Finite Group Databasebeta

← #26370757

Subgroups — #26370757

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689558C211yesmaximal, characteristic, complemented
2C22×#13093964C211yesmaximal, characteristic, complemented
3C2×#15689359C211yesmaximal, characteristic, complemented
4#20245503C212yesmaximal, complemented
5#20245489C212yesmaximal, complemented
6#20245486C212yesmaximal, complemented
7#20245472C212yesmaximal, complemented
8#20245177C211yesmaximal, characteristic, complemented
9#20245103C211yesmaximal, characteristic, complemented
10#20245100C211yesmaximal, characteristic, complemented
11C47×#3050C211yesmaximal, characteristic, complemented
12C22×D376C2211yescharacteristic
13C2×#13093998—21no—
14#15689558—24no—
15C2×#13093967C2212yescomplemented
16C2×#13093965C2212yescomplemented
17C2×#13093964—24no—
18C2×#13093964C2211yescharacteristic
19C2×#13093964C2211yescharacteristic
20C2×#13093964C2212yescomplemented
21C22×C47⋊C16C2212yescomplemented
22#15689359—21no—
23#15689359—21no—
24#15689359C2212yes—
25#15689359C2212yes—
26C2×#13093958C2211yescharacteristic
27C2×#13093946—21no—
28#15688950C2212yescomplemented
29#15688948C2212yescomplemented
30#15688927C2212yescomplemented
31#15688924C2212yescomplemented
32#15688923C2212yescomplemented
33#15688923C2212yescomplemented
34C2×#135×C47C2211yescharacteristic
35C23×C8×C47C2211yescharacteristic
36C47×#525—21no—
37C47×#467C2211yescharacteristic, complemented
38C47×#467C2211yescharacteristic, complemented
39C47×#466C2211yescharacteristic, complemented
40C47×#466C2211yescharacteristic, complemented
41C47×#465—21no—
42C47×#465—21no—
43C47×#427C2211yescharacteristic
44C47×#416—21no—
45C22×D188—21no—
46C22×C22⋊D47—41no—
47C2×D376—22no—
48C2×D376—22no—
49C2×D376—21no—
50C2×D376C2312yes—
51#13093998—22no—
52#13093998—22no—
53#13093998—41no—
54#13093998—41no—
55C2×C4⋊Dic47—21no—
56C2×(C2×D47)⋊C4—41no—
57#13093967—24no—
58#13093965—24no—
59#13093964—22no—
60#13093964—24no—
61#13093964D414yescomplemented
62C2×C47⋊C16—24no—
63C2×C47⋊C16C2312yes—
64C2×C47⋊C16C2312yes—
65#13093958—21no—
66#13093958C2312yes—
67#13093946—22no—
68#13093946—22no—
69C2×#41×C47—21no—
70C23×C4×C47D411yescharacteristic
71#135×C47—22no—
72#135×C47—22no—
73C2×Q16×C47C2311yescharacteristic
74C2×SD16×C47—21no—
75C2×D8×C47C2311yescharacteristic
76C22×C8×C47—22no—
77C22×C8×C47—22no—
78C22×C8×C47D411yescharacteristic
79C22×C8×C47D411yescharacteristic
80C22×C8×C47D411yescharacteristic
81C22×C8×C47C2311yescharacteristic
82C22×C8×C47C2311yescharacteristic
83C22×C8×C47C2311yescharacteristic
84#123×C47—41no—
85C22⋊Q8×C47—21no—
86C22⋊Q8×C47—21no—
87#121×C47—21no—
88#121×C47—21no—
89#120×C47—41no—
90C4×D4×C47—41no—
91C4×D4×C47—41no—
92#117×C47—21no—
93#107×C47C2311yescharacteristic
94#107×C47C2311yescharacteristic
95#106×C47—21no—
96#103×C47—21no—
97#103×C47—21no—
98#102×C47—21no—
99#102×C47—21no—
100C24×C47D811yescharacteristic
101C23×D47—41no—
102#41×C47—42no—
103#41×C47—42no—
104C2×Q8×C47—21no—
105C2×D4×C47—21no—
106C2×D4×C47—41no—
107C2×D4×C47—41no—
108C2×D4×C47—41no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—41no—
112C22×C4×C47D811yescharacteristic
113C22×C4×C47D811yescharacteristic
114C22×C4×C47D811yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C2×C22⋊D47—42no—
119C2×C22⋊D47—42no—
120C2×C22⋊D47—84no—
121C22×Dic47—41no—
122C2×D188—22no—
123C2×D188—42no—
124C2×D188—42no—
125C2×D188—41no—
126Q16×C47—21no—
127Q16×C47—21no—
128SD16×C47—42no—
129D8×C47—21no—
130D8×C47—21no—
131C2×C8×C47—22no—
132C2×C8×C47—22no—
133C2×C8×C47—22no—
134C2×C8×C47—21no—
135C2×C8×C47—21no—
136C2×C8×C47—21no—
137C2×C8×C47—21no—
138C2×C8×C47C2×D411yescharacteristic
139C2×C8×C47C2×D411yescharacteristic
140C2×C8×C47C2×D411yescharacteristic
141C2×C8×C47C2×D412yes—
142C2×C8×C47#4112yes—
143C2×C8×C47C2411yescharacteristic
