Finite Group Databasebeta

← #26249557

Subgroups — #26249557

Order 12032.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689444C212yesmaximal, complemented
2#20245871C211yesmaximal, characteristic, complemented
3#20244718C211yesmaximal, characteristic, complemented
4#20244713C211yesmaximal, characteristic, complemented
5#20244619C211yesmaximal, characteristic, complemented
6#20244619C211yesmaximal, characteristic, complemented
7#20243711C212yesmaximal, complemented
8#20243649C212yesmaximal, complemented
9#20243635C212yesmaximal, complemented
10#20243456C211yesmaximal, characteristic, complemented
11C2×C47×#532C211yesmaximal, characteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14#15689444—28no—
15C2×#13093957C2212yes—
16C2×#13093955C2212yes—
17C2×#13093945C2212yes—
18C2×#13093945C2212yes—
19C2×#13093945C2212yes—
20C2×#13093945C2212yes—
21C2×C22⋊(C47⋊C8)C2212yes—
22#15689021C2211yescharacteristic
23#15689021C2211yescharacteristic
24#15689021C2211yescharacteristic
25#15689021C2211yescharacteristic
26#15688996—21no—
27#15688996—21no—
28C4×Dic47⋊C4—21no—
29C4×Dic47⋊C4—21no—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688878C2211yescharacteristic
33#15688878C2211yescharacteristic
34#15688878C2212yes—
35C22×#41×C47—21no—
36C47×#532—28no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#127×C47C2211yescharacteristic
39C2×#124×C47—21no—
40C2×#121×C47—21no—
41C2×#121×C47C2212yes—
42C2×C4×D4×C47—21no—
43C2×C4×D4×C47C2212yes—
44C22×C42×C47C2211yescharacteristic
45C22×C47⋊C8C2312yes—
46C22×C47⋊C8C2312yes—
47#13093983—22no—
48#13093983—22no—
49#13093983—24no—
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2311yescharacteristic
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2311yescharacteristic
54C2×Dic47⋊C4—21no—
55C2×Dic47⋊C4—21no—
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C2×C22⋊Dic47—21no—
59C2×C22⋊Dic47—21no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47—21no—
62#13093957—24no—
63#13093955—24no—
64#13093945—24no—
65#13093945—24no—
66#13093945—24no—
67#13093945—24no—
68C22⋊(C47⋊C8)—24no—
69C47⋊#95—21no—
70C47⋊#95—21no—
71C47⋊#95—21no—
72C47⋊#95—21no—
73C2×#41×C47—28no—
74C2×#41×C47—44no—
75C2×#41×C47—44no—
76C22×Q8×C47—21no—
77C22×D4×C47—22no—
78C22×D4×C47—21no—
79C22×D4×C47—21no—
80C22×D4×C47C2312yes—
81C23×C4×C47—22no—
82C23×C4×C47D411yescharacteristic
83C23×C4×C47D411yescharacteristic
84C23×C4×C47C2311yescharacteristic
85C4⋊Q8×C47—24no—
86#127×C47—24no—
87#124×C47—28no—
88#121×C47—28no—
89#121×C47—24no—
90#121×C47D418yes—
91C4×D4×C47—28no—
92C4×D4×C47—24no—
93C4×D4×C47—24no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47C2312yes—
96C2×C22⋊C4×C47—22no—
97C2×C22⋊C4×C47—22no—
98C2×C42×C47—24no—
99C2×C42×C47D411yescharacteristic
100C2×C42×C47D411yescharacteristic
101C2×C42×C47C2311yescharacteristic
102C2×C42×C47C2311yescharacteristic
103C24×C47—22no—
104C24×C47C2×D411yescharacteristic
105#41×C47—432no—
106C2×Q8×C47—24no—
107C2×Q8×C47—42no—
108C2×Q8×C47—42no—
109C2×D4×C47—24no—
110C2×D4×C47—24no—
111C2×D4×C47—28no—
112C2×D4×C47—22no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—42no—
118C2×D4×C47—42no—
119C2×D4×C47—44no—
120C2×D4×C47—44no—
121C2×D4×C47—42no—
122C2×D4×C47—42no—
123C2×D4×C47C2×D414yes—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—28no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—41no—
135C22×C4×C47—44no—
136C22×C4×C47—44no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D414yes—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47C2×D414yes—
