Finite Group Databasebeta

← #26215883

Subgroups — #26215883

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094031C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689163C211yesmaximal, characteristic, complemented
4#20245980C211yesmaximal, characteristic
5#20243419C212yesmaximal, complemented
6#20243419C212yesmaximal, complemented
7#20243415C212yesmaximal, complemented
8#20243411C212yesmaximal, complemented
9#20243104C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#2311C211yesmaximal, characteristic
12C2×#13094031—24no—
13C2×#13094031—24no—
14C2×#13094031C418yescomplemented
15C22×#10625329C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C23×C47⋊C8C2212yes—
19C2×#13093983C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21#15689166—24no—
22#15689163—24no—
23C2×#13093957C2212yes—
24C2×#13093957C2212yes—
25C2×#13093955C2212yes—
26C2×#13093955C2212yes—
27#15689030—21no—
28#15689026—21no—
29#15689003—22no—
30#15688983C2212yes—
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38C22×#41×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#391—21no—
42C47×#390—21no—
43C47×#386—22no—
44C47×#380C2212yes—
45C47×#380C2212yes—
46C23×Dic47—21no—
47#13094031—232no—
48C2×#10625329—24no—
49C2×#10625329—24no—
50C2×#10625329C2×C414yes—
51C2×#10625329C2×C418yescomplemented
52C2×#10625329D414yes—
53C2×#10625329D414yes—
54C22×C47⋊C8—22no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—24no—
57C22×C47⋊C8—24no—
58C22×C47⋊C8C2×C414yes—
59C22×C47⋊C8C2×C418yescomplemented
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2312yes—
63#13093983—22no—
64#13093983—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2312yes—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—22no—
73C2×C22⋊Dic47—22no—
74C2×C4×Dic47—21no—
75#13093957—24no—
76#13093957—24no—
77#13093955—24no—
78#13093955—24no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C2×#41×C47—22no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47C2×C414yes—
86C2×#41×C47Q814yes—
87C22×Q8×C47D411yescharacteristic
88C22×D4×C47D412yes—
89C22×D4×C47D411yescharacteristic
90C23×C4×C47—21no—
91C23×C4×C47C2312yes—
92C23×C4×C47C2311yescharacteristic
93#117×C47—22no—
94#117×C47—22no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47C2312yes—
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47—22no—
104C2×C42×C47—21no—
105#95×C47—22no—
106#95×C47—22no—
107C24×C47C2×D412yes—
108C24×C47C2×D411yescharacteristic
109#41×C47—28no—
110#41×C47—28no—
111#41×C47—28no—
112#41×C47—28no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47C4⋊C414yes—
116C2×D4×C47—24no—
117C2×D4×C47—24no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47C4⋊C418yes—
121C2×D4×C47C4⋊C414yes—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—22no—
129C22×C4×C47—42no—
130C22×C4×C47—81no—
131C22×C4×C47—81no—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×Q814yes—
139C22×C4×C47C2×Q812yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—42no—
147C22×Dic47—81no—
148C22×Dic47—81no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—42no—
156C22⋊C4×C47—42no—
157C22⋊C4×C47—82no—
158C22⋊C4×C47—82no—
159C42×C47—42no—
160C22⋊Dic47—42no—
161C22⋊Dic47—82no—
162C22⋊Dic47—82no—
163C4⋊Dic47—21no—
164C4⋊Dic47—22no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—42no—
170C4×Dic47—42no—
171#10625329—216no—
172#10625329C22⋊C4116yescomplemented
173C2×C47⋊C8—28no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—216no—
179C2×C47⋊C8C22×C412yes—
180C2×C47⋊C8C22×C412yes—
181C2×C47⋊C8C22×C414yes—
