Finite Group Databasebeta

← #26369593

Subgroups — #26369593

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689560C211yesmaximal, characteristic
2C4×#13094047C211yesmaximal, characteristic, complemented
3C4×D8×D47C211yesmaximal, characteristic, complemented
4#20245265C211yesmaximal, characteristic, complemented
5#20245262C211yesmaximal, characteristic, complemented
6#20245250C211yesmaximal, characteristic, complemented
7SD32×Dic47C211yesmaximal, characteristic, complemented
8#20245237C211yesmaximal, characteristic, complemented
9#20245234C211yesmaximal, characteristic, complemented
10#20245232C211yesmaximal, characteristic, complemented
11#20245173C211yesmaximal, characteristic, complemented
12#20245102C211yesmaximal, characteristic, complemented
13#20245099C211yesmaximal, characteristic, complemented
14#20244981C211yesmaximal, characteristic, complemented
15C47×#2175C211yesmaximal, characteristic, complemented
16C2×#13094047C2211yescharacteristic
17C2×D8×D47C2211yescharacteristic
18C4×#10625354—21no—
19C4×D4×D47—21no—
20#15689560C418yescomplemented
21C2×#13093967C2211yescharacteristic
22C2×D752C2211yescharacteristic
23C2×#13093965C2211yescharacteristic
24C2×#13093963C2211yescharacteristic
25C2×#13093963C2211yescharacteristic
26Q16×Dic47C2211yescharacteristic, complemented
27C4×#10625336—21no—
28#15689208—21no—
29C4×D376C2211yescharacteristic
30C4×D376C2211yescharacteristic, complemented
31D8×Dic47C2211yescharacteristic
32#15689205C2211yescharacteristic
33#15689205C2211yescharacteristic, complemented
34C4×#10625334—21no—
35#15689203—21no—
36#15689160C2211yescharacteristic
37#107×D47C2211yescharacteristic
38#15689118—21no—
39#102×D47—21no—
40C4×C8×D47C2211yescharacteristic
41#15688951C2211yescharacteristic, complemented
42#15688950C2211yescharacteristic, complemented
43#15688948C2211yescharacteristic, complemented
44#15688927C2211yescharacteristic, complemented
45#15688925C2211yescharacteristic
46#15688925C2211yescharacteristic, complemented
47#15688924C2211yescharacteristic
48#15688923C2211yescharacteristic
49#15688923C2211yescharacteristic, complemented
50#15688899C2211yescharacteristic, complemented
51C16×Dic47C2211yescharacteristic, complemented
52C16×Dic47C2211yescharacteristic, complemented
53C2×SD32×C47C2211yescharacteristic
54C4×Q16×C47C2211yescharacteristic, complemented
55C4×D8×C47C2211yescharacteristic
56C47×#366C2211yescharacteristic, complemented
57C47×#358C2211yescharacteristic, complemented
58C47×#357C2211yescharacteristic
59C47×#346C2211yescharacteristic, complemented
60C2×#10625354—21no—
61C2×D4×D47—21no—
62C22×C4×D47—41no—
63#13094047—22no—
64#13094047C2×C414yescomplemented
65C2×#10625336—21no—
66C2×D376C2311yescharacteristic
67C2×D376C2311yescharacteristic
68C2×#10625334—21no—
69D8×D47—22no—
70D8×D47C2×C414yescomplemented
71C2×C8×D47D411yescharacteristic
72C2×C8×D47C2311yescharacteristic
73Q8×Dic47—21no—
74C4×D188—21no—
75C4×D188—21no—
76C4×D188—41no—
77D4×Dic47—21no—
78#13093986—21no—
79#13093986—21no—
80#13093986—41no—
81C4×C22⋊D47—41no—
82(C22⋊D47)⋊C4—41no—
83#13093982—21no—
84#13093982—41no—
85C4⋊C4×D47—21no—
86C22⋊C4×D47—41no—
87C42×D47—41no—
88C42×D47D411yescharacteristic
89#13093967C2×C414yescomplemented
90D752C2×C414yescomplemented
91#13093965C2×C414yescomplemented
92#13093963—22no—
93#13093963C2×C414yescomplemented
94C2×C47⋊C16C2311yescharacteristic
95C2×C47⋊C16C2311yescharacteristic
96#13093958C2311yescharacteristic
