Finite Group Databasebeta

← #26250122

Subgroups — #26250122

Order 12032.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689451C211yesmaximal, characteristic, complemented
2C2×#15689166C211yesmaximal, characteristic, complemented
3#20247705C211yesmaximal, characteristic
4#20244723C212yesmaximal, complemented
5#20244588C212yesmaximal, complemented
6#20243822C211yesmaximal, characteristic, complemented
7#20243686C211yesmaximal, characteristic
8#20243656C212yesmaximal, complemented
9#20243581C212yesmaximal, complemented
10#20243483C211yesmaximal, characteristic, complemented
11C2×C47×#552C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689451—24no—
15#15689166—24no—
16C2×#13093957C2212yes—
17C2×#13093955C2212yes—
18C2×#13093955C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×C22⋊(C47⋊C8)C2211yescharacteristic
22C2×C22⋊(C47⋊C8)C2212yes—
23#15689062—21no—
24#15689059—21no—
25#15689021C2212yes—
26#15689003—21no—
27#15688995—21no—
28#15688983C2211yescharacteristic
29#15688983C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34C47×#552—24no—
35C47×#552—24no—
36C2×#126×C47—21no—
37C2×#125×C47—21no—
38C2×#125×C47C2211yescharacteristic
39C2×#123×C47—21no—
40C2×C22⋊Q8×C47C2211yescharacteristic
41C2×C22⋊Q8×C47C2211yescharacteristic
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×#117×C47—21no—
45C2×#117×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C23×Dic47—21no—
48C2×#10625329—22no—
49C2×#10625329—24no—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47C2312yes—
56C2×C4⋊Dic47C2312yes—
57C2×Dic47⋊C4—22no—
58C2×C22⋊Dic47—21no—
59C2×C22⋊Dic47—22no—
60#13093957—24no—
61#13093955—24no—
62#13093955—24no—
63#13093948—24no—
64#13093948—24no—
65C22⋊(C47⋊C8)—22no—
66C22⋊(C47⋊C8)—24no—
67C47⋊#95—22no—
68C47⋊#95—21no—
69C47⋊#95—22no—
70C22×Q8×C47C2311yescharacteristic
71C22×D4×C47D411yescharacteristic
72C23×C4×C47—21no—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75#126×C47—28no—
76#125×C47—28no—
77#125×C47—22no—
78#125×C47—22no—
79#123×C47—28no—
80C22⋊Q8×C47—22no—
81C22⋊Q8×C47—22no—
82C22⋊Q8×C47—22no—
83C22⋊Q8×C47D414yes—
84C4×Q8×C47—22no—
85C4×Q8×C47—22no—
86C4×D4×C47—22no—
87C4×D4×C47—22no—
88#117×C47—28no—
89#117×C47—22no—
90#117×C47D414yes—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47D411yescharacteristic
107C2×C22⋊C4×C47D411yescharacteristic
108C2×C42×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47C2311yescharacteristic
111C24×C47C2×D411yescharacteristic
112C24×C47C2×D411yescharacteristic
113C2×Q8×C47—21no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47C2×D412yes—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—41no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—24no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—42no—
143C22×Dic47—82no—
144C4⋊C4×C47—24no—
145C4⋊C4×C47—24no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47C2×D412yes—
155C4⋊C4×C47C2×D412yes—
156C4⋊C4×C47C2×D412yes—
157C22⋊C4×C47—24no—
158C22⋊C4×C47—24no—
159C22⋊C4×C47—24no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C42×C47—24no—
163C42×C47—24no—
164C42×C47—21no—
165C42×C47—21no—
166C22⋊Dic47—42no—
167C22⋊Dic47—82no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—42no—
173Dic47⋊C4—82no—
174#10625329—28no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8#4114yes—
181C23×C47—21no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47#12012yes—
190C23×C47#13712yes—
191C23×C47C22×D411yescharacteristic
192Q8×C47—24no—
193Q8×C47—41no—
194Q8×C47—41no—
195D4×C47—44no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—24no—
200C2×C4×C47—21no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—48no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—42no—
221C2×C4×C47#12012yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C2×#4112yes—
225C2×C4×C472−1+412yes—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—84no—
233C47⋊C8—24no—
234C47⋊C8—42no—
235#24111—471nomaximal, Sylow
236C22×C47—24no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—42no—
242C22×C47—42no—
243C22×C47—42no—
244C22×C47—41no—
245C22×C47—41no—
246C22×C47C2×#12011yescharacteristic
247C22×C47#54712yes—
248C22×C47C2×#13711yescharacteristic
249C22×C47#57511yescharacteristic
250C22×C47#57712yes—
251C4×C47—22no—
252C4×C47—41no—
253C4×C47—42no—
254C4×C47—41no—
255C4×C47—42no—
256C4×C47—42no—
257C4×C47—41no—
258C4×C47—42no—
259C4×C47#52511yescharacteristic
260C4×C47#53611yescharacteristic
261C4×C47#54712yes—
262Dic47—82no—
263Dic47—82no—
264C2×#552D4711yescharacteristic
265C2×#483—471no—
266C2×#428—471no—
267#2487—471no—
268#2059—472no—
269#2036—472no—
