Finite Group Databasebeta

← #26217534

Subgroups — #26217534

Order 12032.

1778 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×#15689694C211yesmaximal, characteristic
2C2×#15689131C211yesmaximal, characteristic, complemented
3C22×#13093947C211yesmaximal, characteristic, complemented
4C2×#15689090C211yesmaximal, characteristic
5#20243700C214yesmaximal, complemented
6#20243695C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243104C211yesmaximal, characteristic, complemented
9C2×C47×#520C211yesmaximal, characteristic
10C23×Dic94C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13#15689694—24no—
14#15689694C418yescomplemented
15C2×#13094022—21no—
16C2×#13094011—21no—
17C2×#13094005—22no—
18C2×#13094003C2214yes—
19C2×#13093983C2211yescharacteristic
20#15689131—24no—
21C2×#13093947—24no—
22C2×#13093947C2214yes—
23C2×#13093947C2212yes—
24C2×#13093947C2212yes—
25#15689090—24no—
26C2×C22⋊(C47⋊C8)C2214yes—
27#15688885C2212yes—
28#15688885C2212yes—
29#15688885C2214yes—
30C22×#41×C47C2211yescharacteristic
31C47×#520—24no—
32C47×#520—24no—
33C2×C4×Q8×C47—21no—
34C2×C4×D4×C47—22no—
35C2×C4×D4×C47—21no—
36C2×#117×C47C2212yes—
37C2×#117×C47C2211yescharacteristic
38C22×C4⋊C4×C47C2212yes—
39C22×C4⋊C4×C47C2211yescharacteristic
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—22no—
44C22×Dic94—21no—
45C22×Dic94—22no—
46C22×Dic94C2×C414yescomplemented
47C22×Dic94C2312yes—
48C23×Dic47—21no—
49C2×#10625329—22no—
50C2×#10625329—22no—
51C2×#10625329—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57#13094022—28no—
58#13094011—28no—
59#13094005—216no—
60#13094003—28no—
61#13094003C2×C4116yescomplemented
62#13093983—22no—
63#13093983C2×C414yescomplemented
64C2×C4⋊Dic47C2312yes—
65C2×Dic47⋊C4—24no—
66C2×C22⋊Dic47—21no—
67C2×C22⋊Dic47—24no—
68C2×C4×Dic47—21no—
69#13093947—28no—
70#13093947—24no—
71#13093947—24no—
72C22⋊(C47⋊C8)—28no—
73C2×#41×C47—22no—
74C2×#41×C47—22no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47C2×C414yes—
78C2×#41×C47D414yes—
79C22×Q8×C47D411yescharacteristic
80C22×D4×C47D412yes—
81C22×D4×C47D411yescharacteristic
82C23×C4×C47—22no—
83C23×C4×C47—21no—
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86C4×Q8×C47—28no—
87C4×D4×C47—216no—
88C4×D4×C47—28no—
89#117×C47—22no—
90#117×C47—24no—
91#117×C47—24no—
92#117×C47D414yes—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—24no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47D414yes—
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—22no—
105C2×C42×C47—21no—
106C24×C47C2×D412yes—
107C24×C47C2×D411yescharacteristic
108#41×C47—28no—
109#41×C47—28no—
110#41×C47—28no—
111#41×C47—28no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22⋊C414yes—
115C2×D4×C47—24no—
116C2×D4×C47—24no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47C22⋊C418yes—
120C2×D4×C47C22⋊C414yes—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47—28no—
129C22×C4×C47—24no—
130C22×C4×C47—82no—
131C22×C4×C47—84no—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C2×Dic94—28no—
149C2×Dic94—28no—
150C2×Dic94—22no—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94—22no—
154C2×Dic94—24no—
155C2×Dic94—22no—
156C2×Dic94—42no—
157C2×Dic94—44no—
158C2×Dic94—44no—
159C2×Dic94—42no—
160C2×Dic94C22×C414yes—
161C4⋊C4×C47—28no—
162C4⋊C4×C47—24no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47C2×D412yes—
167C4⋊C4×C47C2×D414yes—
168C4⋊C4×C47C2×D412yes—
169C22⋊C4×C47—28no—
170C22⋊C4×C47—24no—
171C42×C47—28no—
172C42×C47—24no—
173C22⋊Dic47—24no—
174C22⋊Dic47—216no—
175C4⋊Dic47—22no—
176C4⋊Dic47C22×C414yes—
177Dic47⋊C4—216no—
