Finite Group Databasebeta

← #26226172

Subgroups — #26226172

Order 12032.

1150 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689660C211yesmaximal, characteristic
2C2×C4⋊C8×D47C211yesmaximal, characteristic, complemented
3#20245781C211yesmaximal, characteristic, complemented
4#20244344C212yesmaximal, complemented
5#20244317C212yesmaximal, complemented
6#20243815C212yesmaximal
7#20243808C212yesmaximal, complemented
8#20243403C211yesmaximal, characteristic, complemented
9#20243399C211yesmaximal, characteristic, complemented
10#20243395C211yesmaximal, characteristic, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×C8×D47C2212yes—
13#15689660—24no—
14#15689660C418yescomplemented
15C2×#13094019—21no—
16C2×#13094015C2211yescharacteristic
17C2×#13094013C2211yescharacteristic
18C2×#13094009—21no—
19C2×#13094000C2211yescharacteristic
20C2×#13093999C2211yescharacteristic
21C2×C4×D188—21no—
22C2×C4×D188C2211yescharacteristic
23C2×#13093981—21no—
24C2×C42×D47C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×Dic47⋊C8C2212yes—
27C4⋊C8×D47—24no—
28C2×#13093948C2212yes—
29C2×#13093946C2212yes—
30C2×#13093944C2212yes—
31#15688990—21no—
32#15688989C2211yescharacteristic
33#15688985—21no—
34#15688980C2212yes—
35C4⋊C4×Dic47C2212yes—
36#15688970—21no—
37#15688968—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×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C2×C8×D47—24no—
54C2×C8×D47—42no—
55C2×C8×D47—42no—
56#13094019—28no—
57#13094015—22no—
58#13094015C2×C414yes—
59#13094013—22no—
60#13094013C2×C414yes—
61#13094009—28no—
62#13094000—22no—
63#13094000C2×C414yes—
64#13093999—22no—
65#13093999C2×C414yes—
66C4×D188—28no—
67C4×D188—22no—
68C4×D188C2×C414yes—
69#13093981—28no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47D414yes—
87C4⋊(C47⋊C8)—22no—
88Dic47⋊C8—24no—
89#13093948—42no—
90#13093946—42no—
91#13093944—42no—
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×C414yes—
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×D47C22⋊C414yes—
145C2×C4×D47C22⋊C414yes—
146C2×C4×D47C22×C412yes—
147C2×C4×D47C2×D412yes—
148C2×C8×C47—22no—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C4⋊C4×C47—24no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47C22×C412yes—
156C42×C47—24no—
157C42×C47C22×C412yes—
158C42×C47C2×D412yes—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—24no—
161(C2×D47)⋊C4—24no—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—44no—
165C4⋊Dic47—21no—
166C4⋊Dic47—24no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47C22×C412yes—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172C4×Dic47—44no—
173C4×Dic47C22⋊C414yes—
174C4×Dic47C22⋊C414yes—
175C4×Dic47C22×C412yes—
176C4×Dic47C2×D412yes—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C8×D47—48no—
181C23×C47C22×D411yescharacteristic
182C22×D47—21no—
183C22×D47—21no—
184C22×D47—22no—
185C22×D47—22no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—42no—
189C22×D47—42no—
190C22×D47C2×C22⋊C412yes—
191C22×D47#13514yes—
192Q8×C47—22no—
193Q8×C47—22no—
194Q8×C47—41no—
195Q8×C47—41no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C23×C411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47H(5,𝔽2)11yescharacteristic
211C2×C4×C47H(5,𝔽2)11yescharacteristic
