Finite Group Databasebeta

← #26238120

Subgroups — #26238120

Order 12032.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689425C211yesmaximal, characteristic
2#20246489C211yesmaximal, characteristic, complemented
3#20246293C211yesmaximal, characteristic
4#20244351C211yesmaximal, characteristic, complemented
5#20244157C211yesmaximal, characteristic, complemented
6#20244118C211yesmaximal, characteristic, complemented
7#20244114C211yesmaximal, characteristic, complemented
8#20243978C211yesmaximal, characteristic, complemented
9#20243974C211yesmaximal, characteristic, complemented
10#20243633C211yesmaximal, characteristic
11#20243249C211yesmaximal, characteristic, complemented
12#102×Dic47C211yesmaximal, characteristic
13#102×Dic47C211yesmaximal, characteristic
14C4×#13093945C211yesmaximal, characteristic
15C47×#1864C211yesmaximal, characteristic
16C2×#13094015C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689425C418yescomplemented
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×Dic47⋊C8C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×C4×C47⋊C8C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689019—21no—
31#15689019C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689008—21no—
34#15689004—21no—
35#15688996—21no—
36#15688995—21no—
37#15688988C2211yescharacteristic
38#15688981—21no—
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×Dic47⋊C4C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43#15688974—21no—
44C2×C47⋊#95—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C23×Dic47—21no—
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094015—22no—
65#13094015C2×C414yescomplemented
66D4×Dic47—22no—
67D4×Dic47—22no—
68D4×Dic47—41no—
69D4×Dic47C2×C414yes—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4D411yescharacteristic
80C2×Dic47⋊C4D411yescharacteristic
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093957—41no—
89#13093957C2×C414yescomplemented
90#13093956C2×C414yescomplemented
91Dic47⋊C8C2×C414yescomplemented
92#13093945C2×C414yes—
93#13093945Q814yes—
94C4×C47⋊C8—41no—
95C8×Dic47—22no—
96C8×Dic47—41no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C22×D4×C47C2311yescharacteristic
108C23×C4×C47—21no—
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C4×D4×C47—22no—
112C4×D4×C47—41no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#106×C47—41no—
120#102×C47—22no—
121#102×C47C2×C414yes—
122#95×C47—21no—
123C24×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47C22×C412yes—
128C2×D4×C47C2×Q812yes—
129C2×D4×C47#4112yes—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—81no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—24no—
143C22×Dic47—21no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22×Dic47—81no—
147C22×Dic47—81no—
148C22×Dic47—81no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47C22×C412yes—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—81no—
165C4⋊C4×C47C22×C412yes—
166C22⋊C4×C47—42no—
167C42×C47—21no—
168C42×C47—21no—
169C22⋊Dic47—24no—
170C22⋊Dic47—42no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—24no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47C22×C412yes—
178C4⋊Dic47C22×C412yes—
179C4⋊Dic47C2×Q812yes—
180Dic47⋊C4—22no—
181Dic47⋊C4—24no—
182Dic47⋊C4—22no—
183Dic47⋊C4—42no—
184Dic47⋊C4—81no—
185Dic47⋊C4—81no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—42no—
189C4×Dic47—81no—
190C4×Dic47—81no—
191C4×Dic47C22×C412yes—
192C4×Dic47#4112yes—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8C22×C412yes—
198C2×C47⋊C8C2×Q812yes—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C22×D411yescharacteristic
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—22no—
210D4×C47C4×Q814yes—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—22no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
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—84no—
226C2×C4×C47#13512yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47H(5,𝔽2)11yescharacteristic
232C2×C4×C472−1+411yescharacteristic
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—21no—
242C2×Dic47—22no—
243C2×Dic47—21no—
244C2×Dic47—21no—
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—84no—
255C2×Dic47—84no—
256C2×Dic47C4×D412yes—
257C2×Dic47C4×D412yes—
258C2×Dic47#13512yes—
259C8×C47—22no—
260C8×C47—41no—
261C47⋊C8—22no—
262C47⋊C8—22no—
263#22966—471nomaximal, Sylow
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47C2×C4×D411yescharacteristic
272C22×C47#54611yescharacteristic
273C22×C47D4×Q811yescharacteristic
274C22×C47C2×#13511yescharacteristic
275C22×C47C2×#13511yescharacteristic
276C22×C47#57611yescharacteristic
277C22×C47#57711yescharacteristic
278C4×C47—22no—
279C4×C47—42no—
280C4×C47—41no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47C2×C4×Q811yescharacteristic
284C4×C47#52011yescharacteristic
285C4×C47#55211yescharacteristic
286C4×C47#55411yescharacteristic
287Dic47—22no—
288Dic47—22no—
289Dic47—22no—
290Dic47—42no—
291Dic47—42no—
292Dic47—42no—
293Dic47—41no—
294Dic47—81no—
295C2×#478—471no—
296#2370—471no—
297#2349—471no—
298#1991—471no—
299#1949—471no—
300#1942—471no—
301#1941—471no—
302#1921—471no—