178C4×Dic47—24no—
179#10625329—44no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—44no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47C2×C22⋊C414yes—
194C23×C47C2×C22⋊C412yes—
195C23×C47#12012yes—
196C23×C47#12312yes—
197C23×C47C22×D411yescharacteristic
198Q8×C47—28no—
199D4×C47—216no—
200D4×C47—28no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—44no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—84no—
214C2×C4×C47—88no—
215C2×C4×C47C2×C22⋊C414yes—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C4×D414yes—
218C2×C4×C47C4×D414yes—
219C2×C4×C47#12012yes—
220C2×C4×C47#12114yes—
221C2×C4×C47#12114yes—
222C2×C4×C47#12312yes—
223C2×C4×C47C23×C411yescharacteristic
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×D411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—48no—
236Dic94—24no—
237Dic94—24no—
238Dic94—42no—
239Dic94—42no—
240Dic94—48no—
241C47⋊C8—42no—
242C47⋊C8—42no—
243#20913—471nomaximal, Sylow
244C22×C47—24no—
245C22×C47—24no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—28no—
249C22×C47—24no—
250C22×C47—44no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47#38614yes—
254C22×C47C22×C22⋊C411yescharacteristic
255C22×C47C2×#12011yescharacteristic
256C22×C47C2×#12311yescharacteristic
257C22×C47#57712yes—
258C22×C47#57812yes—
259C4×C47—24no—
260C4×C47—44no—
261C4×C47—42no—
262C4×C47—42no—
263C4×C47#38614yes—
264C4×C47C2×C4×D411yescharacteristic
265C4×C47C2×C4×D411yescharacteristic
266C4×C47C2×#12111yescharacteristic
267C4×C47C2×#12111yescharacteristic
268Dic47—44no—
269Dic47—44no—
270C2×#536—471no—
271C2×#520D4711yescharacteristic
272C2×#419—471no—
273C22×#103—471no—
274C2×#408—471no—
275#1873—474no—
276#1872—474no—
277#1736—471no—
278#1735—471no—
279C2×C47—24no—
280C2×C47—44no—
281C2×C47C2×#38611yescharacteristic
282C2×C47#289812yes—
283C2×C47#300911yescharacteristic
284C2×C47#301111yescharacteristic
285C2×C47#309712yes—
286C22×#41C2×D4711yescharacteristic
287C23×Q8—471no—
288C22×M16—471no—
289C23×C8—471no—
290#536—478no—
291#536—944no—
292C2×C4⋊Q8—941no—
293C2×#124—941no—
294C2×#123—942no—
295C2×C22⋊Q8—474no—
296#520—24no—
297#520—24no—
298C2×C4×Q8—21no—
299C2×C4×D4—22no—
300C2×C4×D4—21no—
301C2×#117—471no—
302C2×#117C2×D4712yes—
303C2×#117C2×D4711yescharacteristic
304C22×C4⋊C4C2×D4712yes—
305C22×C4⋊C4C2×D4711yescharacteristic
306#419—944no—
307C2×#103—474no—
308C2×#103—472no—
309C2×#103—472no—
310C2×#103—944no—
311#408—944no—
312C2×C22⋊C8—474no—
313#340—472no—
314#340—474no—
315#340—472no—
316C47#2091311yescharacteristic, complemented, Sylow
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41C4×D4714yes—
322C2×#41D18814yes—
323C22×Q8—474no—
324C22×Q8—472no—
325C22×Q8—942no—
326C22×Q8—941no—
327C22×Q8—942no—
328C22×Q8—942no—
329C22×Q8—942no—
330C22×Q8—942no—
331C22×Q8C22⋊D4711yescharacteristic
332C22×D4C22⋊D4712yes—
333C22×D4C22⋊D4711yescharacteristic
334C23×C4—22no—
335C23×C4—21no—
336C23×C4—941no—
337C23×C4C22×D4712yes—
338C23×C4C22×D4711yescharacteristic
339C2×M16—942no—
340C2×M16—942no—
341C2×M16—942no—
342C22×C8—472no—
343C22×C8—472no—
344C22×C8—942no—
345C22×C8—942no—
346C22×C8—942no—
347C4⋊Q8—948no—
348#124—948no—
349#123—9416no—
350C22⋊Q8—4716no—
351C22⋊Q8—948no—
352C4×Q8—28no—
353C4×D4—216no—
354C4×D4—28no—
355#117—22no—
356#117—24no—
357#117—24no—
358#117—474no—
359#117—942no—
360#117C22⋊D4714yes—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4C22⋊D4714yes—
369C2×C4⋊C4C22×D4711yescharacteristic
370C2×C4⋊C4C22×D4712yes—
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—941no—
375C2×C22⋊C4—944no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—941no—
