Finite Group Databasebeta

← #26226845

Subgroups — #26226845

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689177C211yesmaximal, characteristic
2#20245791C211yesmaximal, characteristic
3#20245790C211yesmaximal, characteristic
4#20244150C211yesmaximal, characteristic, complemented
5#20244148C211yesmaximal, characteristic, complemented
6#20244145C211yesmaximal, characteristic, complemented
7#20244143C211yesmaximal, characteristic, complemented
8#20243589C211yesmaximal, characteristic
9#20243587C211yesmaximal, characteristic
10#20243585C211yesmaximal, characteristic
11#20243555C211yesmaximal, characteristic, complemented
12#20243553C211yesmaximal, characteristic, complemented
13#20243520C211yesmaximal, characteristic, complemented
14#20243520C211yesmaximal, characteristic, complemented
15C47×#1854C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C2×#13093980—21no—
18C22×C4×Dic47C2211yescharacteristic
19#15689177C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30C2×#13093934C2211yescharacteristic
31#15688992C2211yescharacteristic
32#15688992C2211yescharacteristic
33#15688992C2211yescharacteristic
34#15688992C2211yescharacteristic
35#15688988C2211yescharacteristic
36#15688988C2211yescharacteristic
37#15688987—21no—
38#15688983C2211yescharacteristic
39#15688981—21no—
40#15688981—21no—
41#15688981—21no—
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4×Dic47⋊C4—21no—
45C4×Dic47⋊C4—21no—
46#15688974—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47D411yescharacteristic
61C23×Dic47D411yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13093983—41no—
65#13093983—41no—
66#13093980—44no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
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×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2×C414yes—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47Q814yes—
88C2×C4×Dic47C2311yescharacteristic
89C2×C4×Dic47C2311yescharacteristic
90#13093958—41no—
91#13093957—41no—
92#13093957—41no—
93#13093956—41no—
94#13093955—41no—
95#13093955—41no—
96Dic47⋊C8C2×C414yes—
97Dic47⋊C8C2×C414yes—
98C22⋊(C47⋊C8)C2×C414yes—
99#13093934C2×C414yes—
100#13093934C2×C414yes—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111#117×C47—41no—
112#117×C47—41no—
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120#107×C47—41no—
121#106×C47—41no—
122C22⋊C8×C47C2×C414yes—
123#95×C47—21no—
124C24×C47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×Q812yes—
136C22×C4×C47C2411yescharacteristic
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—41no—
144C22×Dic47—41no—
145C22×Dic47C4⋊C414yes—
146C22×Dic47C4⋊C414yes—
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47C22×C412yes—
156C2×C8×C47C22×C412yes—
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—
165C22⋊C4×C47—42no—
166C22⋊C4×C47—42no—
167C42×C47—42no—
168C22⋊Dic47—42no—
169C22⋊Dic47—42no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178Dic47⋊C4—42no—
179Dic47⋊C4—42no—
180Dic47⋊C4—42no—
181Dic47⋊C4—42no—
182Dic47⋊C4—42no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185Dic47⋊C4—42no—
186C4×Dic47—42no—
187C4×Dic47—42no—
188C4×Dic47—42no—
189C4×Dic47C4⋊C414yes—
190C4×Dic47C4⋊C414yes—
191C4×Dic47C22×C412yes—
192C4×Dic47C22×C412yes—
193C4×Dic47C2×Q812yes—
194C4×Dic47C2×Q812yes—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8C22×C412yes—
200C2×C47⋊C8C22×C412yes—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47C2×C4⋊C412yes—
208C23×C47C22×D411yescharacteristic
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
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×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47C2×C4⋊C412yes—
228C2×C4×C47C23×C411yescharacteristic
229C2×C4×C47C22×Q811yescharacteristic
230C2×C4×C47H(5,𝔽2)11yescharacteristic
231C2×C4×C47H(5,𝔽2)11yescharacteristic
232C2×C4×C472−1+411yescharacteristic
233C2×C4×C472−1+411yescharacteristic
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—28no—
241C2×Dic47—44no—
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×Dic47C2×C4⋊C412yes—
255C2×Dic47C2×C4⋊C412yes—
256C2×Dic47C2×C4⋊C412yes—
257C2×Dic47C2×C4⋊C412yes—
258C2×Dic47AGL(1,ℤ/8)12yes—
259C2×Dic47#13712yes—
260C8×C47—22no—
261C8×C47—22no—
262C47⋊C8—22no—
263C47⋊C8—22no—
264#21988—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—41no—
272C22×C47C22×C4⋊C411yescharacteristic
273C22×C47#53511yescharacteristic
274C22×C47#53611yescharacteristic
275C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C47C2×#13711yescharacteristic
277C22×C47#57511yescharacteristic
278C22×C47#57511yescharacteristic
279C4×C47—22no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47—81no—
284C4×C47#52011yescharacteristic
285C4×C47#52011yescharacteristic
286C4×C47#54411yescharacteristic
287C4×C47#54411yescharacteristic
288Dic47—22no—
289Dic47—22no—
290Dic47—22no—
291Dic47—81no—
292Dic47—81no—
293Dic47—81no—
294Dic47—81no—
295Dic47#42814yes—
296C2×#432—471no—
297#2285—471no—
298#2285—471no—
299#1948—471no—
300#1948—471no—
301#1947—471no—
302#1947—471no—
303#1854—471no—
304#1854—471no—
