Finite Group Databasebeta

← #26226889

Subgroups — #26226889

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246011C211yesmaximal, characteristic
2#20245997C211yesmaximal, characteristic
3#20244158C211yesmaximal, characteristic, complemented
4#20244158C211yesmaximal, characteristic, complemented
5#20244152C211yesmaximal, characteristic, complemented
6#20244147C211yesmaximal, characteristic, complemented
7#20243589C211yesmaximal, characteristic
8#20243589C211yesmaximal, characteristic
9#20243583C211yesmaximal, characteristic
10#20243578C211yesmaximal, characteristic
11#20243556C211yesmaximal, characteristic, complemented
12#20243552C211yesmaximal, characteristic, complemented
13#20243522C211yesmaximal, characteristic, complemented
14#20243516C211yesmaximal, characteristic, complemented
15C2×C47×#433C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093958C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093956C2211yescharacteristic
26#15689022—21no—
27#15689022—21no—
28#15689016—21no—
29#15689016—21no—
30#15689009—21no—
31#15689009—21no—
32#15688998—21no—
33#15688998—21no—
34#15688993C2211yescharacteristic
35#15688993C2211yescharacteristic
36#15688993C2211yescharacteristic
37#15688989C2211yescharacteristic
38#15688989C2211yescharacteristic
39#15688989C2211yescharacteristic
40#15688983C2211yescharacteristic
41#15688983C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C47×#433C418yescomplemented
54C2×C4⋊C8×C47C2211yescharacteristic
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C2×C22⋊C8×C47C2211yescharacteristic
58C2×#97×C47C2211yescharacteristic
59C2×C4×C8×C47C2211yescharacteristic
60C23×Dic47—21no—
61#13093983—41no—
62#13093983—41no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—41no—
71C2×C4⋊Dic47—41no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86#13093958—41no—
87#13093958—41no—
88#13093958—41no—
89#13093958—41no—
90#13093956—41no—
91#13093956—41no—
92#13093956—41no—
93#13093956—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—
104C23×C4×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109#117×C47C2×C414yes—
110#117×C47Q814yes—
111C2×C4⋊C4×C47D411yescharacteristic
112C2×C4⋊C4×C47D411yescharacteristic
113C2×C22⋊C4×C47D411yescharacteristic
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C4⋊C8×C47C2×C414yescomplemented
118C4⋊C8×C47C2×C414yescomplemented
119C22⋊C8×C47C2×C414yes—
120C22⋊C8×C47C2×C414yes—
121#97×C47C2×C414yescomplemented
122C4×C8×C47C2×C414yescomplemented
123C24×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×Q812yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—42no—
145C22×Dic47—81no—
146C22×Dic47—81no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47C22×C412yes—
156C2×C8×C47C22×C412yes—
157C2×C8×C47C22×C412yes—
158C2×C8×C47C22×C412yes—
159C4⋊C4×C47C4⋊C414yes—
160C4⋊C4×C47C4⋊C414yes—
161C22⋊C4×C47C4⋊C414yes—
162C22⋊C4×C47C4⋊C414yes—
163C42×C47C22×C412yes—
164C42×C47C22×C412yes—
165C42×C47C2×Q812yes—
166C42×C47C2×Q812yes—
167C22⋊Dic47—42no—
168C22⋊Dic47—81no—
169C22⋊Dic47—81no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—42no—
187C4⋊Dic47—81no—
188C4⋊Dic47—81no—
189C4⋊Dic47—81no—
190C4⋊Dic47—81no—
191C4⋊Dic47—81no—
192C4⋊Dic47—81no—
193C4×Dic47—42no—
194C4×Dic47—81no—
195C4×Dic47—81no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C2×C4⋊C412yes—
203C23×C47C22×D411yescharacteristic