379#103—948no—
380#103—944no—
381#103—944no—
382C22⋊C8—948no—
383C24C2×C22⋊D4712yes—
384C24C2×C22⋊D4711yescharacteristic
385#41—28no—
386#41—28no—
387#41—28no—
388#41—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—474no—
392C2×Q8—944no—
393C2×Q8—942no—
394C2×Q8—948no—
395C2×Q8—948no—
396C2×Q8—942no—
397C2×Q8—942no—
398C2×Q8—942no—
399C2×Q8—942no—
400C2×Q8—1882no—
401C2×Q8—1884no—
402C2×Q8—1884no—
403C2×Q8—1882no—
404C2×Q8(C2×D47)⋊C414yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4(C2×D47)⋊C418yes—
410C2×D4(C2×D47)⋊C414yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—82no—
421C22×C4—84no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4C2×C4×D4714yes—
428C22×C4C2×C4×D4712yes—
429C22×C4C2×D18814yes—
430C22×C4C2×D18812yes—
431C22×C4#1062535012yes—
432C22×C4#1062535012yes—
433C22×C4C2×C22⋊D4712yes—
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4712yes—
436C22×C4C2×C22⋊D4712yes—
437C22×C4C23×D4711yescharacteristic
438M16—1884no—
439C2×C8—942no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—1884no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—474no—
452C4⋊C4—9416no—
453C4⋊C4—942no—
454C4⋊C4C2×C22⋊D4712yes—
455C4⋊C4C2×C22⋊D4714yes—
456C4⋊C4C2×C22⋊D4712yes—
457C22⋊C4—28no—
458C22⋊C4—24no—
459C22⋊C4—944no—
460C22⋊C4—9416no—
461C42—28no—
462C42—24no—
463C42—944no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23C2×(C2×D47)⋊C414yes—
471C23C2×(C2×D47)⋊C412yes—
472C23#1309399312yes—
473C23#1309400712yes—
474C23C22×C22⋊D4711yescharacteristic
475Q8—28no—
476Q8—944no—
477Q8—944no—
478Q8—1888no—
479Q8—1882no—
480Q8—1882no—
481D4—216no—
482D4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4—944no—
498C2×C4—944no—
499C2×C4—1888no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4C2×(C2×D47)⋊C414yes—
505C2×C4C2×(C2×D47)⋊C412yes—
506C2×C4C4×C22⋊D4714yes—
507C2×C4C4×C22⋊D4714yes—
508C2×C4#1309399312yes—
509C2×C4#1309399814yes—
510C2×C4#1309399814yes—
511C2×C4#1309400712yes—
512C2×C4C22×C4×D4711yescharacteristic
513C2×C4C22×C22⋊D4711yescharacteristic
514C2×C4C22×C22⋊D4711yescharacteristic
515C2×C4C22×D18811yescharacteristic
516C2×C4C2×#1062535011yescharacteristic
517C2×C4C2×#1062535011yescharacteristic
518C8—1882no—
519C8—1882no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—28no—
525C22—24no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22#1568900214yes—
530C22C22×(C2×D47)⋊C411yescharacteristic
531C22C2×#1309399311yescharacteristic
532C22C2×#1309400711yescharacteristic
533C22#1568993412yes—
534C22#1568993812yes—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—1884no—
540C4—1884no—
541C4#1568900214yes—
542C4C2×C4×C22⋊D4711yescharacteristic
543C4C2×C4×C22⋊D4711yescharacteristic
544C4C2×#1309399811yescharacteristic
545C4C2×#1309399811yescharacteristic
546C2—24no—
547C2—44no—
548C2C2×#1568900211yescharacteristic
549C2#2025191812yes—
550C2#2025266511yescharacteristic
551C2#2025268311yescharacteristic
552C2#2025335712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47#20913
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×#15689131C2semidirect
3. C22×#13093947C2semidirect
5. #20243700C2semidirect
6. #20243695C2semidirect
7. #20243107C2semidirect
8. #20243104C2semidirect
14. #15689694C4semidirect
46. C22×Dic94C2×C4semidirect
61. #13094003C2×C4semidirect
63. #13093983C2×C4semidirect
316. C47#20913semidirect

Maximal subgroups

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

subgroupindex
C2×#156896942
C2×#156891312
C22×#130939472
C2×#156890902
#202437002
#202436952
#202431072
#202431042
C2×C47×#5202
#2091347