Finite Group Databasebeta

← #29836904

Subgroups — #29836904

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457562C211yesmaximal, characteristic
2#21448053C211yesmaximal, characteristic, complemented
3#21448027C211yesmaximal, characteristic, complemented
4#21447999C211yesmaximal, characteristic, complemented
5#21447980C211yesmaximal, characteristic, complemented
6#21447917C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447881C211yesmaximal, characteristic, complemented
9#21447757C211yesmaximal, characteristic, complemented
10#21447752C211yesmaximal, characteristic, complemented
11C4×#14519723C211yesmaximal, characteristic, complemented
12D4×#11758185C211yesmaximal, characteristic, complemented
13#21447524C211yesmaximal, characteristic, complemented
14C2×#16839565C211yesmaximal, characteristic, complemented
15C61×#3336C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840421—21no—
19C2×#14519723C2211yescharacteristic
20C2×#14519723C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C22×#11758185C2211yescharacteristic
23C22×#11758185C2211yescharacteristic
24#16840146C2211yescharacteristic
25#16840003C2211yescharacteristic
26#16839998C2211yescharacteristic
27#16839985—21no—
28#16839926—21no—
29#16839903—21no—
30C4×Dic244C2211yescharacteristic
31#16839589C2211yescharacteristic
32#16839587C2211yescharacteristic
33#16839587C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839566C2211yescharacteristic
36#16839565C2211yescharacteristic
37#16839565C2211yescharacteristic
38#16839565C2211yescharacteristic
39#16839565C2211yescharacteristic
40#16839565C2211yescharacteristic
41#16839565C2211yescharacteristic
42#16839562C2211yescharacteristic
43#16839562C2211yescharacteristic
44#16839562C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46C22⋊#11758185C2211yescharacteristic
47C4×#11758185C2211yescharacteristic
48C61×#552—21no—
49C61×#499C2211yescharacteristic
50C61×#485—21no—
51C61×#483—21no—
52C61×#479—21no—
53C61×#467C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic
56C8×D4×C61C2211yescharacteristic
57C61×#427C2211yescharacteristic
58C2×#107×C61C2211yescharacteristic
59C2×#11758222—21no—
60C22×Dic122—21no—
61#14519799—22no—
62#14519799—41no—
63C2×Dic244—22no—
64C2×Dic244—21no—
65C2×Dic244—41no—
66C2×Dic244C2311yescharacteristic
67C2×Dic244C2311yescharacteristic
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×D488C2311yescharacteristic
71#14519765—21no—
72#14519763—41no—
73#14519756—21no—
74#14519755—41no—
75#14519751—21no—
76#14519749—41no—
77C4×Dic122—21no—
78#14519737—41no—
79#14519723—22no—
80#14519723—41no—
81#14519720—41no—
82C2×#11758185—41no—
83C2×#11758185—41no—
84C2×#11758185C2311yescharacteristic
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519702—21no—
90#14519702—21no—
91#14519699—21no—
92C2×Q16×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95#126×C61—41no—
96#125×C61—21no—
97#125×C61—41no—
98#123×C61—41no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×D4×C61D411yescharacteristic
103#117×C61—21no—
104#117×C61—41no—
105C2×C4⋊C4×C61—21no—
106#107×C61—21no—
107#107×C61C2311yescharacteristic
108#107×C61C2311yescharacteristic
109#107×C61C2311yescharacteristic
110#106×C61—21no—
111#106×C61—21no—
112C4⋊C8×C61D411yescharacteristic
113#103×C61—21no—
114#103×C61—21no—
115#103×C61—21no—
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C2×Q8×C61—21no—
120C2×D4×C61D811yescharacteristic
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C22×Dic61—41no—
