Finite Group Databasebeta

← #26235731

Subgroups — #26235731

Order 12032.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094044C211yesmaximal, characteristic
2#20252056C211yesmaximal, characteristic, complemented
3#20251956C211yesmaximal, characteristic, complemented
4#20244100C212yesmaximal, complemented
5#20243990C212yesmaximal, complemented
6#20243713C212yesmaximal
7#20243525C211yesmaximal, characteristic, complemented
8#20243409C211yesmaximal, characteristic, complemented
9#20243257C211yesmaximal, characteristic, complemented
10#20243233C212yesmaximal, complemented
11C47×#1806C211yesmaximal, characteristic, complemented
12C2×#13094044C2211yescharacteristic
13C2×#13094031C2212yes—
14C4×#10625350C2211yescharacteristic
15C4×#10625337C411yescharacteristic, complemented
16C4×#10625337C411yescharacteristic, complemented
17C4×#10625336C411yescharacteristic, complemented
18C4×#10625336C411yescharacteristic, complemented
19C4×#10625335C411yescharacteristic, complemented
20C4×#10625335C411yescharacteristic, complemented
21C4×#10625334C411yescharacteristic, complemented
22C4×#10625334C411yescharacteristic, complemented
23#15689183—22no—
24#15689179—21no—
25#15689175—21no—
26#15689159C2211yescharacteristic
27#15689150—22no—
28#15689150C2212yes—
29#15689147—21no—
30#15689147C2211yescharacteristic
31C2×C4⋊(C47⋊C8)—21no—
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33C2×#13093952C2212yes—
34#15689119C2211yescharacteristic
35#15689117C2211yescharacteristic
36#15689091C2212yes—
37#15689081C2212yes—
38C2×#13093935C2212yes—
39C2×C4×C47⋊C8C2211yescharacteristic
40#15688887C2212yes—
41#15688887C2212yes—
42#15688883C2212yes—
43#15688878C2212yes—
44C4×#41×C47C2211yescharacteristic
45C47×#423C2211yescharacteristic
46C2×#104×C47C2212yes—
47C4×C4⋊C4×C47C2211yescharacteristic
48C47×#339C2212yes—
49C2×#10625350C2311yescharacteristic
50#13094044—22no—
51#13094044—22no—
52C2×#10625337C2×C411yescharacteristic
53C2×#10625335C2×C411yescharacteristic
54C2×#10625336C2×C411yescharacteristic
55C2×#10625334C2×C411yescharacteristic
56#13094031—24no—
57#13094031—42no—
58C2×#10625329—22no—
59C2×#10625329C2312yes—
60C22×C47⋊C8—22no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2312yes—
64C4×Dic94C2×C411yescharacteristic
65C4×Dic94C2×C411yescharacteristic
66C4×D188C2×C411yescharacteristic
67C4×D188C2×C411yescharacteristic
68C4×C22⋊D47—21no—
69C4×C22⋊D47—21no—
70#13093983C2311yescharacteristic
71#13093981—21no—
72C42×D47—21no—
73#13093957C2×C411yescharacteristic
74#13093957C2×C411yescharacteristic
75#13093955C2×C411yescharacteristic
76#13093955C2×C411yescharacteristic
77C4⋊(C47⋊C8)—21no—
78C4⋊(C47⋊C8)—21no—
79C4⋊(C47⋊C8)—22no—
80C4⋊(C47⋊C8)—22no—
81C4⋊(C47⋊C8)D411yescharacteristic
82C4⋊(C47⋊C8)D411yescharacteristic
83C4⋊(C47⋊C8)D412yes—
84#13093952—42no—
85#13093948C2×C411yescharacteristic
86#13093948C2×C411yescharacteristic
87#13093947C2×C411yescharacteristic
88#13093947C2×C411yescharacteristic
89#13093945C2×C411yescharacteristic
90#13093945C2×C411yescharacteristic
91#13093944C2×C411yescharacteristic
92#13093944C2×C411yescharacteristic
93(C2×D47)⋊C8—22no—
94(C2×D47)⋊C8—22no—
95#13093935—42no—
96#13093934—22no—
97C4×C47⋊C8C2×C411yescharacteristic
