Finite Group Databasebeta

← #26216866

Subgroups — #26216866

Order 12032.

898 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#97×D47C211yesmaximal, characteristic
2#20245792C211yesmaximal, characteristic
3#20245784C211yesmaximal, characteristic
4#20243555C211yesmaximal, characteristic, complemented
5#20243553C211yesmaximal, characteristic, complemented
6#20243520C211yesmaximal, characteristic, complemented
7#20243520C211yesmaximal, characteristic, complemented
8#20243416C211yesmaximal, characteristic
9#20243414C211yesmaximal, characteristic
10#20243412C211yesmaximal, characteristic
11#20243410C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13#20243286C211yesmaximal, characteristic, complemented
14#20243285C211yesmaximal, characteristic, complemented
15C47×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D47C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18C2×#13093981—21no—
19C2×C42×D47C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29#97×D47C418yescomplemented
30#15688992C2211yescharacteristic
31#15688992C2211yescharacteristic
32#15688990—21no—
33#15688988C2211yescharacteristic
34#15688988C2211yescharacteristic
35#15688987—21no—
36#15688985—21no—
37#15688982—21no—
38#15688980C2211yescharacteristic
39#15688980C2211yescharacteristic
40#15688980C2211yescharacteristic
41#15688980C2211yescharacteristic
42C4×Dic47⋊C4—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45C4×(C2×D47)⋊C4—21no—
46#15688969—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C2×C8×D47—22no—
66C2×C8×D47—22no—
67C2×C8×D47—22no—
68C2×C8×D47—22no—
69C2×C8×D47C2×C414yes—
70C2×C8×D47C2×C414yes—
71#13093981—44no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89C42×D47C2×C414yes—
90C42×D47Q814yes—
91#13093958—22no—
92#13093957—22no—
93#13093957—22no—
94#13093956—22no—
95#13093955—22no—
96#13093955—22no—
97#13093935C2×C414yes—
98#13093934C2×C414yes—
99#13093934C2×C414yes—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#107×C47—22no—
120#106×C47—22no—
121#97×C47C2×C414yes—
122#95×C47—21no—
123#95×C47—21no—
124C23×D47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×C4×D47—21no—
144C2×C4×D47—21no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47C4⋊C414yes—
148C2×C4×D47C4⋊C414yes—
149C2×C4×D47C22×C412yes—
150C2×C4×D47C2×Q812yes—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47C22×C412yes—
154C2×C8×C47C22×C412yes—
155C2×C8×C47#4112yes—
156C2×C8×C47#4112yes—
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—
165C42×C47—42no—
166C42×C47—42no—
167C42×C47C22×C412yes—
168C42×C47C2×Q812yes—
169(C2×D47)⋊C4—42no—
170(C2×D47)⋊C4—42no—
171(C2×D47)⋊C4—42no—
172(C2×D47)⋊C4—42no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181Dic47⋊C4—42no—
182Dic47⋊C4—42no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—42no—
186C4×Dic47—42no—
187C4×Dic47C4⋊C414yes—
188C4×Dic47C4⋊C414yes—
189C4×Dic47C22×C412yes—
190C4×Dic47C2×Q812yes—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8C22×C412yes—
194C2×C47⋊C8C22×C412yes—
195C2×C47⋊C8#4112yes—
196C2×C47⋊C8#4112yes—
197C8×D47—28no—
198C8×D47—28no—
199C23×C47C22×D411yescharacteristic
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47—21no—
204C22×D47C2×C4⋊C412yes—
205C22×D47AGL(1,ℤ/8)12yes—
206C22×D47#13712yes—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
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×C47C2×C4⋊C412yes—
224C2×C4×C47C2×C4⋊C412yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×Q811yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
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×Dic47C2×C4⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C4×D47—28no—
253C4×D47—22no—
254C4×D47—22no—
255C4×D47—22no—
256C4×D47—22no—
257C4×D47—44no—
258C8×C47—22no—
259C8×C47#11714yescomplemented
260C47⋊C8—22no—
261C47⋊C8#11714yescomplemented
262#20823—471nomaximal, Sylow
263C22×C47C22×C4⋊C411yescharacteristic
264C22×C47#53211yescharacteristic
265C22×C47#53211yescharacteristic
266C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C47C2×#13711yescharacteristic
268C22×C47#57511yescharacteristic
269C22×C47#57511yescharacteristic
270C2×D47—22no—
271C2×D47—22no—
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—22no—
276C2×D47#42814yes—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47C2×#11711yescharacteristic
284C4×C47C2×#11711yescharacteristic
285C4×C47#53311yescharacteristic
286C4×C47#53311yescharacteristic
287Dic47—22no—
288Dic47—22no—
289Dic47—22no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293Dic47—81no—
294C22×#97—471no—
295#2285—471no—
296#2283—471no—
297#1847—471no—
298#1847—471no—
299#1836—471no—
300#1836—471no—
301#1808—471no—
302#1808—471no—
303#1807—471no—
304#1807—471no—
305#1776—471no—
306#1776—471no—
307#1776—471no—
308#1776D4711yescharacteristic, complemented
309C2×C47#228511yescharacteristic
310C2×C47C2×#42811yescharacteristic
311C2×C47#291111yescharacteristic
