Finite Group Databasebeta

← #26225969

Subgroups — #26225969

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689660C211yesmaximal, characteristic
2C2×#15689175C211yesmaximal, characteristic, complemented
3#382×D47C211yesmaximal, characteristic, complemented
4#20244351C212yesmaximal, complemented
5#20244196C212yesmaximal, complemented
6#20243704C212yesmaximal
7#20243692C212yesmaximal, complemented
8#20243403C211yesmaximal, characteristic, complemented
9#20243399C211yesmaximal, characteristic, complemented
10#20243395C211yesmaximal, characteristic, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12#15689660—24no—
13#15689660C418yescomplemented
14C2×#13094019—21no—
15C2×#13094015C2211yescharacteristic
16C2×#13094013C2211yescharacteristic
17C2×#13094009—21no—
18C2×#13094000C2211yescharacteristic
19C2×#13093999C2211yescharacteristic
20C2×C4×D188—21no—
21C2×C4×D188C2211yescharacteristic
22C2×#13093981—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47C2211yescharacteristic
25#15689175—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2212yes—
28C2×#13093946C2212yes—
29C2×#13093944C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×C8×Dic47C2212yes—
32#15688989C2211yescharacteristic
33#15688988C2212yes—
34#15688987—21no—
35#15688987—21no—
36#95×D47—21no—
37#95×D47—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#382C2211yescharacteristic
45C47×#340C2212yes—
46C22×D188—21no—
47C22×D188C2311yescharacteristic
48C22×C4×D47—21no—
49C22×C4×D47—22no—
50C22×C4×D47D411yescharacteristic
51C22×C4×D47D411yescharacteristic
52C22×C4×D47C2311yescharacteristic
53C22×C47⋊C8C2312yes—
54#13094019—28no—
55#13094015—22no—
56#13094015C2×C414yescomplemented
57#13094013—22no—
58#13094013C2×C414yescomplemented
59#13094009—28no—
60#13094000—22no—
61#13094000C2×C414yes—
62#13093999—22no—
63#13093999C2×C414yes—
64C4×D188—28no—
65C4×D188—22no—
66C4×D188C2×C414yescomplemented
67#13093981—28no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—42no—
75C4⋊C4×D47—42no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84C42×D47C2×C414yes—
85C42×D47D414yes—
86C4⋊(C47⋊C8)—22no—
87#13093948—42no—
88#13093946—42no—
89#13093944—42no—
90(C2×D47)⋊C8—24no—
91C8×Dic47—24no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C4×Q8×C47—22no—
99C4×Q8×C47C2×C414yescomplemented
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106#103×C47—42no—
107#95×C47—21no—
108#95×C47—21no—
109C23×D47—21no—
110C23×D47C2×D411yescharacteristic
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C22×C412yes—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2411yescharacteristic
123C22×Dic47—21no—
124C22×Dic47—22no—
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C2×D188—21no—
129C2×D188—21no—
130C2×D188—21no—
131C2×D188—21no—
132C2×D188—24no—
133C2×D188—21no—
134C2×D188—44no—
135C2×D188C22×C412yes—
136C2×C4×D47—24no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C4×D47—24no—
142C2×C4×D47—42no—
143C2×C4×D47—44no—
144C2×C4×D47—44no—
145C2×C4×D47—44no—
146C2×C4×D47—44no—
147C2×C4×D47C22⋊C414yes—
148C2×C4×D47C22⋊C414yes—
149C2×C4×D47C22×C412yes—
150C2×C4×D47C2×D412yes—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47C22×C412yes—
159C42×C47—24no—
160C42×C47C22×C412yes—
161C42×C47C2×D412yes—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166(C2×D47)⋊C4—24no—
167C4⋊Dic47—21no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47C22×C412yes—
172Dic47⋊C4—24no—
173Dic47⋊C4—24no—
174Dic47⋊C4—44no—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22⋊C414yes—
177C4×Dic47C22×C412yes—
178C4×Dic47C2×D412yes—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C23×C47C22×D411yescharacteristic