182C2×C47⋊C8C2×D414yes—
183C2×C47⋊C8C2×D414yes—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47C2×C4⋊C414yes—
195C23×C47C2×C4⋊C412yes—
196C23×C47#12112yes—
197C23×C47#12112yes—
198C23×C47C22×D411yescharacteristic
199Q8×C47—28no—
200D4×C47—216no—
201D4×C47—28no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—42no—
211C2×C4×C47—41no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—84no—
218C2×C4×C47C2×C22⋊C414yes—
219C2×C4×C47C2×C4⋊C414yes—
220C2×C4×C47C2×C4⋊C412yes—
221C2×C4×C47#11714yes—
222C2×C4×C47#12112yes—
223C2×C4×C47#12112yes—
224C2×C4×C47C22⋊Q814yes—
225C2×C4×C47C22⋊Q814yes—
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×Q811yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—84no—
242C47⋊C8—22no—
243C47⋊C8—22no—
244C47⋊C8C2×C22⋊C418yes—
245#20664—471nomaximal, Sylow
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47#38014yes—
260C22×C47C22×C4⋊C411yescharacteristic
261C22×C47C2×#12111yescharacteristic
262C22×C47C2×#12111yescharacteristic
263C22×C47#57611yescharacteristic
264C22×C47#57711yescharacteristic
265C22×C47#57711yescharacteristic
266C22×C47#57811yescharacteristic
267C4×C47—24no—
268C4×C47—81no—
269C4×C47—82no—
270C4×C47—81no—
271C4×C47#38014yes—
272C4×C47C22×C22⋊C411yescharacteristic
273C4×C47C2×#11711yescharacteristic
274C4×C47C2×C22⋊Q811yescharacteristic
275C4×C47C2×C22⋊Q811yescharacteristic
276Dic47—81no—
277Dic47—82no—
278Dic47—81no—
279C22×#131—471no—
280C2×#428—471no—
281C2×#428—471no—
282#2311—471no—
283#2311D4711yescharacteristic
284#1809—472no—
285#1809—472no—
286#1808—472no—
287#1807—472no—
288#1735—471no—
289C2×#340—471no—
290C2×C47—24no—
291C2×C47—44no—
292C2×C47C2×#38011yescharacteristic
293C2×C47#291211yescharacteristic
294C2×C47#291311yescharacteristic
295C2×C47#306211yescharacteristic
296C2×C47#306211yescharacteristic
297C2×C47#306311yescharacteristic
298C2×C47#306311yescharacteristic
299C22×#41C2×D4711yescharacteristic
300C2×#131—478no—
301C2×#131—944no—
302C2×#131—944no—
303C22×M16—471no—
304C22×M16—472no—
305C23×C8—472no—
306C23×C8—471no—
307C2×#117—471no—
308C2×#117C2×D4711yescharacteristic
309C22×C4⋊C4—471no—
310C22×C4⋊C4C2×D4711yescharacteristic
311#428—944no—
312#428—944no—
313C2×#107—472no—
314C2×#107—472no—
315C2×#106—472no—
316C2×#106—472no—
317#391—21no—
318#391—941no—
319#390—21no—
320#390—941no—
321#386—22no—
322#386—942no—
323#380—472no—
324#380—472no—
325#380C2×D4712yes—
326#380C2×D4712yes—
327#340—471no—
328#340—472no—
329#340—471no—
330#340—472no—
331#340—471no—
332#340—471no—
333C47#2066411yescharacteristic, complemented, Sylow
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41Dic9414yes—
339C2×#41C4×D4714yes—
340C22×Q8C22⋊D4711yescharacteristic
341C22×D4C22⋊D4712yes—
342C22×D4C22⋊D4711yescharacteristic
343C23×C4—21no—
344C23×C4—941no—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347#131—9432no—
348C2×M16—474no—
349C2×M16—474no—
350C2×M16—474no—
351C2×M16—478no—
352C2×M16—944no—
353C2×M16—944no—
354C22×C8—472no—
355C22×C8—474no—
356C22×C8—478no—
357C22×C8—471no—
358C22×C8—471no—
359C22×C8—944no—
360C22×C8—944no—
361C22×C8—942no—
362C22×C8—942no—
363#117—22no—
364#117—22no—
365#117—942no—
366#117—942no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4712yes—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—941no—
385C2×C42—21no—
386C2×C42—941no—
387#107—944no—
388#107—944no—
389#106—944no—
390#106—944no—
391#95—22no—
392#95—22no—
393#95—942no—
394#95—942no—
395C24C2×C22⋊D4712yes—
396C24C2×C22⋊D4711yescharacteristic
397#41—28no—