151C22⋊C4×C47—28no—
152C22⋊C4×C47—28no—
153C42×C47—22no—
154C42×C47—22no—
155C42×C47—22no—
156C42×C47—22no—
157C22⋊Dic47—24no—
158C22⋊Dic47—42no—
159C4⋊Dic47—21no—
160C4⋊Dic47—21no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47#4112yescomplemented
164C4⋊Dic47#4112yescomplemented
165Dic47⋊C4—42no—
166Dic47⋊C4—42no—
167Dic47⋊C4—42no—
168Dic47⋊C4—42no—
169C4×Dic47—24no—
170C4×Dic47—42no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47—24no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47#12012yes—
191C23×C47C22×D411yescharacteristic
192Q8×C47—48no—
193D4×C47—24no—
194D4×C47—24no—
195D4×C47—48no—
196D4×C47—48no—
197D4×C47—416no—
198D4×C47—42no—
199D4×C47—42no—
200C2×C4×C47—28no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—21no—
212C2×C4×C47—24no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—44no—
216C2×C4×C47—416no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47#12012yes—
220C2×C4×C47#13512yes—
221C2×C4×C47AGL(1,ℤ/8)12yes—
222C2×C4×C47AGL(1,ℤ/8)12yes—
223C2×C4×C47C22×D412yes—
224C2×C4×C47C2×#4111yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C472−1+412yes—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C47⋊C8—42no—
240C47⋊C8—42no—
241#24062—471nomaximal, Sylow
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—41no—
246C22×C47—41no—
247C22×C47—41no—
248C22×C47—41no—
249C22×C47—44no—
250C22×C47—44no—
251C22×C47—44no—
252C22×C47—82no—
253C22×C47—82no—
254C22×C47—82no—
255C22×C47—82no—
256C22×C47C2×#12011yescharacteristic
257C22×C47#52511yescharacteristic
258C22×C47#53611yescharacteristic
259C22×C47C2×#13511yescharacteristic
260C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C47#57511yescharacteristic
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—22no—
267C4×C47—41no—
268C4×C47—44no—
269C4×C47#54712yes—
270C4×C47#54712yes—
271Dic47—42no—
272Dic47—42no—
273Dic47—81no—
274Dic47—81no—
275C2×#532D4711yescharacteristic
276C2×#482—472no—
277#2296—471no—
278#2058—471no—
279#2057—471no—
280#2041—471no—
281#2041—471no—
282#1875—472no—
283#1864—472no—
284#1862—472no—
285#1817—471no—
286C2×C47—42no—
287C2×C47—84no—
288C2×C47#240611yescharacteristic
289C2×C47#300211yescharacteristic
290C2×C47#300311yescharacteristic
291C2×C47#310611yescharacteristic
292C2×C47#311511yescharacteristic
293C2×C47#322811yescharacteristic
294C2×C47#323511yescharacteristic
295C22×#41—21no—
296#532—28no—
297C2×C4⋊Q8C2×D4711yescharacteristic
298C2×#127C2×D4711yescharacteristic
299C2×#124—21no—
300C2×#121—21no—
301C2×#121C2×D4712yes—
302C2×C4×D4—21no—
303C2×C4×D4C2×D4712yes—
304C2×#117—471no—
305C2×#117—471no—
306C22×C42C2×D4711yescharacteristic
307#482—948no—
308C2×#107—472no—
309C2×#106—472no—
310C2×#102—472no—
311C2×#102—472no—
312C2×#102—472no—
313C2×#102—472no—
314C2×C22⋊C8—472no—
315#389—471no—
316#389—471no—
317#389—471no—
318#389—471no—
319#385—941no—
320#385—941no—
321C4×C4⋊C4—941no—
322C4×C4⋊C4—941no—
323#340—472no—
324#340—472no—
325#336—471no—
326#336—471no—
327#336—472no—
328C47#2406211yescharacteristic, complemented, Sylow
329C2×#41—28no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C22×D4712yes—
337C23×C4—22no—
338C23×C4C22⋊D4711yescharacteristic
339C23×C4C22⋊D4711yescharacteristic
340C23×C4C22×D4711yescharacteristic
341C22×C8—472no—
342C22×C8—472no—
343C4⋊Q8—24no—
344#127—24no—
345#124—28no—
346#121—28no—
347#121—24no—
348#121C22⋊D4718yes—
349C4×D4—28no—
350C4×D4—24no—