183C22×D47—21no—
184C22×D47—21no—
185C22×D47—22no—
186C22×D47—22no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—42no—
190C22×D47—42no—
191C22×D47C2×C22⋊C412yes—
192C22×D47#13514yes—
193Q8×C47—22no—
194Q8×C47—22no—
195Q8×C47—41no—
196Q8×C47—41no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47C23×C411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47H(5,𝔽2)11yescharacteristic
212C2×C4×C47H(5,𝔽2)11yescharacteristic
213C2×C4×C472−1+411yescharacteristic
214C2×C4×C472−1+411yescharacteristic
215C2×Dic47—21no—
216C2×Dic47—21no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47C2×C22⋊C412yes—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227D188—22no—
228D188—22no—
229D188—48no—
230D188—41no—
231D188—41no—
232C4×D47—28no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—44no—
238C4×D47—48no—
239C4×D47—88no—
240C8×C47—42no—
241C47⋊C8—42no—
242#21861—471nomaximal, Sylow
243C22×C47C22×C22⋊C411yescharacteristic
244C22×C47#53511yescharacteristic
245C22×C47#53611yescharacteristic
246C22×C47C2×#13512yes—
247C22×C47#57512yes—
248C2×D47—22no—
249C2×D47—22no—
250C2×D47—24no—
251C2×D47—24no—
252C2×D47—42no—
253C2×D47—42no—
254C2×D47—42no—
255C2×D47—82no—
256C2×D47—82no—
257C2×D47#41614yes—
258C4×C47—22no—
259C4×C47—22no—
260C4×C47—42no—
261C4×C47—42no—
262C4×C47—82no—
263C4×C47#52011yescharacteristic
264C4×C47#52011yescharacteristic
265C4×C47#53711yescharacteristic
266C4×C47#53711yescharacteristic
267Dic47—22no—
268Dic47—22no—
269Dic47—22no—
270Dic47—42no—
271Dic47—82no—
272C2×#529—471no—
273C2×#431—471no—
274C2×#382—471no—
275#1991—472no—
276#1958—472no—
277#1874—472no—
278#1872—472no—
279#1805—471no—
280#1805D4711yescharacteristic, complemented
281#1804—471no—
282#1804—471no—
283C2×C47#231111yescharacteristic
284C2×C47C2×#41611yescharacteristic
285C2×C47#289611yescharacteristic
286C2×C47#321112yes—
287C2×C47#323212yes—
288D47—24no—
289D47—82no—
290#529—478no—
291#529—944no—
292C2×#126—941no—
293C2×#125—471no—
294C2×#124—471no—
295C2×#123—941no—
296C2×C22⋊Q8—471no—
297C2×#121—471no—
298C2×C4×Q8C2×D4711yescharacteristic
299C2×C4×D4—471no—
300C2×C4×D4—941no—
301C2×#117—941no—
302C22×C4⋊C4—472no—
303C22×C42—471no—
304#431—944no—
305C2×C4⋊C8—471no—
306C2×C4⋊C8C2×D4711yescharacteristic
307C2×#103—472no—
308C2×#103C2×D4712yes—
309C2×#102—472no—
310C2×#102—472no—
311C2×C22⋊C8—472no—
312C2×C4×C8—472no—
313#382—472no—
314#382—471no—
315#382—941no—
316#382—941no—
317#382C2×D4711yescharacteristic
318C2×#95—941no—
319C2×#95—941no—
320#340—472no—
321#340—472no—
322#340—472no—
323#340C2×D4712yes—
324C47#2186111yescharacteristic, complemented, Sylow
325C22×Q8C22×D4711yescharacteristic
326C22×D4—471no—
327C22×D4—941no—
328C23×C4—471no—
329C23×C4—471no—
330C23×C4—471no—
331C23×C4—942no—
332C23×C4—941no—
333C22×C8—472no—
334C22×C8C22×D4712yes—
335#126—948no—
336#125—474no—
337#125—942no—
338#124—474no—
339#124—942no—
340#123—948no—
341C22⋊Q8—474no—
342C22⋊Q8—942no—
343#121—474no—
344#121—942no—
345C4×Q8—22no—
346C4×Q8C4×D4714yescomplemented
347C4×D4—474no—
348C4×D4—948no—
349C4×D4—942no—
350#117—948no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—471no—
353C2×C4⋊C4—472no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—1882no—
359C2×C4⋊C4—1882no—
360C2×C4⋊C4C22×D4712yes—
361C2×C4⋊C4C22×D4711yescharacteristic
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—941no—
365C2×C22⋊C4—941no—
366C2×C22⋊C4—941no—
367C2×C42—21no—
368C2×C42—471no—
369C2×C42—471no—
370C2×C42—471no—
371C2×C42—474no—
372C2×C42—474no—
373C2×C42C22×D4711yescharacteristic
374C4⋊C8—22no—
375C4⋊C8—942no—
376#103—42no—
377#103—1882no—
378#102—1882no—
379#102—1882no—
380C22⋊C8—944no—