98C4×C47⋊C8C2×C411yescharacteristic
99C4×C47⋊C8C2×C411yescharacteristic
100C4×C47⋊C8C2×C411yescharacteristic
101C8×Dic47—22no—
102C2×#41×C47C2311yescharacteristic
103C2×M16×C47C2312yes—
104C4×Q8×C47C2×C411yescharacteristic
105C4×Q8×C47C2×C411yescharacteristic
106C4×D4×C47—21no—
107C4×D4×C47—21no—
108C4×D4×C47C2×C411yescharacteristic
109C4×D4×C47C2×C411yescharacteristic
110#117×C47—21no—
111#117×C47C2311yescharacteristic
112C2×C4⋊C4×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47C2312yes—
115C2×C42×C47C2311yescharacteristic
116C4⋊C8×C47—21no—
117C4⋊C8×C47—21no—
118#104×C47—24no—
119#95×C47—21no—
120#41×C47—21no—
121#41×C47—21no—
122#41×C47#4112yes—
123#41×C47#4112yes—
124C2×Q8×C47C22×C411yescharacteristic
125C2×D4×C47—21no—
126C2×D4×C47C22×C411yescharacteristic
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—22no—
131C22×C4×C47#4111yescharacteristic
132C22×C4×C47#4111yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—21no—
135#10625350—21no—
136#10625350—21no—
137#10625350—21no—
138#10625350—21no—
139C2×D188C22×C411yescharacteristic
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×Dic94C22×C411yescharacteristic
144M16×C47—22no—
145M16×C47—22no—
146C2×C8×C47—22no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—42no—
150C4⋊C4×C47C22×C411yescharacteristic
151C4⋊C4×C47C22×C411yescharacteristic
152C22⋊C4×C47—21no—
153C22⋊C4×C47—21no—
154C22⋊C4×C47—21no—
155C42×C47—21no—
156C42×C47—21no—
157C42×C47—21no—
158C42×C47—22no—
159C42×C47—22no—
160C42×C47C22×C411yescharacteristic
161C42×C47C22×C411yescharacteristic
162C42×C47C2×D411yescharacteristic
163C42×C47C2×D411yescharacteristic
164C22⋊Dic47—21no—
165#10625337—22no—
166#10625336—22no—
167#10625335—22no—
168#10625334—22no—
169(C2×D47)⋊C4—21no—
170(C2×D47)⋊C4—21no—
171C4⋊Dic47C22×C411yescharacteristic
172C4⋊Dic47C22×C411yescharacteristic
173Dic47⋊C4—21no—
174Dic47⋊C4—21no—
175C4×Dic47—21no—
176C4×Dic47—21no—
177C4×Dic47—21no—
178#10625329—22no—
179#10625329—22no—
180#10625329—44no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—44no—
189C2×C47⋊C8C22×C411yescharacteristic
190C2×C47⋊C8C22×C411yescharacteristic
191C2×C47⋊C8C22×C411yescharacteristic
192C2×C47⋊C8C22×C411yescharacteristic
193C2×C47⋊C8C2×D412yes—
194C2×C47⋊C8C2×D412yes—
195C23×C47—21no—
196C23×C47C2×#4111yescharacteristic
197C22×D47—21no—
198Q8×C47—21no—
199Q8×C47#11712yes—
200D4×C47—22no—
201D4×C47—22no—
202D4×C47—21no—
203D4×C47#11712yes—
204C2×C4×C47—21no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—81no—
216C2×C4×C47C4×D411yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47H(5,𝔽2)11yescharacteristic
225C2×C4×C472−1+411yescharacteristic
226C22⋊D47—41no—
227C22⋊D47—41no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231D188—21no—
232D188—21no—
233C4×D47—41no—
234C4×D47—41no—
235C4×D47—42no—
236C4×D47—81no—
237Dic94—21no—
238Dic94—21no—
239C8×C47—42no—
240C47⋊C8—21no—
241C47⋊C8—21no—
242C47⋊C8—21no—
243C47⋊C8—21no—
244C47⋊C8—22no—
245C47⋊C8—22no—
246C47⋊C8—42no—
247#22782—471nomaximal, Sylow
248C22×C47—21no—
249C22×C47—21no—
250C22×C47—22no—
251C22×C47—41no—
252C22×C47—41no—
253C22×C47C2×C4×D411yescharacteristic
254C22×C47#55211yescharacteristic
255C22×C47#55311yescharacteristic
256C2×D47—41no—
257C2×D47—41no—
