Finite Group Databasebeta

← #26318865

Subgroups — #26318865

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094016C211yesmaximal, characteristic, complemented
2#20244869C211yesmaximal, characteristic, complemented
3#20244801C211yesmaximal, characteristic
4#20244726C211yesmaximal, characteristic, complemented
5#20244534C211yesmaximal, characteristic
6#20244012C211yesmaximal, characteristic
7#20244005C211yesmaximal, characteristic
8#20243980C211yesmaximal, characteristic, complemented
9#20243966C211yesmaximal, characteristic, complemented
10#20243963C211yesmaximal, characteristic, complemented
11#20243956C211yesmaximal, characteristic, complemented
12#20243649C211yesmaximal, characteristic
13#20243635C211yesmaximal, characteristic
14#20243531C211yesmaximal, characteristic, complemented
15C47×#2293C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093945C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25#15689021C2211yescharacteristic
26#15689021C2211yescharacteristic
27#15689007—21no—
28#15689007—21no—
29#15688989C2211yescharacteristic
30C4×Dic47⋊C4—21no—
31C4×Dic47⋊C4—21no—
32C4×C4⋊Dic47C2211yescharacteristic
33C4×C4⋊Dic47C2211yescharacteristic
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4×D4×C47C2211yescharacteristic
49C2×#117×C47—21no—
50C47×#390C2211yescharacteristic
51C47×#390C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#388—21no—
55C47×#388—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47—21no—
58C43×C47C2211yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63#13094016—22no—
64#13094016—22no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47—21no—
75#13093957—22no—
76#13093957—22no—
77#13093955—22no—
78#13093955—22no—
79#13093945—22no—
80#13093945—22no—
81#13093945—22no—
82#13093945—22no—
83C47⋊#95—21no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C22×D4×C47C2311yescharacteristic
90C23×C4×C47—21no—
91C4×D4×C47—22no—
92C4×D4×C47—22no—
93#117×C47—44no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105C2×C42×C47D411yescharacteristic
106C2×C42×C47D411yescharacteristic
107C2×C42×C47D411yescharacteristic
108C2×C42×C47D411yescharacteristic
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C2×C42×C47C2311yescharacteristic
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115C24×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47#4112yes—
122C22×C4×C47—24no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—42no—
145C4⋊C4×C47—42no—
146C4⋊C4×C47#4112yes—
147C22⋊C4×C47—24no—
148C22⋊C4×C47—42no—
149C22⋊C4×C47—42no—
150C22⋊C4×C47—42no—
151C22⋊C4×C47—42no—
152C42×C47—22no—
153C42×C47—22no—
154C42×C47—21no—
155C42×C47—21no—
156C42×C47—21no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—41no—
161C42×C47—41no—
162C42×C47—42no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47#4112yes—
170C4⋊Dic47#4112yes—
171Dic47⋊C4—24no—
172Dic47⋊C4—24no—
173Dic47⋊C4—42no—
174Dic47⋊C4—42no—
175C4×Dic47—42no—
176C4×Dic47—42no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C23×C47—22no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47C22×D411yescharacteristic
197D4×C47—22no—
198D4×C47—22no—
199D4×C47—41no—
200D4×C47—41no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—22no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—48no—
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×C47AGL(1,ℤ/8)12yes—
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C472−1+411yescharacteristic
234C2×C4×C472−1+411yescharacteristic
235C2×Dic47—22no—
236C2×Dic47—22no—
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—
247C47⋊C8—41no—
248C47⋊C8—41no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251#29396—471nomaximal, Sylow
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—81no—
256C22×C47—81no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47#52511yescharacteristic
260C22×C47#52511yescharacteristic
261C22×C47#53611yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C47#57511yescharacteristic
265C22×C47#57511yescharacteristic
266C4×C47—22no—
267C4×C47—22no—
268C4×C47—41no—
269C4×C47—42no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47—41no—
273C4×C47—81no—
274C4×C47—81no—
275C4×C47#55211yescharacteristic
276C4×C47#55211yescharacteristic
277C4×C47#55611yescharacteristic
278C4×C47#55611yescharacteristic
279Dic47—42no—
280Dic47—42no—
281Dic47—81no—
282Dic47—81no—
283C4×#125—471no—
284#2293D4711yescharacteristic
285#2085—471no—
286#2073—471no—
287#2059—471no—
288#2025—471no—
289#1926—471no—
290#1925—471no—
291#1921—471no—
292#1919—471no—
293#1919—471no—
294#1918—471no—
295#1864—471no—
296#1862—471no—
297#1839—471no—
298C2×C47—82no—
299C2×C47#259911yescharacteristic
300C2×C47#310211yescharacteristic
301C2×C47#310511yescharacteristic