381C4×C8—944no—
382#95—21no—
383#95—21no—
384#95—942no—
385#95—942no—
386#95—941no—
387#95—941no—
388C24—471no—
389C24—941no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×C4×D4712yes—
396C2×D4—472no—
397C2×D4—944no—
398C2×D4—941no—
399C2×D4—941no—
400C2×D4—941no—
401C2×D4—941no—
402C2×D4—941no—
403C2×D4—1884no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—471no—
408C22×C4—471no—
409C22×C4—471no—
410C22×C4—474no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—474no—
414C22×C4—944no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—941no—
418C22×C4—941no—
419C22×C4—941no—
420C22×C4—944no—
421C22×C4—942no—
422C22×C4—1882no—
423C22×C4—1884no—
424C22×C4—1884no—
425C22×C4—1884no—
426C22×C4—1884no—
427C22×C4D4×D4711yescharacteristic
428C22×C4D4×D4711yescharacteristic
429C22×C4C23×D4711yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—942no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—472no—
441C4⋊C4—944no—
442C4⋊C4—944no—
443C4⋊C4—944no—
444C4⋊C4—941no—
445C4⋊C4—942no—
446C4⋊C4—942no—
447C4⋊C4—1884no—
448C4⋊C4C2×C4×D4712yes—
449C22⋊C4—944no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C22⋊C4—944no—
453C22⋊C4—944no—
454C42—24no—
455C42—474no—
456C42—474no—
457C42—472no—
458C42—472no—
459C42C2×C4×D4712yes—
460C42D4×D4712yes—
461C23—472no—
462C23—474no—
463C23—942no—
464C23—941no—
465C23—941no—
466C23—942no—
467C23—1882no—
468C23—1882no—
469C23—1881no—
470C23—1881no—
471C23C2×D4×D4711yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476D4—942no—
477D4—942no—
478D4—1888no—
479D4—1881no—
480D4—1881no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—472no—
494C2×C4—948no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—941no—
498C2×C4—941no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—942no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—3768no—
505C2×C4—1884no—
506C2×C4—1888no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4C22⋊C4×D4712yes—
512C2×C4C22⋊C4×D4712yes—
513C2×C4C22×C4×D4711yescharacteristic
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4#1309406911yescharacteristic
516C2×C4#1309407011yescharacteristic
517C2×C4#1309407111yescharacteristic
518C2×C4#1309407211yescharacteristic
519C8—42no—
520C8—1882no—
521C22—474no—
522C22—942no—
523C22—944no—
524C22—944no—
525C22—942no—
526C22—3762no—
527C22—3762no—
528C22—1882no—
529C22—1882no—
530C22—1882no—
531C22C2×C22⋊C4×D4711yescharacteristic
532C22#1568968911yescharacteristic
533C22#1568969611yescharacteristic
534C22#135×D4712yes—
535C22#1568992212yes—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—942no—
542C4—942no—
543C4—942no—
544C4—3762no—
545C4—1882no—
546C4#1568960311yescharacteristic
547C4#1568960411yescharacteristic
548C4#1568970211yescharacteristic
549C4#1568970311yescharacteristic
550C2—944no—
551C2—3762no—
552C2#2024597711yescharacteristic
553C2#416×D4711yescharacteristic
554C2#2025190611yescharacteristic
555C2#2025435712yes—
556C2#2025456012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#21861
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#15689175C2semidirect
3. #382×D47C2semidirect
4. #20244351C2semidirect
5. #20244196C2semidirect
7. #20243692C2semidirect
8. #20243403C2semidirect
9. #20243399C2semidirect
10. #20243395C2semidirect
11. C47×#1805C2semidirect
13. #15689660C4semidirect
56. #13094015C2×C4semidirect
58. #13094013C2×C4semidirect
66. C4×D188C2×C4semidirect
99. C4×Q8×C47C2×C4semidirect
280. #1805D47semidirect
324. C47#21861semidirect
346. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896602
C2×#156891752
#382×D472
#202443512
#202441962
#202437042
#202436922
#202434032
#202433992
#202433952
C47×#18052
#2186147