258C2×D47—41no—
259C4×C47—22no—
260C4×C47—21no—
261C4×C47—21no—
262C4×C47—21no—
263C4×C47—41no—
264C4×C47—41no—
265C4×C47—82no—
266C4×C47#44412yes—
267C4×C47C2×#11711yescharacteristic
268C4×C47#52011yescharacteristic
269C4×C47#54611yescharacteristic
270C4×C47#54811yescharacteristic
271Dic47—41no—
272Dic47—41no—
273Dic47—41no—
274C4×#135—471no—
275#2925—471no—
276#2906—471no—
277#1938—472no—
278#1922—472no—
279#1876—472no—
280#1837—471no—
281#1806—471no—
282#1806D4711yescharacteristic, complemented
283#1770—471no—
284#1765—472no—
285C2×C47—21no—
286C2×C47—21no—
287C2×C47—42no—
288C2×C47#235211yescharacteristic
289C2×C47C2×#44411yescharacteristic
290C2×C47#295711yescharacteristic
291D47—81no—
292C2×#135—471no—
293C2×#131—472no—
294C4×#41—471no—
295C4×#41C2×D4711yescharacteristic
296C4×Q16—471no—
297C4×Q16—471no—
298C4×SD16—471no—
299C4×SD16—471no—
300C4×SD16—471no—
301C4×SD16—471no—
302C4×D8—471no—
303C4×D8—471no—
304#433—942no—
305#432—941no—
306#431—941no—
307#427—471no—
308#424—472no—
309#424—942no—
310#423—471no—
311#423—941no—
312#423C2×D4711yescharacteristic
313C2×C4⋊C8—471no—
314C2×C4⋊C8—941no—
315C2×#104—472no—
316C2×#104C2×D4712yes—
317#416—471no—
318#416—471no—
319#408—472no—
320#405—472no—
321C2×#97—472no—
322C2×C4×C8—471no—
323C4×C4⋊C4C2×D4711yescharacteristic
324#341—472no—
325#341—472no—
326#339—472no—
327#339C2×D4712yes—
328#336—472no—
329C47#2278211yescharacteristic, complemented, Sylow
330C2×#41—471no—
331C2×#41C22×D4711yescharacteristic
332#135—942no—
333#135—942no—
334C2×Q16—471no—
335C2×SD16—471no—
336C2×SD16—471no—
337C2×D8—471no—
338#131—944no—
339#131—1882no—
340C2×M16—472no—
341C2×M16—942no—
342C2×M16C22×D4712yes—
343C22×C8—472no—
344C22×C8—471no—
345C22×C8—471no—
346C22×C8—942no—
347C4×Q8—471no—
348C4×Q8—471no—
349C4×Q8C4×D4711yescharacteristic
350C4×Q8C4×D4711yescharacteristic
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—471no—
354C4×D4—471no—
355C4×D4—941no—
356C4×D4—941no—
357C4×D4C4×D4711yescharacteristic
358C4×D4C4×D4711yescharacteristic
359#117—21no—
360#117—471no—
361#117—941no—
362#117C22×D4711yescharacteristic
363C2×C4⋊C4—21no—
364C2×C42—21no—
365C2×C42—941no—
366C2×C42C22×D4712yes—
367C2×C42C22×D4711yescharacteristic
368#107—471no—
369#107—471no—
370#106—471no—
371#106—471no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—471no—
375C4⋊C8—471no—
376C4⋊C8—472no—
377C4⋊C8—942no—
378C4⋊C8—942no—
379C4⋊C8—941no—
380C4⋊C8—941no—
381#104—24no—
382#104—1882no—
383#103—471no—
384#103—471no—
385#103—471no—
386#103—471no—
387#102—471no—
388#102—471no—
389#102—471no—
390#102—471no—
391C22⋊C8—942no—
392C22⋊C8—942no—
393#97—942no—
394#97—1882no—
395C4×C8—471no—
396C4×C8—471no—
397C4×C8—471no—
398C4×C8—471no—
399C4×C8—942no—
400#95—21no—
401#41—21no—
402#41—21no—
403#41—941no—
404#41—941no—
405#41—941no—
406#41—941no—
407#41#1062535212yes—
408#41#1062535412yes—
409C2×Q8—471no—
410C2×Q8C2×C4×D4711yescharacteristic
411C2×D4—21no—
412C2×D4—471no—
413C2×D4—941no—
414C2×D4C2×C4×D4711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4#1062535211yescharacteristic
423C22×C4#1062535211yescharacteristic
424C22×C4C23×D4711yescharacteristic
425Q16—942no—
426SD16—942no—
427SD16—942no—
428D8—942no—
429M16—22no—
