Finite Group Databasebeta

← #29836912

Subgroups — #29836912

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#21458188C211yesmaximal, characteristic, complemented
2#21457562C211yesmaximal, characteristic
3#21448051C211yesmaximal, characteristic, complemented
4#21448028C211yesmaximal, characteristic, complemented
5#21448000C211yesmaximal, characteristic, complemented
6#21447984C211yesmaximal, characteristic, complemented
7#21447915C211yesmaximal, characteristic, complemented
8#21447910C211yesmaximal, characteristic, complemented
9#21447882C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12C4×Dic488C211yesmaximal, characteristic, complemented
13#21447527C211yesmaximal, characteristic, complemented
14C2×#16839567C211yesmaximal, characteristic, complemented
15D4×C16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840452—21no—
19#16840421—21no—
20C2×Dic488C2211yescharacteristic
21C2×Dic488C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23#16840164C2211yescharacteristic
24#16840146C2211yescharacteristic
25#16840104—21no—
26#16840093—21no—
27#16840070—21no—
28#16840003C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840003C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839985—21no—
33#16839926—21no—
34#16839903—21no—
35C4×Dic244C2211yescharacteristic
36C4×Dic244C2211yescharacteristic
37#16839800C2211yescharacteristic
38C2×#14519711C2211yescharacteristic
39#16839590C2211yescharacteristic
40#16839588C2211yescharacteristic
41#16839588C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839567C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic
52C22×C16×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C8×D4×C61C2211yescharacteristic
55C61×C4⋊C16C2211yescharacteristic
56C61×C22⋊C16C2211yescharacteristic
57C61×C22⋊C16C2211yescharacteristic
58C4×C16×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×Dic244C2311yescharacteristic
69C2×#11758190—21no—
70C2×#11758190—21no—
71C2×D488C2311yescharacteristic
72#14519770—41no—
73#14519769—21no—
74#14519767—41no—
75#14519765—21no—
76#14519763—41no—
77#14519758—41no—
78#14519756—21no—
79#14519756—21no—
80#14519756—21no—
81#14519755—41no—
82#14519751—21no—
83#14519749—41no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86#14519737—41no—
87#14519736—21no—
88#14519733—41no—
89C2×C4⋊Dic61—21no—
90Dic488—22no—
91Dic488—41no—
92#14519722—41no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519709—21no—
99#14519709—21no—
100#14519702—21no—
101#14519702—21no—
102#14519702—21no—
103#14519702—21no—
104#14519702—21no—
105#14519699—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61D411yescharacteristic
109C2×C16×C61—41no—
110C2×C16×C61—41no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115C4⋊C8×C61D411yescharacteristic
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C2×D4×C61D811yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126#11758222—42no—
127#11758222—81no—
128C2×D244—21no—
129C2×C4×D61—41no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—42no—
134C2×Dic122—41no—
135C2×Dic122—81no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61D811yescharacteristic
145C22⋊C4×C61D811yescharacteristic
146C22⋊C4×C61D811yescharacteristic
147C42×C61C2×D411yescharacteristic
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170Dic244—21no—
171Dic244—21no—
172Dic244—21no—
173Dic244—21no—
174Dic244—21no—
175Dic244—42no—
176Dic244#4112yes—
177D488—21no—
178D488—21no—
179#11758190—42no—
180#11758190—81no—
181C16×C61—21no—
182C16×C61—21no—
183C16×C61—21no—
184C16×C61—21no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188D4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—82no—
199C22⋊D61—82no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—82no—
208C2×Dic61—82no—
209D244—41no—
210D244—41no—
211C4×D61—82no—
212Dic122—22no—
213Dic122—41no—
214Dic122—41no—
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#54611yescharacteristic
240C4×C61#57711yescharacteristic
241Dic61—42no—
242Dic61—81no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248#3336—611no—
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×C16D6111yescharacteristic, complemented
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—1221no—
272#548—1221no—
273C2×Q32—611no—
274C2×Q32—611no—
275C2×SD32—611no—
276C22×C16C2×D6111yescharacteristic
277C22×C16C2×D6111yescharacteristic
278#499—611no—