398#41—28no—
399#41—28no—
400#41—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8Dic47⋊C414yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4Dic47⋊C418yes—
409C2×D4Dic47⋊C414yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—942no—
423C22×C4—3761no—
424C22×C4—3761no—
425C22×C4—1882no—
426C22×C4C2×Dic9414yes—
427C22×C4C2×Dic9412yes—
428C22×C4C2×C4×D4714yes—
429C22×C4C2×C4×D4712yes—
430C22×C4D4×D4712yes—
431C22×C4D4×D4712yes—
432C22×C4#1062535212yes—
433C22×C4#1062535212yes—
434C22×C4C2×C22⋊D4712yes—
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C23×D4711yescharacteristic
437M16—4716no—
438M16—9416no—
439C2×C8—474no—
440C2×C8—474no—
441C2×C8—474no—
442C2×C8—472no—
443C2×C8—472no—
444C2×C8—9416no—
445C2×C8—948no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—941no—
458C4⋊C4—942no—
459C4⋊C4—941no—
460C4⋊C4—942no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—1882no—
464C22⋊C4—42no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—3762no—
468C22⋊C4—3762no—
469C22⋊C4—1882no—
470C42—42no—
471C42—1882no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23C2×Dic47⋊C414yes—
483C23C2×Dic47⋊C412yes—
484C23#1309399612yes—
485C23#1309399612yes—
486C23C22×C22⋊D4711yescharacteristic
487Q8—28no—
488D4—216no—
489D4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—84no—
506C2×C4—3764no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1882no—
510C2×C4—1881no—
511C2×C4—1882no—
512C2×C4—1881no—
513C2×C4—1882no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4C22⋊C4×D4714yes—
517C2×C4C2×Dic47⋊C414yes—
518C2×C4C2×Dic47⋊C412yes—
519C2×C4#1309398014yes—
520C2×C4#1309399612yes—
521C2×C4#1309399612yes—
522C2×C4#1309400114yes—
523C2×C4#1309400114yes—
524C2×C4C22×C4×D4711yescharacteristic
525C2×C4C2×D4×D4711yescharacteristic
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4C22×Dic9411yescharacteristic
528C2×C4C2×#1062535211yescharacteristic
529C2×C4C2×#1062535211yescharacteristic
530C8—478no—
531C8—942no—
532C8—942no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22#1568898114yes—
547C22C22×Dic47⋊C411yescharacteristic
548C22C2×#1309399611yescharacteristic
549C22C2×#1309399611yescharacteristic
550C22#1568992611yescharacteristic
551C22#1568993111yescharacteristic
552C22#1568993411yescharacteristic
553C22#1568993811yescharacteristic
554C4—24no—
555C4—82no—
556C4—81no—
557C4—81no—
558C4—3761no—
559C4—3762no—
560C4—3761no—
561C4#1568898114yes—
562C4C2×C22⋊C4×D4711yescharacteristic
563C4C2×#1309398011yescharacteristic
564C4C2×#1309400111yescharacteristic
565C4C2×#1309400111yescharacteristic
566C2—24no—
567C2—44no—
568C2C2×#1568898111yescharacteristic
569C2#2025198111yescharacteristic
570C2#2025198611yescharacteristic
571C2#2025306411yescharacteristic
572C2#2025307011yescharacteristic
573C2#2025307511yescharacteristic
574C2#2025308111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenter#2311D47
hypocenterC47#20664
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689163C2semidirect
5. #20243419C2semidirect
6. #20243419C2semidirect
7. #20243415C2semidirect
8. #20243411C2semidirect
9. #20243104C2semidirect
10. C2×#15688885C2semidirect
14. C2×#13094031C4semidirect
51. C2×#10625329C2×C4semidirect
59. C22×C47⋊C8C2×C4semidirect
172. #10625329C22⋊C4semidirect
333. C47#20664semidirect

Maximal subgroups

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

subgroupindex
C22×#130940312
C2×#156891662
C2×#156891632
#202459802
#202434192
#202434152
#202434112
#202431042
C2×#156888852
C47×#23112
#2066447