302C2×C47#311111yescharacteristic
303C2×C47#311511yescharacteristic
304C2×C47#323311yescharacteristic
305C2×C47#323411yescharacteristic
306C2×#125—471no—
307C2×C4×D4C2×D4711yescharacteristic
308C2×#117—21no—
309C2×#107—471no—
310C2×#107—471no—
311C2×#106—471no—
312C2×#106—471no—
313C2×#102—471no—
314C2×#102—471no—
315C2×#102—471no—
316C2×#102—471no—
317#390C2×D4711yescharacteristic
318#390C2×D4711yescharacteristic
319#389—471no—
320#389—471no—
321#389C2×D4711yescharacteristic
322#389C2×D4711yescharacteristic
323#388—21no—
324#388—21no—
325#387—941no—
326#387—941no—
327#382—471no—
328C4×C4⋊C4—471no—
329C4×C4⋊C4—471no—
330C4×C4⋊C4—941no—
331C4×C4⋊C4—941no—
332C4×C4⋊C4C2×D4711yescharacteristic
333C4×C22⋊C4—21no—
334C43C2×D4711yescharacteristic
335#340—471no—
336#340—471no—
337#340—471no—
338#340—471no—
339#340—471no—
340#340—471no—
341#340—471no—
342#340—471no—
343#336—471no—
344#336—471no—
345#336—471no—
346#336—471no—
347#336—471no—
348#336—471no—
349C47#2939611yescharacteristic, complemented, Sylow
350C22×D4C22×D4711yescharacteristic
351C23×C4—21no—
352C22×C8—471no—
353C22×C8—471no—
354C22×C8—471no—
355C22×C8—471no—
356#125—942no—
357#125—942no—
358C4×D4—22no—
359C4×D4—22no—
360#117—44no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C4⋊C4C22×D4711yescharacteristic
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42—21no—
380C2×C42—941no—
381C2×C42—941no—
382C2×C42C22⋊D4711yescharacteristic
383C2×C42C22⋊D4711yescharacteristic
384C2×C42C22⋊D4711yescharacteristic
385C2×C42C22⋊D4711yescharacteristic
386C2×C42C22×D4711yescharacteristic
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389#107—942no—
390#107—942no—
391#106—942no—
392#106—942no—
393#102—942no—
394#102—942no—
395#102—942no—
396#102—942no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—941no—
401#95—941no—
402#95—941no—
403#95—941no—
404#95—941no—
405#95—941no—
406C24—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4#1062535012yes—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4C2×C22⋊D4711yescharacteristic
424C22×C4C2×C22⋊D4711yescharacteristic
425C22×C4C2×C22⋊D4711yescharacteristic
426C22×C4C2×C22⋊D4711yescharacteristic
427C22×C4C2×C22⋊D4711yescharacteristic
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C23×D4711yescharacteristic
430C2×C8—941no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—941no—
434C2×C8—941no—
435C2×C8—941no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—472no—
450C4⋊C4—472no—
451C4⋊C4—944no—
452C4⋊C4—944no—
453C4⋊C4—941no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—1882no—
460C4⋊C4—1882no—
461C4⋊C4#1062535012yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—41no—
477C42—42no—
478C42—1882no—
479C42—1882no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C22×C22⋊D4711yescharacteristic
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—942no—
519C2×C4—942no—
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#1309404412yes—
531C2×C4#1309405012yes—
532C2×C4#41×D4711yescharacteristic
533C2×C4#41×D4711yescharacteristic
534C2×C4C2×#1062535011yescharacteristic
535C2×C4C2×#1062535011yescharacteristic
536C2×C4#1309407111yescharacteristic
537C2×C4#1309407111yescharacteristic
538C8—1881no—
539C8—1881no—
540C8—1881no—
541C8—1881no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22#1568963011yescharacteristic
550C22#1568963411yescharacteristic
551C22#1568969411yescharacteristic
552C22C2×#1309404411yescharacteristic
553C22C2×#1309405011yescharacteristic
554C22#1568992111yescharacteristic
555C22#1568992111yescharacteristic
556C4—22no—
557C4—22no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—3761no—
566C4—3761no—
567C4—1882no—
568C4—1882no—
569C4#1568980611yescharacteristic
570C4#1568980611yescharacteristic
571C4#1568984111yescharacteristic
572C4#1568984111yescharacteristic
573C2—82no—
574C2#2024896511yescharacteristic
575C2#2025340011yescharacteristic
576C2#2025343511yescharacteristic
577C2#2025349711yescharacteristic
578C2#2025353311yescharacteristic
579C2#2025456311yescharacteristic
580C2#2025457411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2293C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2293D47
hypocenterC47#29396
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. C4×#13094016C2semidirect
2. #20244869C2semidirect
4. #20244726C2semidirect
8. #20243980C2semidirect
9. #20243966C2semidirect
10. #20243963C2semidirect
11. #20243956C2semidirect
14. #20243531C2semidirect
349. C47#29396semidirect

Maximal subgroups

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

subgroupindex
C4×#130940162
#202448692
#202448012
#202447262
#202445342
#202440122
#202440052
#202439802
#202439662
#202439632
#202439562
#202436492
#202436352
#202435312
C47×#22932
#2939647