270#1894—471no—
271#1870—471no—
272#1865—472no—
273#1853—472no—
274#1825—471no—
275C2×C47—42no—
276C2×C47—42no—
277C2×C47#240611yescharacteristic
278C2×C47#300711yescharacteristic
279C2×C47#301011yescharacteristic
280C2×C47#334412yes—
281C2×C47#335112yes—
282C22×M16—471no—
283#552—24no—
284#552—24no—
285C2×#126—21no—
286C2×#125—21no—
287C2×#125C2×D4711yescharacteristic
288C2×#123—21no—
289C2×C22⋊Q8C2×D4711yescharacteristic
290C2×C22⋊Q8C2×D4711yescharacteristic
291C2×C4×Q8C2×D4711yescharacteristic
292C2×C4×D4C2×D4711yescharacteristic
293C2×#117—21no—
294C2×#117C2×D4711yescharacteristic
295C22×C4⋊C4—471no—
296C22×C4⋊C4C2×D4711yescharacteristic
297#483—944no—
298#428—944no—
299C2×#107—472no—
300C2×#106—472no—
301C2×#106—472no—
302C2×#103—472no—
303C2×#103—472no—
304C2×C22⋊C8—471no—
305C2×C22⋊C8—472no—
306#400—941no—
307#399—941no—
308#389—472no—
309#386—941no—
310#385—941no—
311#380—471no—
312#380—472no—
313#340—472no—
314#340—472no—
315#340—472no—
316#336—472no—
317C47#2411111yescharacteristic, complemented, Sylow
318C22×Q8C22×D4711yescharacteristic
319C22×D4C22⋊D4711yescharacteristic
320C23×C4—21no—
321C23×C4—941no—
322C23×C4C22×D4711yescharacteristic
323C23×C4C22×D4711yescharacteristic
324C2×M16—944no—
325C2×M16—942no—
326C22×C8—472no—
327C22×C8—472no—
328#126—28no—
329#125—28no—
330#125—22no—
331#125—22no—
332#123—28no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8C22⋊D4714yes—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—28no—
342#117—22no—
343#117C22⋊D4714yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—472no—
353C2×C4⋊C4—472no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4C22⋊D4711yescharacteristic
359C2×C4⋊C4C22×D4711yescharacteristic
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C4⋊C4C22×D4711yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—942no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4C22⋊D4711yescharacteristic
368C2×C22⋊C4C22⋊D4711yescharacteristic
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C22×D4711yescharacteristic
372#107—944no—
373#106—944no—
374#106—944no—
375#103—944no—
376#103—944no—
377C22⋊C8—944no—
378C22⋊C8—942no—
379#95—942no—
380#95—941no—
381#95—942no—
382C24C2×C22⋊D4711yescharacteristic
383C24C2×C22⋊D4711yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D4712yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—3762no—
407C22×C4—1882no—
408C22×C4C2×C22⋊D4711yescharacteristic
409C22×C4C2×C22⋊D4711yescharacteristic
410C22×C4C2×C22⋊D4711yescharacteristic
411C22×C4C2×C22⋊D4711yescharacteristic
412C22×C4C2×C22⋊D4712yes—
413C22×C4C2×C22⋊D4712yes—
414C22×C4C23×D4711yescharacteristic
415M16—948no—
416C2×C8—474no—
417C2×C8—942no—
418C2×C8—942no—
419C2×C8—942no—
420C2×C8—942no—
421C2×C8—942no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—942no—
433C4⋊C4—942no—
434C4⋊C4—942no—
435C4⋊C4—942no—
436C4⋊C4—3762no—
437C4⋊C4—1882no—
438C4⋊C4C2×C22⋊D4712yes—
439C4⋊C4C2×C22⋊D4712yes—
440C4⋊C4C2×C22⋊D4712yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—3762no—
447C22⋊C4—1882no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23#1309399312yes—
461C23#1309405412yes—
462C23C22×C22⋊D4711yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—3764no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4#1309399312yes—
500C2×C4C22×C22⋊D4711yescharacteristic
501C2×C4C22×C22⋊D4711yescharacteristic
502C2×C4#41×D4712yes—
503C2×C4#1309407112yes—
504C8—944no—
505C8—1882no—
506C22—22no—
507C22—24no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22C2×#1309399311yescharacteristic
517C22#1568977212yes—
518C22C2×#1309405411yescharacteristic
519C22#1568992111yescharacteristic
520C22#1568993112yes—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—3762no—
530C4—3762no—
531C4#1568963011yescharacteristic
532C4#1568969411yescharacteristic
533C4#1568977212yes—
534C2—42no—
535C2—42no—
536C2#2024671911yescharacteristic
537C2#2025264811yescharacteristic
538C2#2025267411yescharacteristic
539C2#2025584312yes—
540C2#2025594812yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#552C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenterC2×#552D47
hypocenterC47#24111
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#15689451C2semidirect
2. C2×#15689166C2semidirect
4. #20244723C2semidirect
5. #20244588C2semidirect
6. #20243822C2semidirect
8. #20243656C2semidirect
9. #20243581C2semidirect
10. #20243483C2semidirect
317. C47#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
C2×#156891662
#202477052
#202447232
#202445882
#202438222
#202436862
#202436562
#202435812
#202434832
C2×C47×#5522
#2411147