312C2×C47#318011yescharacteristic
313C2×C47#318011yescharacteristic
314C2×C47#318111yescharacteristic
315C2×C47#318111yescharacteristic
316D47—24no—
317C23×C8—471no—
318C23×C8—471no—
319C2×#117—941no—
320C22×C42—471no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#107—471no—
324C2×#107C2×D4711yescharacteristic
325C2×#106—471no—
326C2×#106—471no—
327C2×#106—471no—
328C2×#106C2×D4711yescharacteristic
329C2×#97—471no—
330C2×#97—471no—
331C2×#97—471no—
332C2×#97—478no—
333C2×#97C2×D4711yescharacteristic
334#384—471no—
335#384—471no—
336#383—941no—
337#382—471no—
338#382—471no—
339#382—941no—
340#381—941no—
341#380—471no—
342#380—471no—
343#380—471no—
344#380—471no—
345#380—941no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—941no—
349C4×C4⋊C4C2×D4711yescharacteristic
350C4×C4⋊C4C2×D4711yescharacteristic
351C4×C22⋊C4—941no—
352#376—941no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340C2×D4711yescharacteristic
360#340C2×D4711yescharacteristic
361C47#2082311yescharacteristic, complemented, Sylow
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—471no—
365C22×C8—471no—
366C22×C8—471no—
367C22×C8—474no—
368C22×C8—474no—
369C22×C8—942no—
370C22×C8—942no—
371C22×C8—942no—
372C22×C8—942no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375#117—1884no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—474no—
398C2×C42—474no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42C22×D4711yescharacteristic
402#107—22no—
403#107—942no—
404#107—942no—
405#107—942no—
406#106—22no—
407#106—942no—
408#106—942no—
409#106—942no—
410#97—474no—
411#97—474no—
412#97—474no—
413#97C4×D4714yes—
414#95—21no—
415#95—21no—
416#95—941no—
417#95—941no—
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—471no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—471no—
434C22×C4—474no—
435C22×C4—472no—
436C22×C4—472no—
437C22×C4—474no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—1881no—
449C22×C4—1881no—
450C22×C4D4×D4711yescharacteristic
451C22×C4D4×D4711yescharacteristic
452C22×C4C23×D4711yescharacteristic
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—472no—
456C2×C8—472no—
457C2×C8—472no—
458C2×C8—472no—
459C2×C8—948no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—948no—
463C2×C8C2×C4×D4712yes—
464C2×C8C2×C4×D4712yes—
465C2×C8#1062535012yes—
466C2×C8#1062535012yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—1882no—
484C4⋊C4—1882no—
485C4⋊C4—1882no—
486C4⋊C4—1882no—
487C22⋊C4—1882no—
488C22⋊C4—1882no—
489C22⋊C4—1882no—
490C22⋊C4—1882no—
491C42—42no—
492C42—42no—
493C42—474no—
494C42—474no—
495C42—472no—
496C42—472no—
497C42—1882no—
498C42—1882no—
499C42C2×C4×D4712yes—
500C42Q8×D4712yes—
501C23—472no—
502C23—472no—
503C23—472no—
504C23—941no—
505C23—941no—
506C23—941no—
507C23—941no—
508C23C2×D4×D4711yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—472no—
528C2×C4—948no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—1884no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4C4⋊C4×D4712yes—
553C2×C4C4⋊C4×D4712yes—
554C2×C4C22×C4×D4711yescharacteristic
555C2×C4C2×Q8×D4711yescharacteristic
556C2×C4#41×D4711yescharacteristic
557C2×C4#41×D4711yescharacteristic
558C2×C4C2×#1062535011yescharacteristic
559C2×C4C2×#1062535011yescharacteristic
560C8—22no—
561C8—474no—
562C8—942no—
563C8#1309398114yescomplemented
564C22—474no—
565C22—942no—
566C22—942no—
567C22—942no—
568C22—942no—
569C22—942no—
570C22—942no—
571C22C2×C4⋊C4×D4711yescharacteristic
572C22#1568967011yescharacteristic
573C22#1568967011yescharacteristic
574C22AGL(1,ℤ/8)×D4711yescharacteristic
575C22#137×D4711yescharacteristic
576C22#1568991911yescharacteristic
577C22#1568992211yescharacteristic
578C4—22no—
579C4—22no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—942no—
585C4—942no—
586C4—942no—
587C4—3761no—
588C4—3761no—
589C4—3761no—
590C4—3761no—
591C4#117×D4711yescharacteristic
592C4C2×#1309398111yescharacteristic
593C4#1568967711yescharacteristic
594C4#1568967711yescharacteristic
595C2—944no—
596C2#2024579211yescharacteristic
597C2#428×D4711yescharacteristic
598C2#2025197511yescharacteristic
599C2#2025408211yescharacteristic
600C2#2025408411yescharacteristic
601C2#2025409311yescharacteristic
602C2#2025409511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#20823
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20243555C2semidirect
5. #20243553C2semidirect
6. #20243520C2semidirect
7. #20243520C2semidirect
12. #20243286C2semidirect
13. #20243286C2semidirect
14. #20243285C2semidirect
15. C47×#1776C2semidirect
29. #97×D47C4semidirect
259. C8×C47#117semidirect
261. C47⋊C8#117semidirect
308. #1776D47semidirect
361. C47#20823semidirect
563. C8#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×#97×D472
#202457922
#202457842
#202435552
#202435532
#202435202
#202434162
#202434142
#202434122
#202434102
#202432862
#202432852
C47×#17762
#2082347