303#1920—471no—
304#1864D4711yescharacteristic
305#1862—471no—
306C8×C4⋊C4—471no—
307C4×#102—471no—
308C4×#102—471no—
309C4×#102—471no—
310C2×C47—44no—
311C2×C47#236011yescharacteristic
312C2×C47C4×#13511yescharacteristic
313C2×C47#294611yescharacteristic
314C2×C47#329611yescharacteristic
315C2×C47#329711yescharacteristic
316C2×C47#338011yescharacteristic
317C2×C47#338111yescharacteristic
318C2×#125—471no—
319C2×C4×D4—471no—
320C2×C4×D4—471no—
321C2×C4×D4C2×D4711yescharacteristic
322#478—478no—
323C2×#107—471no—
324C2×#107—471no—
325C2×#106—471no—
326C2×#106C2×D4711yescharacteristic
327C2×C4⋊C8—471no—
328C2×#102—471no—
329C2×#102—471no—
330C2×#102C2×D4711yescharacteristic
331C2×#102C2×D4711yescharacteristic
332C2×C4×C8—471no—
333C2×C4×C8—471no—
334C2×C4×C8—471no—
335#389—471no—
336#389—471no—
337#389—471no—
338#389—941no—
339#389C2×D4711yescharacteristic
340#387—941no—
341#387—941no—
342#385—941no—
343#385—941no—
344#382—471no—
345#380—941no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—941no—
351C2×#95—941no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340C2×D4711yescharacteristic
360#336—471no—
361#336C2×D4711yescharacteristic
362C47#2296611yescharacteristic, complemented, Sylow
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C23×C4—941no—
366C23×C4—941no—
367C22×C8—471no—
368C22×C8—471no—
369C22×C8C22×D4711yescharacteristic
370C22×C8C22×D4711yescharacteristic
371#125—474no—
372#125—942no—
373C4×D4—22no—
374C4×D4—41no—
375C4×D4—474no—
376C4×D4—942no—
377C4×D4—942no—
378C4×D4—1881no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—941no—
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—941no—
396C2×C22⋊C4—941no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402C2×C42C22×D4711yescharacteristic
403#107—474no—
404#107—1881no—
405#106—41no—
406#106—474no—
407C4⋊C8—474no—
408#102—22no—
409#102—474no—
410#102—474no—
411#102C4×D4714yes—
412C4×C8—942no—
413C4×C8—1881no—
414C4×C8—1881no—
415#95—21no—
416#95—471no—
417#95—471no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D4712yes—
431C2×D4#1062535212yes—
432C2×D4Q8×D4712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—944no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—3761no—
449C22×C4—3761no—
450C22×C4—3761no—
451C22×C4—1882no—
452C22×C4—1882no—
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—472no—
462C2×C8—472no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8C2×C4×D4712yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—81no—
474C4⋊C4—472no—
475C4⋊C4—472no—
476C4⋊C4—472no—
477C4⋊C4—942no—
478C4⋊C4—944no—
479C4⋊C4—944no—
480C4⋊C4—942no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—3761no—
487C4⋊C4—3761no—
488C4⋊C4—1882no—
489C4⋊C4C2×C4×D4712yes—
490C22⋊C4—42no—
491C22⋊C4—944no—
492C22⋊C4—1882no—
493C42—21no—
494C42—21no—
495C42—472no—
496C42—472no—
497C42—941no—
498C42—941no—
499C42—3761no—
500C42—3761no—
501C42—1882no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D4711yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4#1309398914yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—472no—
530C2×C4—472no—
531C2×C4—472no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—3764no—
545C2×C4—3764no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4#1309404512yes—
556C2×C4C22×C4×D4711yescharacteristic
557C2×C4C2×Q8×D4711yescharacteristic
558C2×C4#41×D4711yescharacteristic
559C2×C4C2×#1062535211yescharacteristic
560C2×C4#1309406911yescharacteristic
561C2×C4#1309407111yescharacteristic
562C8—22no—
563C8—41no—
564C8—942no—
565C8—942no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C4×D4×D4711yescharacteristic
574C22#1568976011yescharacteristic
575C22#1568978611yescharacteristic
576C22#135×D4711yescharacteristic
577C22C2×#1309404511yescharacteristic
578C22#1568992711yescharacteristic
579C22#1568993311yescharacteristic
580C4—22no—
581C4—42no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—942no—
586C4—942no—
587C4—942no—
588C4—3761no—
589C4—1882no—
590C4—1881no—
591C4—1882no—
592C4—1882no—
593C4C2×#1309398911yescharacteristic
594C4#1568960211yescharacteristic
595C4#1568981011yescharacteristic
596C4#1568982711yescharacteristic
597C2—44no—
598C2#2024638811yescharacteristic
599C2C4×#1309404511yescharacteristic
600C2#2025221711yescharacteristic
601C2#2025522711yescharacteristic
602C2#2025523111yescharacteristic
603C2#2025625411yescharacteristic
604C2#2025625811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1864C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1864D47
hypocenterC47#22966
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #20246489C2semidirect
4. #20244351C2semidirect
5. #20244157C2semidirect
6. #20244118C2semidirect
7. #20244114C2semidirect
8. #20243978C2semidirect
9. #20243974C2semidirect
11. #20243249C2semidirect
19. #15689425C4semidirect
65. #13094015C2×C4semidirect
89. #13093957C2×C4semidirect
90. #13093956C2×C4semidirect
91. Dic47⋊C8C2×C4semidirect
362. C47#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#156894252
#202464892
#202462932
#202443512
#202441572
#202441182
#202441142
#202439782
#202439742
#202436332
#202432492
#102×Dic472
C4×#130939452
C47×#18642
#2296647