127#11758222—42no—
128#11758222—81no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—42no—
134C2×Dic122—41no—
135C2×Dic122—81no—
136Q16×C61—21no—
137Q16×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2411yescharacteristic
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61—41no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61D811yescharacteristic
156C22⋊C4×C61—41no—
157C22⋊C4×C61—41no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61D811yescharacteristic
160C22⋊C4×C61D811yescharacteristic
161C42×C61—41no—
162C42×C61—41no—
163C42×C61C2×D411yescharacteristic
164C22⋊Dic61—41no—
165#11758198—41no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169#11758196—41no—
170#11758196—41no—
171C4×Dic61—41no—
172Dic244—21no—
173Dic244—21no—
174Dic244—21no—
175Dic244—42no—
176Dic244#4112yes—
177D488—21no—
178D488—21no—
179#11758190—42no—
180#11758190—81no—
181#11758185—21no—
182#11758185—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188Q8×C61—41no—
189Q8×C61—41no—
190D4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—82no—
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—82no—
206C22⋊D61—82no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—82no—
211D244—41no—
212D244—41no—
213C4×D61—82no—
214Dic122—22no—
215Dic122—41no—
216Dic122—41no—
217Dic122—41no—
218Dic122—82no—
219C8×C61—21no—
220C8×C61—21no—
221C8×C61—21no—
222C8×C61C2×#4111yescharacteristic
223C8×C61H(5,𝔽2)11yescharacteristic
224#33867—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61#50812yes—
231C22×C61C22×D811yescharacteristic
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61—81no—
243C4×C61#54611yescharacteristic
244C4×C61#57711yescharacteristic
245Dic61—42no—
246Dic61—81no—
247Dic61—81no—
248#3336D6111yescharacteristic, complemented
249#3285—611no—
250#2242—611no—
251#2237—611no—
252#2232—611no—
253#2229—611no—
254#2217—611no—
255#2216—611no—
256#2211—611no—
257#2189—611no—
258#2188—611no—
259C4×Q32—611no—
260D4×C16—611no—
261#2138—611no—
262C2×#358—611no—
263C2×C61—22no—
264C2×C61—41no—
265C2×C61#329411yescharacteristic
266C2×C61C2×#50811yescharacteristic
267C2×C61#341711yescharacteristic
268D61—161no—
269C2×#135—611no—
270C22×Q16—611no—
271#552—21no—
272#548—1221no—
273C2×Q32—611no—
274C2×Q32—611no—
275C2×SD32—611no—
276C22×C16—611no—
277C22×C16—611no—
278#499C2×D6111yescharacteristic
279#495—611no—
280#485—21no—
281#483—21no—
282#479—21no—
283#467—611no—
284#467C2×D6111yescharacteristic
285#467C2×D6111yescharacteristic
286#466—611no—
287#464—1221no—
288#452—1221no—
289#448—1221no—
290C4×Q16—611no—
291C4×Q16C2×D6111yescharacteristic
292C8×D4C2×D6111yescharacteristic
293#427C2×D6111yescharacteristic
294C2×#107C2×D6111yescharacteristic
295#367—611no—
296#366—611no—
297#366—611no—
298C4⋊C16—611no—
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358—611no—
304#358—611no—
305#358—611no—
306#357—611no—
307#357—611no—
308#357—611no—
309C22⋊C16—611no—
310C22⋊C16—611no—
311C4×C16—611no—
312C61#3386711yescharacteristic, complemented, Sylow
313C2×#41—1221no—
314C22×Q8—1221no—
315#135—1222no—
316#135—2441no—
317C2×Q16—611no—
318C2×Q16—611no—
319C2×Q16—1222no—
320C2×Q16—1221no—
321C2×Q16—2441no—
322C2×Q16C22×D6111yescharacteristic
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—611no—
326C22×C8C22×D6111yescharacteristic
327C22×C8C22×D6111yescharacteristic
328#126—41no—
329#125—21no—
330#125—41no—
331#124—1221no—
332#124—2441no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—1221no—
337C22⋊Q8—2441no—
