Finite Group Databasebeta

← #29838089

Subgroups — #29838089

Order 15616.

516 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458832C211yesmaximal, characteristic, complemented
2#21448088C211yesmaximal, characteristic, complemented
3#21448086C211yesmaximal, characteristic
4#21448078C211yesmaximal, characteristic, complemented
5#21447881C211yesmaximal, characteristic, complemented
6#21447878C211yesmaximal, characteristic, complemented
7#21447874C211yesmaximal, characteristic, complemented
8#21447871C211yesmaximal, characteristic, complemented
9#21447861C211yesmaximal, characteristic, complemented
10#21447859C211yesmaximal, characteristic, complemented
11#21447728C211yesmaximal, characteristic, complemented
12#21447693C211yesmaximal, characteristic, complemented
13#21447688C211yesmaximal, characteristic, complemented
14C4×#14519720C211yesmaximal, characteristic, complemented
15C61×#3393C211yesmaximal, characteristic, complemented
16#16840497—21no—
17C2×#14519723C2211yescharacteristic
18C2×#14519723C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C2×#14519720C2211yescharacteristic
23C2×#14519720C2211yescharacteristic
24C2×#14519718C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26#16840146C2211yescharacteristic
27#16840146C2211yescharacteristic
28#16840139C2211yescharacteristic
29#16839978—21no—
30#16839978—21no—
31#16839963—21no—
32C4×Dic244C2211yescharacteristic
33C4×D488C2211yescharacteristic
34C4×D488C2211yescharacteristic
35#16839589C2211yescharacteristic
36C4⋊#11758185C2211yescharacteristic
37C4⋊#11758185C2211yescharacteristic
38C4⋊#11758185C2211yescharacteristic
39#16839566C2211yescharacteristic
40#16839566C2211yescharacteristic
41#16839565C2211yescharacteristic
42#16839563C2211yescharacteristic
43#16839562C2211yescharacteristic
44#16839562C2211yescharacteristic
45#16839538C2211yescharacteristic
46#16839538C2211yescharacteristic
47C4×#11758185C2211yescharacteristic
48C61×#556—21no—
49C61×#495C2211yescharacteristic
50C61×#495C2211yescharacteristic
51C61×#494C2211yescharacteristic
52C61×#464—21no—
53C61×#464—21no—
54C61×#461—21no—
55C8×Q8×C61C2211yescharacteristic
56C4×Q16×C61C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic
58C4×D8×C61C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758190—21no—
61C2×#11758190—21no—
62C2×D488—21no—
63C2×D488C2311yescharacteristic
64C2×D488C2311yescharacteristic
65#14519774—21no—
66#14519766—41no—
67#14519765—21no—
68#14519765—21no—
69#14519762—41no—
70#14519762—41no—
71C4×Dic122—21no—
72C4×D244—21no—
73C4×D244—21no—
74#14519739—41no—
75#14519723—22no—
76#14519723—41no—
77#14519721—41no—
78#14519720—22no—
79#14519720—41no—
80#14519720—41no—
81#14519720—41no—
82#14519718—22no—
83#14519718—41no—
84C2×#11758185C2311yescharacteristic
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519702—21no—
90#14519699—21no—
91#14519699—21no—
92C2×Q16×C61—21no—
93C2×Q16×C61C2311yescharacteristic
94C2×Q16×C61C2311yescharacteristic
95C2×SD16×C61—21no—
96C2×SD16×C61—21no—
97C2×D8×C61C2311yescharacteristic
98C4⋊Q8×C61—21no—
99#126×C61—41no—
100#126×C61—41no—
101#124×C61—21no—
102#124×C61—21no—
103C22⋊Q8×C61—41no—
104C4×Q8×C61—21no—
105C4×Q8×C61—21no—
106C4×Q8×C61D411yescharacteristic
107C4×D4×C61—21no—
108#117×C61—41no—
109#107×C61C2311yescharacteristic
110C4⋊C8×C61D411yescharacteristic
111C4⋊C8×C61D411yescharacteristic
112C4⋊C8×C61D411yescharacteristic
113#103×C61—21no—
114#103×C61—21no—
115#102×C61—21no—
116C4×C8×C61C2311yescharacteristic
117C4×C8×C61C2311yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61D811yescharacteristic
122C2×D4×C61—21no—