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47C2×C4⋊C412yes—
219C2×C4×C47C2×C4⋊C412yes—
220C2×C4×C47C2×C4⋊C412yes—
221C2×C4×C47C2×C4⋊C412yes—
222C2×C4×C47C2×C4⋊C412yes—
223C2×C4×C47C23×C411yescharacteristic
224C2×C4×C47C22×Q811yescharacteristic
225C2×C4×C47H(5,𝔽2)11yescharacteristic
226C2×C4×C47H(5,𝔽2)11yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×C4×C472−1+411yescharacteristic
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
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—84no—
254C8×C47—22no—
255C8×C47—22no—
256C8×C47—22no—
257C8×C47—22no—
258#21992—471nomaximal, Sylow
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47C22×C4⋊C411yescharacteristic
267C22×C47#53511yescharacteristic
268C22×C47#53611yescharacteristic
269C22×C47#57611yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57811yescharacteristic
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47#52011yescharacteristic
279C4×C47#52011yescharacteristic
280C4×C47#54411yescharacteristic
281C4×C47#54411yescharacteristic
282Dic47—81no—
283Dic47—81no—
284Dic47—81no—
285Dic47—81no—
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289Dic47—81no—
290C2×#433D4711yescharacteristic
291#2317—471no—
292#2314—471no—
293#1949—471no—
294#1949—471no—
295#1948—471no—
296#1947—471no—
297#1854—471no—
298#1854—471no—
299#1853—471no—
300#1852—471no—
301#1847—471no—
302#1846—471no—
303#1836—471no—
304#1834—471no—
305C2×C47—42no—
306C2×C47#231711yescharacteristic
307C2×C47#291211yescharacteristic
308C2×C47#291311yescharacteristic
309C2×C47#322411yescharacteristic
310C2×C47#322511yescharacteristic
311C2×C47#324111yescharacteristic
312C2×C47#324211yescharacteristic
313C2×#117—471no—
314C2×#117C2×D4711yescharacteristic
315C22×C4⋊C4—471no—
316#433Dic4718yescomplemented
317C2×#107—471no—
318C2×#107—471no—
319C2×#107—471no—
320C2×#107—471no—
321C2×#106—471no—
322C2×#106—471no—
323C2×#106—471no—
324C2×#106—471no—
325C2×C4⋊C8C2×D4711yescharacteristic
326C2×C4⋊C8C2×D4711yescharacteristic
327C2×C22⋊C8C2×D4711yescharacteristic
328C2×C22⋊C8C2×D4711yescharacteristic
329C2×#97C2×D4711yescharacteristic
330C2×C4×C8C2×D4711yescharacteristic
331#389—941no—
332#389—941no—
333#388—941no—
334#388—941no—
335#387—941no—
336#387—941no—
337#385—941no—
338#385—941no—
339#384—471no—
340#384—471no—
341#384—471no—
342#382—471no—
343#382—471no—
344#382—471no—
345#380—471no—
346#380—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357C47#2199211yescharacteristic, complemented, Sylow
358C23×C4—941no—
359C23×C4C22×D4711yescharacteristic
360C22×C8C22×D4711yescharacteristic
361C22×C8C22×D4711yescharacteristic
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4711yescharacteristic
364#117—1881no—
365#117—1881no—
366#117Dic9414yes—
367#117C2×Dic4714yes—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—1881no—
384C2×C4⋊C4—1881no—
385C2×C4⋊C4D18811yescharacteristic
386C2×C4⋊C4D18811yescharacteristic
387C2×C22⋊C4—941no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4D18811yescharacteristic
391C2×C22⋊C4D18811yescharacteristic
392C2×C42—941no—
393C2×C42—941no—
394C2×C42—941no—
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397#107—1881no—
398#107—1881no—
399#107—1881no—
400#107—1881no—
401#106—1881no—
402#106—1881no—
403#106—1881no—
404#106—1881no—
405C4⋊C8C2×Dic4714yescomplemented
406C4⋊C8C2×Dic4714yescomplemented
407C22⋊C8C2×Dic4714yes—
408C22⋊C8C2×Dic4714yes—
409#97C2×Dic4714yescomplemented