212C2×C4×C472−1+411yescharacteristic
213C2×C4×C472−1+411yescharacteristic
214C2×Dic47—21no—
215C2×Dic47—21no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47C2×C22⋊C412yes—
224C2×Dic47C2×C22⋊C412yes—
225C2×Dic47C2×C22⋊C412yes—
226D188—22no—
227D188—22no—
228D188—48no—
229D188—41no—
230D188—41no—
231C4×D47—28no—
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—22no—
236C4×D47—44no—
237C4×D47—48no—
238C8×C47—42no—
239C47⋊C8—42no—
240#21886—471nomaximal, Sylow
241C22×C47C22×C22⋊C411yescharacteristic
242C22×C47#53711yescharacteristic
243C22×C47#53711yescharacteristic
244C22×C47C2×#13512yes—
245C22×C47#57512yes—
246C2×D47—22no—
247C2×D47—22no—
248C2×D47—24no—
249C2×D47—24no—
250C2×D47—42no—
251C2×D47—42no—
252C2×D47—42no—
253C2×D47—82no—
254C2×D47—82no—
255C2×D47#41614yes—
256C4×C47—22no—
257C4×C47—22no—
258C4×C47—42no—
259C4×C47—42no—
260C4×C47—82no—
261C4×C47#52011yescharacteristic
262C4×C47#52011yescharacteristic
263C4×C47#53511yescharacteristic
264C4×C47#53611yescharacteristic
265Dic47—22no—
266Dic47—22no—
267Dic47—22no—
268Dic47—42no—
269Dic47—82no—
270C2×#529—471no—
271C22×C4⋊C8—471no—
272#2283—471no—
273#1989—472no—
274#1983—472no—
275#1893—472no—
276#1892—472no—
277#1805—471no—
278#1805D4711yescharacteristic, complemented
279#1804—471no—
280#1804—471no—
281C2×C47#231111yescharacteristic
282C2×C47C2×#41611yescharacteristic
283C2×C47#289611yescharacteristic
284C2×C47#325012yes—
285C2×C47#325012yes—
286D47—24no—
287D47—82no—
288C23×C8—472no—
289#529—478no—
290#529—944no—
291C2×#126—941no—
292C2×#125—471no—
293C2×#124—471no—
294C2×#123—941no—
295C2×C22⋊Q8—471no—
296C2×#121—471no—
297C2×C4×Q8C2×D4711yescharacteristic
298C2×C4×D4—471no—
299C2×C4×D4—941no—
300C2×#117—941no—
301C22×C42—471no—
302C2×C4⋊C8—472no—
303C2×C4⋊C8—471no—
304C2×C4⋊C8—944no—
305C2×C4⋊C8C2×D4711yescharacteristic
306C2×#103—472no—
307C2×#103C2×D4712yes—
308C2×#102—472no—
309C2×#102—472no—
310#383—941no—
311#382—471no—
312#382C2×D4711yescharacteristic
313#381—941no—
314#380—472no—
315C4×C4⋊C4—472no—
316C4×C22⋊C4—941no—
317#376—941no—
318#340—472no—
319#340—472no—
320#340—472no—
321#340C2×D4712yes—
322C47#2188611yescharacteristic, complemented, Sylow
323C22×Q8C22×D4711yescharacteristic
324C22×D4—471no—
325C22×D4—941no—
326C23×C4—471no—
327C23×C4—471no—
328C23×C4—471no—
329C23×C4—941no—
330C22×C8—472no—
331C22×C8—944no—
332C22×C8—1882no—
333C22×C8—1882no—
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×D4714yes—
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⋊C4C22×D4712yes—
358C2×C4⋊C4C22×D4711yescharacteristic
359C2×C22⋊C4—941no—
360C2×C22⋊C4—941no—
361C2×C22⋊C4—941no—
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—942no—
365C2×C42—21no—
366C2×C42—471no—
367C2×C42—471no—
368C2×C42—471no—
369C2×C42—474no—
370C2×C42—474no—
371C2×C42—942no—
372C2×C42C22×D4711yescharacteristic
373C4⋊C8—22no—
374C4⋊C8—944no—
375C4⋊C8—942no—
376#103—42no—
377#103—1882no—
378#102—1882no—
379#102—1882no—
380#95—21no—
381#95—21no—
382#95—942no—
383#95—942no—
384#95—941no—
385#95—941no—
386C24—471no—
387C24—941no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C2×C4×D4712yes—
394C2×D4—472no—
395C2×D4—944no—
396C2×D4—941no—
397C2×D4—941no—
398C2×D4—941no—
399C2×D4—941no—