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—41no—
147C4⋊C4×C47—41no—
148C22⋊C4×C47—41no—
149C22⋊C4×C47—41no—
150C22⋊C4×C47—41no—
151C22⋊C4×C47—41no—
152C22⋊C4×C47—41no—
153C42×C47—41no—
154(C2×D47)⋊C4—42no—
155(C2×D47)⋊C4—42no—
156C4⋊Dic47—22no—
157C4⋊Dic47—41no—
158D376—22no—
159D376—22no—
160D376—42no—
161C47⋊C16—22no—
162C47⋊C16C2×D414yescomplemented
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—41no—
166C23×C47—41no—
167C23×C47C2×D811yescharacteristic
168C23×C47C2×D811yescharacteristic
169C23×C47C2×D811yescharacteristic
170C22×D47—42no—
171C22×D47—82no—
172C22×D47—82no—
173C22×D47—82no—
174Q8×C47—41no—
175Q8×C47—41no—
176D4×C47—41no—
177D4×C47—41no—
178D4×C47—82no—
179D4×C47—82no—
180D4×C47—82no—
181C2×C4×C47—22no—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—82no—
192C2×C4×C47#12112yes—
193C2×C4×C47C2×D811yescharacteristic
194C2×C4×C47C2×D811yescharacteristic
195C2×C4×C47C2×D811yescharacteristic
196C2×C4×C47#13512yes—
197C2×C4×C47C22×D411yescharacteristic
198C22⋊D47—84no—
199C22⋊D47—84no—
200C2×Dic47—42no—
201C2×Dic47—82no—
202D188—42no—
203D188—42no—
204D188—82no—
205C8×C47—21no—
206C8×C47—21no—
207C8×C47#12112yes—
208C8×C47C22×D411yescharacteristic
209C8×C47C2×#4111yescharacteristic
210#33605—471nomaximal, Sylow
211C22×C47—21no—
212C22×C47—21no—
213C22×C47—21no—
214C22×C47—21no—
215C22×C47—21no—
216C22×C47—21no—
217C22×C47—21no—
218C22×C47—44no—
219C22×C47—81no—
220C22×C47—81no—
221C22×C47—81no—
222C22×C47#46612yes—
223C22×C47#50912yescomplemented
224C22×C47C22×D811yescharacteristic
225C2×D47—82no—
226C2×D47—82no—
227C2×D47—82no—
228C2×D47—164no—
229C4×C47—21no—
230C4×C47—21no—
231C4×C47—81no—
232C4×C47—81no—
233C4×C47—81no—
234C4×C47#46612yes—
235C4×C47C2×#12111yescharacteristic
236C4×C47C2×#13511yescharacteristic
237Dic47—82no—
238C2×#509—471no—
239C22×M32—471no—
240#3050D4711yescharacteristic, complemented
241C2×#466—471no—
242#2220—472no—
243#2218—472no—
244#2217—472no—
245#2215—472no—
246#2159—471no—
247#2140—471no—
248#2139—471no—
249C2×C47—22no—
250C2×C47—22no—
251C2×C47—41no—
252C2×C47—161no—
253C2×C47C2×#46611yescharacteristic
254C2×C47C2×#50911yescharacteristic
255C2×C47#341411yescharacteristic
256D47—162no—
257C2×#135C2×D4711yescharacteristic
258C22×D8—471no—
259C23×C8C2×D4711yescharacteristic
260#525—21no—
261C2×#121—941no—
262#509—944no—
263C2×SD32—472no—
264C2×D16—472no—
265C2×M32—471no—
266C2×M32—471no—
267C2×M32—472no—
268C2×M32—944no—
269C22×C16—472no—
270#467C2×D4711yescharacteristic, complemented
271#467C2×D4711yescharacteristic, complemented
272#466—472no—
273#466—472no—
274#466—941no—
275#466—941no—
276#466C2×D4711yescharacteristic, complemented
277#466C2×D4711yescharacteristic, complemented
278#465—21no—
279#465—21no—
280#427C2×D4711yescharacteristic
281C2×#107—471no—
282#416—21no—
283C2×#102—941no—
284#367—472no—
285#366—472no—
286#358—472no—
287#357—472no—
288#357—472no—
289#357—472no—
290C47#3360511yescharacteristic, complemented, Sylow
291C2×#41—21no—
292C22×D4—941no—
293C22×D4—1881no—
294C23×C4C22⋊D4711yescharacteristic
295#135—22no—
296#135—22no—
297C2×Q16C22×D4711yescharacteristic
298C2×SD16—21no—
299C2×D8—472no—
300C2×D8—942no—
301C2×D8—942no—
302C2×D8—941no—
303C2×D8C22×D4711yescharacteristic
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D4711yescharacteristic
307C22×C8C22⋊D4711yescharacteristic
308C22×C8C22⋊D4711yescharacteristic
309C22×C8C22×D4711yescharacteristic
310C22×C8C22×D4711yescharacteristic
311C22×C8C22×D4711yescharacteristic
312#123—41no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—21no—
316#121—21no—
317#121—942no—
318#121—942no—
319#121—1881no—
320#121—1881no—
321#120—41no—
322C4×D4—41no—
323C4×D4—41no—
324#117—21no—
325C2×C4⋊C4—941no—
326C2×C22⋊C4—1881no—
327SD32—944no—
328D16—944no—
329M32—474no—
330M32—944no—
331M32—942no—
332C2×C16—472no—