97#13093958C2311yescharacteristic
98#13093957—21no—
99#13093955—21no—
100#13093948—21no—
101#13093946—21no—
102#13093946—21no—
103#13093945—21no—
104#13093944—21no—
105#13093944—21no—
106C4×C47⋊C8D411yescharacteristic
107C8×Dic47D411yescharacteristic
108C8×Dic47C2311yescharacteristic
109C2×Q16×C47C2311yescharacteristic
110C2×D8×C47C2311yescharacteristic
111C4×Q8×C47—21no—
112C4×D4×C47—21no—
113SD32×C47C2×C414yescomplemented
114C2×C16×C47C2311yescharacteristic
115C2×C16×C47C2311yescharacteristic
116#107×C47C2311yescharacteristic
117#107×C47C2311yescharacteristic
118#103×C47—21no—
119#102×C47—21no—
120C4×C8×C47C2311yescharacteristic
121C23×D47—41no—
122C2×Q8×C47—21no—
123C2×D4×C47—21no—
124C22×C4×C47—41no—
125C2×C22⋊D47—41no—
126C22×Dic47—41no—
127#10625354—24no—
128#10625354—42no—
129D4×D47—24no—
130D4×D47—42no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—41no—
134C2×C4×D47—41no—
135C2×C4×D47—41no—
136C2×C4×D47—41no—
137C2×C4×D47—41no—
138C2×C4×D47—41no—
139C2×C4×D47—41no—
140C2×C4×D47—84no—
141C2×C4×D47D811yescharacteristic
142C2×C4×D47D811yescharacteristic
143C2×C4×D47C2×D411yescharacteristic
144Q16×C47—21no—
145Q16×C47C22×C412yescomplemented
146D8×C47—21no—
147D8×C47C22×C412yes—
148C2×C8×C47C2×D411yescharacteristic
149C2×C8×C47C2411yescharacteristic
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—41no—
153C22⋊C4×C47—41no—
154C42×C47—41no—
155C42×C47C2×D411yescharacteristic
156C22⋊Dic47—41no—
157#10625336—24no—
158#10625334—24no—
159(C2×D47)⋊C4—41no—
160(C2×D47)⋊C4—41no—
161(C2×D47)⋊C4—41no—
162(C2×D47)⋊C4—41no—
163(C2×D47)⋊C4—41no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—41no—
167Dic47⋊C4—41no—
168C4×Dic47—41no—
169C4×Dic47—41no—
170C4×Dic47D811yescharacteristic
171C4×Dic47D811yescharacteristic
172C4×Dic47C2×D411yescharacteristic
173C2×C47⋊C8C2×D411yescharacteristic
174C2×C47⋊C8C2×D411yescharacteristic
175D376—21no—
176D376—21no—
177D376C22×C412yes—
178D376C22×C412yescomplemented
179C8×D47—22no—
180C8×D47C22×C412yes—
181C8×D47#4112yes—
182C16×C47—21no—
183C16×C47C22×C412yescomplemented
184C47⋊C16—21no—
185C47⋊C16C22×C412yescomplemented
186C23×C47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—44no—
190C22×D47—82no—
191C22×D47—82no—
192C22×D47C2×D811yescharacteristic
193Q8×C47—22no—
194Q8×C47—41no—
195D4×C47—22no—
196D4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—82no—
201C2×C4×C47C2×D811yescharacteristic
202C2×C4×C47C2×D811yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C22⋊D47—44no—
205C2×Dic47—41no—
206C2×Dic47—41no—
207C2×Dic47—82no—
208C2×Dic47C2×D811yescharacteristic
209C2×Dic47C2×D811yescharacteristic
210C2×Dic47C2×D811yescharacteristic
211D188—22no—
212D188—22no—
213D188—44no—
214D188—41no—
215D188—41no—
216C4×D47—22no—
217C4×D47—41no—
218C4×D47—44no—
219C4×D47—82no—
220C4×D47—82no—
221C4×D47—82no—
222C4×D47—82no—
223C4×D47—82no—
224C4×D47C4×D412yes—
225C4×D47#13512yes—
226C8×C47—21no—
227C8×C47C23×C411yescharacteristic
228C8×C47C2×#4111yescharacteristic
229C47⋊C8—21no—
230C47⋊C8C4×D412yes—
231C4×#509—471nomaximal, Sylow
232C22×C47—42no—
233C22×C47—81no—
234C22×C47—81no—
235C22×C47C22×D811yescharacteristic
236C2×D47—21no—
237C2×D47—21no—
238C2×D47—42no—
239C2×D47—42no—
240C2×D47—88no—
241C2×D47—81no—
242C2×D47—81no—
243C2×D47—81no—
244C2×D47—81no—
245C2×D47C4×D812yes—
246C4×C47—21no—
247C4×C47—21no—
248C4×C47—42no—
249C4×C47—81no—
250C4×C47—81no—
251C4×C47C2×C4×D411yescharacteristic
252C4×C47C2×#13511yescharacteristic