430M16—22no—
431M16—942no—
432M16—942no—
433M16—1884no—
434C2×C8—22no—
435C2×C8—471no—
436C2×C8—471no—
437C2×C8—471no—
438C2×C8—471no—
439C2×C8—472no—
440C2×C8—472no—
441C2×C8—941no—
442C2×C8—942no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—942no—
446C2×C8—941no—
447C2×C8—942no—
448C2×C8—1884no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—471no—
453C4⋊C4—471no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4C2×C4×D4711yescharacteristic
457C4⋊C4C2×C4×D4711yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—941no—
462C22⋊C4—941no—
463C22⋊C4—941no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—941no—
470C42—941no—
471C42—941no—
472C42C2×C4×D4711yescharacteristic
473C42C2×C4×D4711yescharacteristic
474C42D4×D4711yescharacteristic
475C42D4×D4711yescharacteristic
476C23—21no—
477C23—941no—
478C23C2×#1062535211yescharacteristic
479Q8—21no—
480Q8—941no—
481Q8—941no—
482Q8#1309398212yes—
483D4—22no—
484D4—22no—
485D4—21no—
486D4—941no—
487D4—941no—
488D4—1881no—
489D4—1881no—
490D4#1309398212yes—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—941no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—3761no—
507C2×C4—1882no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4(C22⋊D47)⋊C411yescharacteristic
511C2×C4(C22⋊D47)⋊C411yescharacteristic
512C2×C4(C22⋊D47)⋊C411yescharacteristic
513C2×C4(C22⋊D47)⋊C411yescharacteristic
514C2×C4C22×C4×D4711yescharacteristic
515C2×C4C2×D4×D4711yescharacteristic
516C2×C4C2×#1062535211yescharacteristic
517C2×C4C2×#1062535411yescharacteristic
518C2×C4#1309407011yescharacteristic
519C2×C4#1309407111yescharacteristic
520C8—42no—
521C8—942no—
522C8—942no—
523C8—941no—
524C8—941no—
525C8—941no—
526C8—941no—
527C8—1882no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—1881no—
534C22—1881no—
535C22—1881no—
536C22C2×(C22⋊D47)⋊C411yescharacteristic
537C22#1568981511yescharacteristic
538C22#1568982511yescharacteristic
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—1881no—
547C4—1881no—
548C4—1881no—
549C4#1568924212yes—
550C4C2×#1309398211yescharacteristic
551C4#1568960311yescharacteristic
552C4#1568976411yescharacteristic
553C4#1568977911yescharacteristic
554C2—21no—
555C2—21no—
556C2—42no—
557C2—3761no—
558C2#2024631911yescharacteristic
559C2C2×#1568924211yescharacteristic
560C2#2025230111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1806C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094044C22
cosocleC22×C4×C47C24
hypercenter#1806D47
hypocenterC47#22782
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20252056C2semidirect
3. #20251956C2semidirect
4. #20244100C2semidirect
5. #20243990C2semidirect
7. #20243525C2semidirect
8. #20243409C2semidirect
9. #20243257C2semidirect
10. #20243233C2semidirect
11. C47×#1806C2semidirect
15. C4×#10625337C4semidirect
16. C4×#10625337C4semidirect
17. C4×#10625336C4semidirect
18. C4×#10625336C4semidirect
19. C4×#10625335C4semidirect
20. C4×#10625335C4semidirect
21. C4×#10625334C4semidirect
22. C4×#10625334C4semidirect
282. #1806D47semidirect
329. C47#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#130940442
#202520562
#202519562
#202441002
#202439902
#202437132
#202435252
#202434092
#202432572
#202432332
C47×#18062
#2278247