351C4×D4—24no—
352#117—942no—
353#117—942no—
354#117—944no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—471no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4C22×D4712yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—941no—
368C2×C22⋊C4—941no—
369C2×C42—24no—
370C2×C42—941no—
371C2×C42—941no—
372C2×C42C22⋊D4711yescharacteristic
373C2×C42C22⋊D4711yescharacteristic
374C2×C42C22×D4711yescharacteristic
375C2×C42C22×D4711yescharacteristic
376#107—944no—
377#106—944no—
378#102—944no—
379#102—944no—
380#102—944no—
381#102—944no—
382C22⋊C8—944no—
383#95—941no—
384#95—941no—
385#95—941no—
386#95—941no—
387C24—22no—
388C24C2×C22⋊D4711yescharacteristic
389#41—432no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×C22⋊D4714yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4C2×C22⋊D4711yescharacteristic
426C22×C4C2×C22⋊D4711yescharacteristic
427C22×C4C2×C22⋊D4714yes—
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C23×D4711yescharacteristic
432C2×C8—942no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—942no—
436C2×C8—942no—
437C2×C8—942no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—472no—
441C4⋊C4—472no—
442C4⋊C4—941no—
443C4⋊C4—941no—
444C4⋊C4—942no—
445C4⋊C4—942no—
446C4⋊C4—1882no—
447C4⋊C4—1882no—
448C4⋊C4—1882no—
449C4⋊C4—1882no—
450C4⋊C4C2×C22⋊D4714yes—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—944no—
454C22⋊C4—1882no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—944no—
460C42—1882no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23#1309399312yes—
475C23C22×C22⋊D4711yescharacteristic
476Q8—48no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—416no—
482D4—42no—
483D4—42no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—942no—
504C2×C4—942no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4#1309399312yes—
516C2×C4#1309404412yes—
517C2×C4#1309405012yes—
518C2×C4#1309405012yes—
519C2×C4C22×C22⋊D4712yes—
520C2×C4#41×D4711yescharacteristic
521C2×C4#41×D4711yescharacteristic
522C2×C4#1309407112yes—
523C8—1882no—
524C8—1882no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22C2×#1309399311yescharacteristic
540C22#1568963011yescharacteristic
541C22#1568969411yescharacteristic
542C22C2×#1309404411yescharacteristic
543C22C2×#1309405011yescharacteristic
544C22C2×#1309405011yescharacteristic
545C22#1568992111yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—44no—
552C4—3761no—
553C4—3761no—
554C4—1882no—
555C4—1882no—
556C4#1568977212yes—
557C4#1568977212yes—
558C2—42no—
559C2—84no—
560C2#2024671911yescharacteristic
561C2#2025260611yescharacteristic
562C2#2025261511yescharacteristic
563C2#2025344211yescharacteristic
564C2#2025353311yescharacteristic
565C2#2025452011yescharacteristic
566C2#2025458211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#532C2
socleC23×C47C22×D4
hypercenterC2×#532D47
hypocenterC47#24062
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#15689444C2semidirect
2. #20245871C2semidirect
3. #20244718C2semidirect
4. #20244713C2semidirect
5. #20244619C2semidirect
6. #20244619C2semidirect
7. #20243711C2semidirect
8. #20243649C2semidirect
9. #20243635C2semidirect
10. #20243456C2semidirect
163. C4⋊Dic47#41semidirect
164. C4⋊Dic47#41semidirect
328. C47#24062semidirect

Maximal subgroups

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

subgroupindex
C2×#156894442
#202458712
#202447182
#202447132
#202446192
#202437112
#202436492
#202436352
#202434562
C2×C47×#5322
#2406247