253Dic47—21no—
254Dic47—81no—
255Dic47—81no—
256Dic47C4×D812yes—
257Dic47#50912yescomplemented
258C2×#509—471no—
259C4×#135—471no—
260C2×C4×D8—471no—
261#2177—471no—
262#2177—471no—
263#2175—471no—
264#2175D4711yescharacteristic, complemented
265C4×SD32—471no—
266C4×SD32—471no—
267C4×D16—471no—
268C4×D16—471no—
269#2159—471no—
270#2140—471no—
271#2139—471no—
272C4×M32—471no—
273C2×C47—82no—
274C2×C47C2×C4×D811yescharacteristic
275C2×C47C2×#50911yescharacteristic
276C2×C47#341411yescharacteristic
277D47—22no—
278D47—82no—
279D47—82no—
280C2×#135—471no—
281C22×D8—471no—
282C4×#41—941no—
283C2×C4×D4—941no—
284#509—478no—
285C2×SD32—471no—
286C2×SD32C2×D4711yescharacteristic
287C2×D16—471no—
288C2×D16—471no—
289C2×M32—471no—
290C2×M32—471no—
291C4×Q16—471no—
292C4×Q16C2×D4711yescharacteristic, complemented
293C4×SD16—941no—
294C4×SD16—941no—
295C4×D8—471no—
296C4×D8—471no—
297C4×D8—471no—
298C4×D8—471no—
299C4×D8—471no—
300C4×D8—941no—
301C4×D8—941no—
302C4×D8C2×D4711yescharacteristic
303#427—471no—
304C2×#107—471no—
305#416—941no—
306C2×#102—941no—
307C2×C4×C8—471no—
308#367—471no—
309#367—471no—
310#366—471no—
311#366C2×D4711yescharacteristic, complemented
312#358—471no—
313#358C2×D4711yescharacteristic, complemented
314#357—471no—
315#357—471no—
316#357—471no—
317#357—471no—
318#357—471no—
319#357C2×D4711yescharacteristic
320#346—471no—
321#346C2×D4711yescharacteristic, complemented
322C4×C16—471no—
323C4×C16—471no—
324C47C4×#50911yescharacteristic, complemented, Sylow
325C2×#41—941no—
326C22×D4—941no—
327C23×C4—1881no—
328#135—474no—
329#135—942no—
330C2×Q16C22×D4711yescharacteristic
331C2×SD16—941no—
332C2×D8—474no—
333C2×D8—471no—
334C2×D8—471no—
335C2×D8—942no—
336C2×D8—941no—
337C2×D8C22×D4711yescharacteristic
338C22×C8—471no—
339C22×C8—471no—
340C4×Q8—21no—
341C4×Q8—941no—
342C4×D4—21no—
343C4×D4—941no—
344C4×D4—941no—
345C4×D4—941no—
346C4×D4—941no—
347C4×D4—941no—
348C4×D4—1881no—
349C4×D4—1881no—
350C4×D4—1881no—
351C4×D4—1881no—
352#117—941no—
353#117—1881no—
354C2×C4⋊C4—941no—
355C2×C22⋊C4—1881no—
356C2×C42—471no—
357C2×C42—1881no—
358SD32—474no—
359SD32C4×D4714yescomplemented
360D16—474no—
361D16—474no—
362M32—474no—
363M32—942no—
364C2×C16—471no—
365C2×C16—471no—
366C2×C16C22×D4711yescharacteristic
367C2×C16C22×D4711yescharacteristic
368#107—471no—
369#107—471no—
370#107—941no—
371#107C22×D4711yescharacteristic
372#107C22×D4711yescharacteristic
373#106—941no—
374#103—21no—
375#103—941no—
376#102—21no—
377#102—941no—
378#102—941no—
379#102—941no—
380#102—941no—
381#102—941no—
382C4×C8—471no—
383C4×C8—471no—
384C4×C8—471no—
385C4×C8C22×D4711yescharacteristic
386C24—1881no—
387#41—944no—
388#41—1882no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—944no—
392C2×D4—941no—
393C2×D4—941no—
394C2×D4—1881no—
395C2×D4—1882no—
396C2×D4—1881no—
397C22×C4—41no—
398C22×C4—471no—
399C22×C4—471no—
400C22×C4—471no—
401C22×C4—3764no—
402C22×C4—1881no—
403C22×C4—1881no—
404C22×C4—1881no—
405C22×C4—1881no—
406C22×C4—1881no—
407C22×C4—1881no—
408C22×C4—1881no—
409Q16—21no—
410Q16C2×C4×D4712yescomplemented
411SD16—944no—
412D8—21no—
413D8—472no—
414D8—472no—
415D8—944no—
416D8—941no—
417D8—941no—
418D8C2×C4×D4712yes—
419C2×C8—471no—
420C2×C8—472no—
421C2×C8—472no—
422C2×C8—471no—
423C2×C8—942no—
424C2×C8D4×D4711yescharacteristic
425C2×C8C23×D4711yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—941no—
430C4⋊C4—941no—
431C4⋊C4—1881no—
432C4⋊C4—1881no—
