22no—
573C8—22no—
574C8—942no—
575C8—942no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×D4×D4711yescharacteristic
584C22#124×D4711yescharacteristic
585C22#1568966811yescharacteristic
586C22#135×D4711yescharacteristic
587C22#135×D4711yescharacteristic
588C22C2×#1309404311yescharacteristic
589C22C2×#1309404511yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—474no—
596C4—942no—
597C4—942no—
598C4—942no—
599C4—3761no—
600C4—3761no—
601C4—1882no—
602C4—1882no—
603C4C4×#1062535211yescharacteristic
604C4#1568959111yescharacteristic
605C4#1568965011yescharacteristic
606C4#1568965611yescharacteristic
607C2—82no—
608C2#2024592911yescharacteristic
609C2C4×#1309404311yescharacteristic
610C2C4×#1309404511yescharacteristic
611C2#488×D4711yescharacteristic
612C2#2025370711yescharacteristic
613C2#2025395111yescharacteristic
614C2#2025395311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1934C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1934D47
hypocenterC47C4×#488
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #20245947C2semidirect
4. #20244351C2semidirect
7. #20244064C2semidirect
8. #20244063C2semidirect
9. #20243517C2semidirect
10. #20243207C2semidirect
11. #20243207C2semidirect
14. C4×C8×Dic47C2semidirect
20. #15689476C4semidirect
65. #13094015C2×C4semidirect
92. #13093949C2×C4semidirect
94. #13093947C2×C4semidirect
101. C8×Dic47C2×C4semidirect
299. Dic47#488semidirect
367. C47C4×#488semidirect

Maximal subgroups

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

subgroupindex
C2×#156894762
#202459472
#202459352
#202443512
#202441982
#202440742
#202440642
#202440632
#202435172
#202432072
#102×Dic472
C4×C8×Dic472
C47×#19342
C4×#48847