279#495—611no—
280#485—1221no—
281#483—1221no—
282#479—1221no—
283#467—611no—
284#467—611no—
285#467—611no—
286#466—611no—
287#464—1221no—
288#452—1221no—
289#448—1221no—
290C4×Q16—611no—
291C4×Q16—611no—
292C8×D4C2×D6111yescharacteristic
293#427—611no—
294C2×#107—611no—
295#367—611no—
296#366—611no—
297#366—611no—
298C4⋊C16C2×D6111yescharacteristic
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⋊C16C2×D6111yescharacteristic
310C22⋊C16C2×D6111yescharacteristic
311C4×C16C2×D6111yescharacteristic
312C61#3386711yescharacteristic, complemented, Sylow
313C2×#41—1221no—
314C22×Q8—1221no—
315#135—1222no—
316#135—2441no—
317C2×Q16—611no—
318C2×Q16—611no—
319C2×Q16—611no—
320C2×Q16—1222no—
321C2×Q16—1221no—
322C2×Q16—2441no—
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—611no—
326C22×C8C22×D6111yescharacteristic
327C22×C8C22×D6111yescharacteristic
328#126—2441no—
329#125—1221no—
330#125—2441no—
331#124—1221no—
332#124—2441no—
333#123—2441no—
334C22⋊Q8—1221no—
335C22⋊Q8—1221no—
336C22⋊Q8—1221no—
337C22⋊Q8—2441no—
338#121—1221no—
339#121—2441no—
340C4×Q8—1221no—
341C4×Q8—1221no—
342C4×D4—2441no—
343C4×D4D24411yescharacteristic
344#117—1221no—
345#117—2441no—
346C2×C4⋊C4—1221no—
347Q32—1222no—
348Q32—2441no—
349SD32—2441no—
350C2×C16—41no—
351C2×C16—41no—
352C2×C16C22×D6111yescharacteristic
353C2×C16C22×D6111yescharacteristic
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—611no—
358#107—611no—
359#107—611no—
360#107—1221no—
361#106—1221no—
362#106—1221no—
363C4⋊C8D24411yescharacteristic
364#103—1221no—
365#103—1221no—
366#103—1221no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370C22⋊C8D24411yescharacteristic
371C22⋊C8D24411yescharacteristic
372C4×C8C22×D6111yescharacteristic
373#41—4881no—
374#41—2442no—
375C2×Q8—1221no—
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×D4D48811yescharacteristic
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4C2×D24411yescharacteristic
389C22×C4C2×D24411yescharacteristic
390Q16—612no—
391Q16—1221no—
392Q16—1221no—
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×D24411yescharacteristic
406C2×C8C2×D24411yescharacteristic
407C2×C8C2×D24411yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4D48811yescharacteristic
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4D48811yescharacteristic
430C22⋊C4D48811yescharacteristic
431C42—2441no—
432C42—2441no—
433C42—2441no—
434C42C2×D24411yescharacteristic
435C16—21no—
436C16—21no—
437C16—21no—
438C16—21no—
439C23—2441no—
440C23C2×D48811yescharacteristic
441C23C2×D48811yescharacteristic
442Q8—1222no—
443Q8—4882no—
444Q8—2441no—
445Q8—2441no—
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—4882no—
459C2×C4—4882no—
460C2×C4—4882no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4C2×D48811yescharacteristic
469C2×C4C2×D48811yescharacteristic
470C2×C4C2×D48811yescharacteristic
471C2×C4C2×D48811yescharacteristic
472C2×C4C22×D24411yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D6111yescharacteristic
477C8#1451982211yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—4881no—
484C22—4881no—
485C22—4881no—
486C22#1684019412yes—
487C22C22×D48811yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—4881no—
493C4—4881no—
494C4—4881no—
495C4—4881no—
496C4—4881no—
497C4—4881no—
498C4—2442no—
499C4#1684040711yescharacteristic
500C4#1684057411yescharacteristic
501C2—22no—
502C2—41no—
503C2—9761no—
504C2#2145763311yescharacteristic
505C2C2×#1684019411yescharacteristic
506C2#2145897211yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×C16×C61C2
socleC22×C61C22×D8
prime radical#21457562C2
hypercenterD4×C16D61
hypocenterC61#33867
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #21458188C2semidirect
3. #21448051C2semidirect
4. #21448028C2semidirect
5. #21448000C2semidirect
6. #21447984C2semidirect
7. #21447915C2semidirect
8. #21447910C2semidirect
9. #21447882C2semidirect
10. #21447761C2semidirect
11. #21447755C2semidirect
12. C4×Dic488C2semidirect
13. #21447527C2semidirect
14. C2×#16839567C2semidirect
15. D4×C16×C61C2semidirect
260. D4×C16D61semidirect
312. C61#33867semidirect

Maximal subgroups

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

subgroupindex
#214581882
#214575622
#214480512
#214480282
#214480002
#214479842
#214479152
#214479102
#214478822
#214477612
#214477552
C4×Dic4882
#214475272
C2×#168395672
D4×C16×C612
#3386761