123C22×C4×C61—41no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—41no—
127C2×D244—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×Dic122—21no—
131Q16×C61—21no—
132Q16×C61—21no—
133Q16×C61—21no—
134Q16×C61—21no—
135Q16×C61—81no—
136SD16×C61—81no—
137SD16×C61—81no—
138D8×C61—21no—
139D8×C61—21no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61D811yescharacteristic
150C4⋊C4×C61D811yescharacteristic
151C4⋊C4×C61D811yescharacteristic
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155C22⋊C4×C61—41no—
156C22⋊C4×C61—41no—
157C42×C61—41no—
158C42×C61—41no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162#11758198—41no—
163#11758198—41no—
164#11758198—41no—
165#11758198—41no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169#11758196—41no—
170C4×Dic61—41no—
171Dic244—21no—
172Dic244—21no—
173D488—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177D488—81no—
178#11758190—81no—
179#11758190—81no—
180#11758185—21no—
181#11758185—21no—
182#11758185—21no—
183#11758185#4112yes—
184C23×C61—41no—
185C22×D61—41no—
186C22×D61—41no—
187Q8×C61—41no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—82no—
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×Dic61—41no—
206C2×Dic61—41no—
207D244—41no—
208D244—41no—
209D244—41no—
210D244—41no—
211D244—82no—
212D244—161no—
213C4×D61—161no—
214C4×D61—161no—
215Dic122—41no—
216Dic122—41no—
217C8×C61—21no—
218C8×C61—21no—
219C8×C61—21no—
220C8×C61C2×#4111yescharacteristic
221C8×C61H(5,𝔽2)11yescharacteristic
222#33968—611nomaximal, Sylow
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C22×D811yescharacteristic
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61#55011yescharacteristic
243C4×C61#57611yescharacteristic
244Dic61—81no—
245Dic61—81no—
246#3396—611no—
247#3393D6111yescharacteristic, complemented
248#2251—611no—
249#2250—611no—
250#2249—611no—
251#2211—611no—
252#2211—611no—
253#2210—611no—
254#2210—611no—
255#2208—611no—
256#2207—611no—
257#2183—611no—
258#2177—611no—
259#2176—611no—
260C4×SD32—611no—
261C2×C61—161no—
262C2×C61#339111yescharacteristic
263C2×C61#341411yescharacteristic
264C2×C61#341611yescharacteristic
265D61—161no—
266D61—161no—
267#559—1221no—
268#556—21no—
269C2×Q32—611no—
270C2×Q32—611no—
271C2×SD32—611no—
272C2×SD32—611no—
273C2×SD32—611no—
274C2×SD32—611no—
275C2×SD32—611no—
276C2×D16—611no—
277C2×D16—611no—
278#495—611no—
279#495—611no—
280#495C2×D6111yescharacteristic
281#495C2×D6111yescharacteristic
282#494C2×D6111yescharacteristic
283#493—611no—
284#464—21no—
285#464—21no—
286#463—1221no—
287#463—1221no—
288#461—21no—
289#460—1221no—
290C8×Q8C2×D6111yescharacteristic
291C4×Q16—611no—
292C4×Q16C2×D6111yescharacteristic
293C4×Q16C2×D6111yescharacteristic
294C4×D8—611no—
295C4×D8—611no—
296C4×D8C2×D6111yescharacteristic
297#367—611no—
298C4⋊C16—611no—
299C4⋊C16—611no—
300C4⋊C16—611no—
301#358—611no—
302#358—611no—
303#358—611no—
304#357—611no—
305#357—611no—
306#357—611no—
307#346—611no—
308#346—611no—
309C4×C16—611no—
310C61#3396811yescharacteristic, complemented, Sylow
311C2×Q16—21no—
312C2×Q16—611no—
313C2×Q16C22×D6111yescharacteristic
314C2×Q16C22×D6111yescharacteristic
315C2×SD16—21no—
316C2×SD16—21no—
317C2×SD16—1221no—
318C2×SD16—1221no—
319C2×D8—611no—
320C2×D8—611no—
321C2×D8—1221no—
322C2×D8C22×D6111yescharacteristic
323C4⋊Q8—21no—
324#127—1221no—
325#126—41no—
326#126—41no—
327#124—21no—
328#124—21no—
329#124—1221no—
330#124—1221no—
331#124—2441no—
332#123—2441no—
333#123—2441no—
334C22⋊Q8—41no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—1221no—
338C4×Q8C22⋊D6111yescharacteristic
339C4×D4—21no—
340C4×D4—1221no—
341C4×D4—1221no—