333C2×C16—472no—
334C2×C16—944no—
335#107—472no—
336#107—941no—
337#107C22×D4711yescharacteristic
338#107C22×D4711yescharacteristic
339#106—21no—
340#103—21no—
341#103—21no—
342#102—21no—
343#102—21no—
344#102—942no—
345#102—942no—
346C24—1881no—
347C24#1062533411yescharacteristic
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—942no—
356C2×D4—3764no—
357C2×D4—1882no—
358C2×D4—1882no—
359C2×D4—1882no—
360C2×D4—1881no—
361C2×D4—1882no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—1881no—
366C22×C4#1062533411yescharacteristic
367C22×C4#1062533411yescharacteristic
368C22×C4#1062533411yescharacteristic
369C22×C4C2×C22⋊D4711yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371C22×C4C2×C22⋊D4711yescharacteristic
372Q16—21no—
373Q16—21no—
374SD16—42no—
375D8—21no—
376D8—21no—
377D8—942no—
378D8—942no—
379D8—1882no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D4712yes—
388C2×C8#1062535212yes—
389C2×C8C2×C22⋊D4711yescharacteristic
390C2×C8C2×C22⋊D4711yescharacteristic
391C2×C8C2×C22⋊D4711yescharacteristic
392C2×C8C23×D4711yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—942no—
398C4⋊C4—1881no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—1882no—
405C22⋊C4—1882no—
406C42—41no—
407C16—474no—
408C16—942no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—3762no—
414C23—3762no—
415C23—3762no—
416C23—1882no—
417C23C2×#1062533411yescharacteristic
418C23C2×#1062533411yescharacteristic
419C23C2×#1062533411yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—41no—
423D4—41no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—3764no—
428D4—3764no—
429D4—3762no—
430D4—1882no—
431D4—1882no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—3762no—
444C2×C4—1882no—
445C2×C4#1309399612yes—
446C2×C4C2×#1062533411yescharacteristic
447C2×C4C2×#1062533411yescharacteristic
448C2×C4C2×#1062533411yescharacteristic
449C2×C4#1309404412yes—
450C2×C4C22×C22⋊D4711yescharacteristic
451C8—21no—
452C8—21no—
453C8#1309399612yes—
454C8C2×D4×D4711yescharacteristic
455C8C2×#1062535211yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—3762no—
468C22—3762no—
469C22—3762no—
470C22—7524no—
471C22#1568935712yes—
472C22#1568955812yescomplemented
473C22C22×#1062533411yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—3762no—
480C4#1568935712yes—
481C4C2×#1309399611yescharacteristic
482C4C2×#1309404411yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—161no—
487C2—7522no—
488C2C2×#1568935711yescharacteristic
489C2C2×#1568955811yescharacteristic
490C2#2025652211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3050C2
socleC22×C47C22×D8
hypercenter#3050D47
hypocenterC47#33605
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#15689558C2semidirect
2. C22×#13093964C2semidirect
3. C2×#15689359C2semidirect
4. #20245503C2semidirect
5. #20245489C2semidirect
6. #20245486C2semidirect
7. #20245472C2semidirect
8. #20245177C2semidirect
9. #20245103C2semidirect
10. #20245100C2semidirect
11. C47×#3050C2semidirect
15. C2×#13093967C22semidirect
16. C2×#13093965C22semidirect
20. C2×#13093964C22semidirect
21. C22×C47⋊C16C22semidirect
28. #15688950C22semidirect
29. #15688948C22semidirect
30. #15688927C22semidirect
31. #15688924C22semidirect
32. #15688923C22semidirect
33. #15688923C22semidirect
37. C47×#467C22semidirect
38. C47×#467C22semidirect
39. C47×#466C22semidirect
40. C47×#466C22semidirect
61. #13093964D4semidirect
162. C47⋊C16C2×D4semidirect
223. C22×C47#509semidirect
240. #3050D47semidirect
270. #467C2×D47semidirect
271. #467C2×D47semidirect
276. #466C2×D47semidirect
277. #466C2×D47semidirect
290. C47#33605semidirect
472. C22#15689558semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
C22×#130939642
C2×#156893592
#202455032
#202454892
#202454862
#202454722
#202451772
#202451032
#202451002
C47×#30502
#3360547