433C22⋊C4—41no—
434C22⋊C4—1881no—
435C22⋊C4—1881no—
436C22⋊C4—1881no—
437C22⋊C4—1881no—
438C22⋊C4—1881no—
439C22⋊C4—1881no—
440C42—41no—
441C42—471no—
442C42—471no—
443C42—471no—
444C42—1881no—
445C42—1881no—
446C42D4×D4711yescharacteristic
447C16—21no—
448C16—472no—
449C16—941no—
450C16C2×C4×D4712yescomplemented
451C23—41no—
452C23—471no—
453C23—3762no—
454C23—3762no—
455C23—1884no—
456C23—1881no—
457C23—1881no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—942no—
463D4—942no—
464D4—1884no—
465D4—1884no—
466D4—1881no—
467D4—1881no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—471no—
473C2×C4—471no—
474C2×C4—472no—
475C2×C4—472no—
476C2×C4—471no—
477C2×C4—942no—
478C2×C4—3762no—
479C2×C4—3762no—
480C2×C4—3762no—
481C2×C4—3762no—
482C2×C4—3762no—
483C2×C4—3762no—
484C2×C4—1881no—
485C2×C4—1884no—
486C2×C4—1881no—
487C2×C4—1881no—
488C2×C4D8×D4711yescharacteristic
489C2×C4D8×D4711yescharacteristic
490C2×C4C2×D4×D4711yescharacteristic
491C8—21no—
492C8—472no—
493C8—941no—
494C8C22×C4×D4711yescharacteristic
495C8#41×D4711yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—472no—
500C22—941no—
501C22—941no—
502C22—3768no—
503C22—3761no—
504C22—3761no—
505C22—3761no—
506C22—3761no—
507C22—1882no—
508C22—1882no—
509C22C2×D8×D4711yescharacteristic
510C4—21no—
511C4—21no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—472no—
516C4—472no—
517C4—941no—
518C4—3761no—
519C4—3761no—
520C4C4×D4×D4711yescharacteristic
521C4#135×D4711yescharacteristic
522C2—82no—
523C2—942no—
524C2—3762no—
525C2—3762no—
526C2C4×D8×D4711yescharacteristic
527C2#509×D4711yescharacteristic
528C2#2025652311yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2175C2
socleC22×C47C22×D8
prime radicalC2×#15689560C2
cosocleC2×C8×C47C24
hypercenter#2175D47
hypocenterC47C4×#509
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#13094047C2semidirect
3. C4×D8×D47C2semidirect
4. #20245265C2semidirect
5. #20245262C2semidirect
6. #20245250C2semidirect
7. SD32×Dic47C2semidirect
8. #20245237C2semidirect
9. #20245234C2semidirect
10. #20245232C2semidirect
11. #20245173C2semidirect
12. #20245102C2semidirect
13. #20245099C2semidirect
14. #20244981C2semidirect
15. C47×#2175C2semidirect
20. #15689560C4semidirect
26. Q16×Dic47C22semidirect
30. C4×D376C22semidirect
33. #15689205C22semidirect
41. #15688951C22semidirect
42. #15688950C22semidirect
43. #15688948C22semidirect
44. #15688927C22semidirect
46. #15688925C22semidirect
49. #15688923C22semidirect
50. #15688899C22semidirect
51. C16×Dic47C22semidirect
52. C16×Dic47C22semidirect
54. C4×Q16×C47C22semidirect
56. C47×#366C22semidirect
57. C47×#358C22semidirect
59. C47×#346C22semidirect
64. #13094047C2×C4semidirect
70. D8×D47C2×C4semidirect
89. #13093967C2×C4semidirect
90. D752C2×C4semidirect
91. #13093965C2×C4semidirect
93. #13093963C2×C4semidirect
113. SD32×C47C2×C4semidirect
145. Q16×C47C22×C4semidirect
178. D376C22×C4semidirect
183. C16×C47C22×C4semidirect
185. C47⋊C16C22×C4semidirect
257. Dic47#509semidirect
264. #2175D47semidirect
292. C4×Q16C2×D47semidirect
311. #366C2×D47semidirect
313. #358C2×D47semidirect
321. #346C2×D47semidirect
324. C47C4×#509semidirect
359. SD32C4×D47semidirect
410. Q16C2×C4×D47semidirect
450. C16C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895602
C4×#130940472
C4×D8×D472
#202452652
#202452622
#202452502
SD32×Dic472
#202452372
#202452342
#202452322
#202451732
#202451022
#202450992
#202449812
C47×#21752
C4×#50947