305#1854—471no—
306#1854D4711yescharacteristic
307#1847—471no—
308#1847—471no—
309#1836—471no—
310#1836—471no—
311C2×C47—42no—
312C2×C47#231711yescharacteristic
313C2×C47C2×#42811yescharacteristic
314C2×C47#291111yescharacteristic
315C2×C47#321411yescharacteristic
316C2×C47#321611yescharacteristic
317C2×C47#323511yescharacteristic
318C2×C47#323711yescharacteristic
319C2×#117—471no—
320C2×#117—941no—
321C2×#117C2×D4711yescharacteristic
322C22×C42—471no—
323#432—478no—
324C2×#107—471no—
325C2×#107—471no—
326C2×#107—471no—
327C2×#107C2×D4711yescharacteristic
328C2×#106—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×#106C2×D4711yescharacteristic
332C2×C4⋊C8—471no—
333C2×C4⋊C8—471no—
334C2×C22⋊C8—471no—
335C2×C22⋊C8C2×D4711yescharacteristic
336C2×#97—471no—
337C2×#97—471no—
338#384—471no—
339#384—471no—
340#384—471no—
341#384—471no—
342#382—471no—
343#382—471no—
344#382—941no—
345#380—471no—
346#380—941no—
347#380—941no—
348#380—941no—
349#380C2×D4711yescharacteristic
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—941no—
353C4×C4⋊C4—941no—
354C4×C4⋊C4—941no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340C2×D4711yescharacteristic
362#340C2×D4711yescharacteristic
363C47#2198811yescharacteristic, complemented, Sylow
364C23×C4—471no—
365C23×C4—471no—
366C23×C4C22×D4711yescharacteristic
367C22×C8—471no—
368C22×C8—471no—
369C22×C8C22×D4711yescharacteristic
370C22×C8C22×D4711yescharacteristic
371#117—41no—
372#117—41no—
373#117—1884no—
374#117—1881no—
375#117—1881no—
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⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C42—21no—
397C2×C42—474no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—474no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42—941no—
406#107—41no—
407#107—1881no—
408#107—1881no—
409#107—1881no—
410#106—41no—
411#106—1881no—
412#106—1881no—
413#106—1881no—
414C4⋊C8—474no—
415C4⋊C8—474no—
416C22⋊C8—474no—
417C22⋊C8C4×D4714yes—
418#97—474no—
419#97—474no—
420#95—21no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428C24D4×D4711yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—471no—
438C22×C4—474no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—474no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—1881no—
450C22×C4—1881no—
451C22×C4C2×C4×D4712yes—
452C22×C4D4×D4711yescharacteristic
453C22×C4Q8×D4712yes—
454C22×C4C23×D4711yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—472no—
460C2×C8—472no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8C2×C4×D4712yes—
466C2×C8C2×C4×D4712yes—
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—
487C4⋊C4—1882no—
488C4⋊C4—1882no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—1882no—
494C22⋊C4—1882no—
495C42—42no—
496C42—474no—
497C42—474no—
498C42—472no—
499C42—472no—
500C42—472no—
501C42—472no—
502C42—1882no—
503C42—1882no—
504C42—1882no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D4712yes—
512C23C2×D4×D4711yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
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—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—948no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—1884no—
545C2×C4—1881no—
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—1881no—
556C2×C4—1881no—
557C2×C4C4⋊C4×D4712yes—
558C2×C4C22×C4×D4711yescharacteristic
559C2×C4C2×Q8×D4711yescharacteristic
560C2×C4#1309406911yescharacteristic
561C2×C4#1309406911yescharacteristic
562C2×C4#1309407111yescharacteristic
563C2×C4#1309407111yescharacteristic
564C8—22no—
565C8—22no—
566C8—942no—
567C8—942no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D4711yescharacteristic
576C22#1568968811yescharacteristic
577C22#1568969511yescharacteristic
578C22AGL(1,ℤ/8)×D4711yescharacteristic
579C22#137×D4711yescharacteristic
580C22#1568992011yescharacteristic
581C22#1568992011yescharacteristic
582C4—22no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—474no—
588C4—942no—
589C4—942no—
590C4—942no—
591C4—3761no—
592C4—3761no—
593C4—3761no—
594C4—3761no—
595C4#1568960211yescharacteristic
596C4#1568960211yescharacteristic
597C4#1568975211yescharacteristic
598C4#1568975211yescharacteristic
599C2—42no—
600C2#2024600711yescharacteristic
601C2#428×D4711yescharacteristic
602C2#2025197611yescharacteristic
603C2#2025438011yescharacteristic
604C2#2025439611yescharacteristic
605C2#2025458311yescharacteristic
606C2#2025459911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1854C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1854D47
hypocenterC47#21988
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #20244150C2semidirect
5. #20244148C2semidirect
6. #20244145C2semidirect
7. #20244143C2semidirect
11. #20243555C2semidirect
12. #20243553C2semidirect
13. #20243520C2semidirect
14. #20243520C2semidirect
19. #15689177C4semidirect
363. C47#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#156891772
#202457912
#202457902
#202441502
#202441482
#202441452
#202441432
#202435892
#202435872
#202435852
#202435552
#202435532
#202435202
C47×#18542
#2198847