342C4×D4—2441no—
343#117—41no—
344Q32—1222no—
345Q32—2441no—
346SD32—1222no—
347SD32—2441no—
348SD32—2441no—
349SD32—2441no—
350SD32—2441no—
351D16—1222no—
352D16—2441no—
353C2×C16—611no—
354C2×C16—611no—
355C2×C16—611no—
356C2×C16—611no—
357#107—611no—
358#107C22×D6111yescharacteristic
359C4⋊C8C22⋊D6111yescharacteristic
360C4⋊C8C22⋊D6111yescharacteristic
361C4⋊C8C22⋊D6111yescharacteristic
362#103—21no—
363#103—21no—
364#103—1221no—
365#102—21no—
366#102—1221no—
367#102—1221no—
368C4×C8C22×D6111yescharacteristic
369C4×C8C22×D6111yescharacteristic
370C4×C8C22×D6111yescharacteristic
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—1221no—
374C2×Q8#1175819911yescharacteristic
375C2×D4—21no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—2441no—
379C2×D4—2441no—
380C22×C4—41no—
381C22×C4—2441no—
382C22×C4—2441no—
383Q16—21no—
384Q16—21no—
385Q16—21no—
386Q16—21no—
387Q16—81no—
388Q16—1221no—
389Q16—1221no—
390SD16—81no—
391SD16—81no—
392SD16—4881no—
393SD16—4881no—
394D8—21no—
395D8—21no—
396D8—1221no—
397D8—1221no—
398D8—1221no—
399D8—1221no—
400D8—4881no—
401C2×C8C2×C22⋊D6111yescharacteristic
402C2×C8C2×C22⋊D6111yescharacteristic
403C2×C8C2×C22⋊D6111yescharacteristic
404C2×C8C23×D6111yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—1221no—
411C4⋊C4—2441no—
412C4⋊C4#1175819911yescharacteristic
413C4⋊C4#1175819911yescharacteristic
414C4⋊C4#1175819911yescharacteristic
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C42—41no—
427C42—41no—
428C42—2441no—
429C42C2×C22⋊D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C16—612no—
433C16—1221no—
434C16—1221no—
435C16—1221no—
436C23—41no—
437C23—2441no—
438C23—2441no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—2441no—
445Q8—2441no—
446D4—41no—
447D4—41no—
448D4—9761no—
449D4—4882no—
450D4—2441no—
451D4—2441no—
452D4—2441no—
453D4—2441no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—82no—
458C2×C4—9761no—
459C2×C4—9761no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4C2×#1175819911yescharacteristic
463C2×C4C2×#1175819911yescharacteristic
464C2×C4C2×#1175819911yescharacteristic
465C2×C4C2×#1175819911yescharacteristic
466C2×C4C2×#1175819911yescharacteristic
467C2×C4C2×#1175819911yescharacteristic
468C2×C4C22×C22⋊D6111yescharacteristic
469C8—21no—
470C8—21no—
471C8—21no—
472C8#41×D6111yescharacteristic
473C8#1451982311yescharacteristic
474C22—81no—
475C22—81no—
476C22—81no—
477C22—4881no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22C22×#1175819911yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—4881no—
494C4—4881no—
495C4#1684043111yescharacteristic
496C4#1684056511yescharacteristic
497C2—161no—
498C2—9761no—
499C2—9761no—
500C2#2145879111yescharacteristic
501C2#2145895311yescharacteristic
502C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3393C2
socleC22×C61C22×D8
prime radical#21448086C2
hypercenter#3393D61
hypocenterC61#33968
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #21458832C2semidirect
2. #21448088C2semidirect
4. #21448078C2semidirect
5. #21447881C2semidirect
6. #21447878C2semidirect
7. #21447874C2semidirect
8. #21447871C2semidirect
9. #21447861C2semidirect
10. #21447859C2semidirect
11. #21447728C2semidirect
12. #21447693C2semidirect
13. #21447688C2semidirect
14. C4×#14519720C2semidirect
15. C61×#3393C2semidirect
247. #3393D61semidirect
310. C61#33968semidirect

Maximal subgroups

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

subgroupindex
#214588322
#214480882
#214480862
#214480782
#214478812
#214478782
#214478742
#214478712
#214478612
#214478592
#214477282
#214476932
#214476882
C4×#145197202
C61×#33932
#3396861