400C2×D4—941no—
401C2×D4—1884no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—471no—
406C22×C4—471no—
407C22×C4—471no—
408C22×C4—474no—
409C22×C4—472no—
410C22×C4—472no—
411C22×C4—474no—
412C22×C4—944no—
413C22×C4—941no—
414C22×C4—941no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—941no—
418C22×C4—944no—
419C22×C4—942no—
420C22×C4—1882no—
421C22×C4—1884no—
422C22×C4D4×D4711yescharacteristic
423C22×C4D4×D4711yescharacteristic
424C22×C4C23×D4711yescharacteristic
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—942no—
429C2×C8—942no—
430C2×C8—942no—
431C2×C8—1888no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—472no—
437C4⋊C4—944no—
438C4⋊C4—944no—
439C4⋊C4—944no—
440C4⋊C4—941no—
441C4⋊C4—942no—
442C4⋊C4—942no—
443C4⋊C4C2×C4×D4712yes—
444C22⋊C4—944no—
445C22⋊C4—944no—
446C22⋊C4—944no—
447C22⋊C4—944no—
448C22⋊C4—944no—
449C22⋊C4—1884no—
450C42—24no—
451C42—474no—
452C42—474no—
453C42—472no—
454C42—472no—
455C42—1884no—
456C42C2×C4×D4712yes—
457C42D4×D4712yes—
458C23—472no—
459C23—474no—
460C23—942no—
461C23—941no—
462C23—941no—
463C23—942no—
464C23—1882no—
465C23—1882no—
466C23—1881no—
467C23—1881no—
468C23C2×D4×D4711yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—942no—
474D4—942no—
475D4—1888no—
476D4—1881no—
477D4—1881no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—472no—
489C2×C4—472no—
490C2×C4—472no—
491C2×C4—948no—
492C2×C4—942no—
493C2×C4—942no—
494C2×C4—941no—
495C2×C4—941no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—1884no—
502C2×C4—1888no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4C22⋊C4×D4712yes—
508C2×C4C22⋊C4×D4712yes—
509C2×C4C22×C4×D4711yescharacteristic
510C2×C4C2×D4×D4711yescharacteristic
511C2×C4#1309406911yescharacteristic
512C2×C4#1309407011yescharacteristic
513C2×C4#1309407111yescharacteristic
514C2×C4#1309407211yescharacteristic
515C8—42no—
516C8—1882no—
517C22—474no—
518C22—942no—
519C22—944no—
520C22—944no—
521C22—942no—
522C22—3762no—
523C22—3762no—
524C22—1882no—
525C22—1882no—
526C22—1882no—
527C22C2×C22⋊C4×D4711yescharacteristic
528C22#1568970211yescharacteristic
529C22#1568970311yescharacteristic
530C22#135×D4712yes—
531C22#1568992212yes—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—942no—
538C4—942no—
539C4—942no—
540C4—3762no—
541C4—1882no—
542C4#1568960311yescharacteristic
543C4#1568960411yescharacteristic
544C4#1568968911yescharacteristic
545C4#1568969611yescharacteristic
546C2—944no—
547C2—3762no—
548C2#2024597711yescharacteristic
549C2#416×D4711yescharacteristic
550C2#2025190611yescharacteristic
551C2#2025473812yes—
552C2#2025474212yes—
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#21886
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×C4⋊C8×D47C2semidirect
3. #20245781C2semidirect
4. #20244344C2semidirect
5. #20244317C2semidirect
7. #20243808C2semidirect
8. #20243403C2semidirect
9. #20243399C2semidirect
10. #20243395C2semidirect
11. C47×#1805C2semidirect
14. #15689660C4semidirect
278. #1805D47semidirect
322. C47#21886semidirect

Maximal subgroups

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

subgroupindex
C2×#156896602
C2×C4⋊C8×D472
#202457812
#202443442
#202443172
#202438152
#202438082
#202434032
#202433992
#202433952
C47×#18052
#2188647