410C4×C8C2×Dic4714yescomplemented
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421C24C2×D18811yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—3761no—
435C22×C4—3761no—
436C22×C4—1882no—
437C22×C4C2×Dic9412yes—
438C22×C4C2×D18811yescharacteristic
439C22×C4C2×D18811yescharacteristic
440C22×C4C2×D18811yescharacteristic
441C22×C4C2×D18811yescharacteristic
442C22×C4C2×D18811yescharacteristic
443C22×C4C22×Dic4712yes—
444C22×C4C23×D4711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8C22×Dic4712yes—
454C2×C8C22×Dic4712yes—
455C2×C8C22×Dic4712yes—
456C2×C8C22×Dic4712yes—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—3761no—
474C4⋊C4—3761no—
475C4⋊C4—3761no—
476C4⋊C4—3761no—
477C4⋊C4—3761no—
478C4⋊C4—3761no—
479C4⋊C4—1882no—
480C4⋊C4C4⋊Dic4714yes—
481C4⋊C4C4⋊Dic4714yes—
482C22⋊C4—3761no—
483C22⋊C4—3761no—
484C22⋊C4—1882no—
485C22⋊C4C4⋊Dic4714yes—
486C22⋊C4C4⋊Dic4714yes—
487C42—3761no—
488C42—3761no—
489C42—1882no—
490C42C2×Dic9412yes—
491C42C2×Dic9412yes—
492C42C22×Dic4712yes—
493C42C22×Dic4712yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×C4⋊Dic4712yes—
501C23C22×D18811yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—3764no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1881no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4C2×C4⋊Dic4712yes—
542C2×C4C2×C4⋊Dic4712yes—
543C2×C4C2×C4⋊Dic4712yes—
544C2×C4C2×C4⋊Dic4712yes—
545C2×C4C2×C4⋊Dic4712yes—
546C2×C4C23×Dic4711yescharacteristic
547C2×C4C22×Dic9411yescharacteristic
548C2×C4#1309407011yescharacteristic
549C2×C4#1309407011yescharacteristic
550C2×C4#1309407111yescharacteristic
551C2×C4#1309407111yescharacteristic
552C8—22no—
553C8—22no—
554C8—22no—
555C8—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×C4⋊Dic4711yescharacteristic
564C22#1568969211yescharacteristic
565C22#1568969911yescharacteristic
566C22#1568992811yescharacteristic
567C22#1568993511yescharacteristic
568C22#1568993511yescharacteristic
569C22#1568994011yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—3761no—
576C4—3761no—
577C4—3761no—
578C4—3761no—
579C4—3761no—
580C4—3761no—
581C4—3761no—
582C4—3761no—
583C4#1568960611yescharacteristic
584C4#1568960611yescharacteristic
585C4#1568975111yescharacteristic
586C4#1568975111yescharacteristic
587C2—42no—
588C2#2024601111yescharacteristic
589C2#2025198311yescharacteristic
590C2#2025198811yescharacteristic
591C2#2025448411yescharacteristic
592C2#2025449911yescharacteristic
593C2#2025464311yescharacteristic
594C2#2025465811yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#433C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC2×#433D47
hypocenterC47#21992
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #20244158C2semidirect
4. #20244158C2semidirect
5. #20244152C2semidirect
6. #20244147C2semidirect
11. #20243556C2semidirect
12. #20243552C2semidirect
13. #20243522C2semidirect
14. #20243516C2semidirect
53. C47×#433C4semidirect
117. C4⋊C8×C47C2×C4semidirect
118. C4⋊C8×C47C2×C4semidirect
121. #97×C47C2×C4semidirect
122. C4×C8×C47C2×C4semidirect
316. #433Dic47semidirect
357. C47#21992semidirect
405. C4⋊C8C2×Dic47semidirect
406. C4⋊C8C2×Dic47semidirect
409. #97C2×Dic47semidirect
410. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202460112
#202459972
#202441582
#202441522
#202441472
#202435892
#202435832
#202435782
#202435562
#202435522
#202435222
#202435162
C2×C47×#4332
#2199247