338#121—1221no—
339#121—2441no—
340C4×Q8—21no—
341C4×Q8—1221no—
342C4×D4—2441no—
343C4×D4C22⋊D6111yescharacteristic
344#117—21no—
345#117—41no—
346C2×C4⋊C4—21no—
347Q32—1222no—
348Q32—2441no—
349SD32—2441no—
350C2×C16—611no—
351C2×C16—611no—
352C2×C16—611no—
353C2×C16—611no—
354C2×C16—2441no—
355C2×C16—2441no—
356#107—21no—
357#107—611no—
358#107C22×D6111yescharacteristic
359#107C22×D6111yescharacteristic
360#107C22×D6111yescharacteristic
361#106—21no—
362#106—21no—
363C4⋊C8C22⋊D6111yescharacteristic
364#103—21no—
365#103—21no—
366#103—21no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370C22⋊C8C22⋊D6111yescharacteristic
371C22⋊C8C22⋊D6111yescharacteristic
372C4×C8C22×D6111yescharacteristic
373#41—4881no—
374#41—2442no—
375C2×Q8—21no—
376C2×Q8—1221no—
377C2×Q8—1221no—
378C2×Q8—4881no—
379C2×Q8—2442no—
380C2×Q8—2441no—
381C2×D4—1221no—
382C2×D4—2441no—
383C2×D4—2441no—
384C2×D4#1175819911yescharacteristic
385C22×C4—41no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—612no—
393Q16—1221no—
394Q16—1221no—
395Q16—1221no—
396Q16—2442no—
397SD16—4881no—
398SD16—2442no—
399D8—1221no—
400D8—1221no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D6111yescharacteristic
406C2×C8C2×C22⋊D6111yescharacteristic
407C2×C8C2×C22⋊D6111yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—1221no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4#1175819911yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4#1175819911yescharacteristic
430C22⋊C4#1175819911yescharacteristic
431C42—41no—
432C42—41no—
433C42—2441no—
434C42C2×C22⋊D6111yescharacteristic
435C16—1221no—
436C16—1221no—
437C16—1221no—
438C16—1221no—
439C23—2441no—
440C23C2×#1175819911yescharacteristic
441C23C2×#1175819911yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—1222no—
445Q8—4882no—
446Q8—2441no—
447Q8—2441no—
448Q8—2441no—
449D4—22no—
450D4—4882no—
451D4—4882no—
452D4—2441no—
453D4—2441no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4C2×#1175819911yescharacteristic
469C2×C4C2×#1175819911yescharacteristic
470C2×C4C2×#1175819911yescharacteristic
471C2×C4C2×#1175819911yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D6111yescharacteristic
477C8#1451982311yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—4881no—
484C22—4881no—
485C22—4881no—
486C22#1684019212yes—
487C22C22×#1175819911yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—4881no—
497C4—4881no—
498C4—2442no—
499C4#1684040011yescharacteristic
500C4#1684057011yescharacteristic
501C2—22no—
502C2—41no—
503C2—9761no—
504C2#2145762611yescharacteristic
505C2C2×#1684019211yescharacteristic
506C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3336C2
socleC22×C61C22×D8
prime radical#21457562C2
hypercenter#3336D61
hypocenterC61#33867
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21448053C2semidirect
3. #21448027C2semidirect
4. #21447999C2semidirect
5. #21447980C2semidirect
6. #21447917C2semidirect
7. #21447908C2semidirect
8. #21447881C2semidirect
9. #21447757C2semidirect
10. #21447752C2semidirect
11. C4×#14519723C2semidirect
12. D4×#11758185C2semidirect
13. #21447524C2semidirect
14. C2×#16839565C2semidirect
15. C61×#3336C2semidirect
248. #3336D61semidirect
312. C61#33867semidirect

Maximal subgroups

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

subgroupindex
#214575622
#214480532
#214480272
#214479992
#214479802
#214479172
#214479082
#214478812
#214477572
#214477522
C4×#145197232
D4×#117581852
#214475242
C2×#168395652
C61×#33362
#3386761