Finite Group Databasebeta

← #29838098

Subgroups — #29838098

Order 15616.

504 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
1#21458808C211yesmaximal, characteristic, complemented
2#21448091C211yesmaximal, characteristic, complemented
3#21448088C211yesmaximal, characteristic, complemented
4#21448078C211yesmaximal, characteristic
5#21447881C211yesmaximal, characteristic, complemented
6#21447878C211yesmaximal, characteristic, complemented
7#21447874C211yesmaximal, characteristic
8#21447871C211yesmaximal, characteristic
9#21447869C211yesmaximal, characteristic, complemented
10#21447866C211yesmaximal, characteristic, complemented
11#21447728C211yesmaximal, characteristic, complemented
12#21447683C211yesmaximal, characteristic, complemented
13#21447681C211yesmaximal, characteristic, complemented
14C4×#14519723C211yesmaximal, characteristic, complemented
15C61×#3393C211yesmaximal, characteristic, complemented
16#16840485—21no—
17C2×#14519723C2211yescharacteristic
18C2×#14519723C2211yescharacteristic
19C2×#14519723C2211yescharacteristic
20C2×#14519723C2211yescharacteristic
21C2×#14519721C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C2×#14519720C2211yescharacteristic
24C2×#14519720C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26#16840146C2211yescharacteristic
27#16840146C2211yescharacteristic
28#16840142C2211yescharacteristic
29#16839985—21no—
30#16839985—21no—
31#16839968—21no—
32C4×Dic244C2211yescharacteristic
33C4×Dic244C2211yescharacteristic
34C4×D488C2211yescharacteristic
35#16839587C2211yescharacteristic
36C4⋊#11758185C2211yescharacteristic
37C4⋊#11758185C2211yescharacteristic
38C4⋊#11758185C2211yescharacteristic
39#16839566C2211yescharacteristic
40#16839566C2211yescharacteristic
41#16839565C2211yescharacteristic
42#16839565C2211yescharacteristic
43#16839563C2211yescharacteristic
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×Dic244—21no—
60C2×Dic244C2311yescharacteristic
61C2×Dic244C2311yescharacteristic
62C2×#11758190—21no—
63C2×#11758190—21no—
64C2×D488C2311yescharacteristic
65#14519777—21no—
66#14519772—41no—
67#14519772—41no—
68#14519765—21no—
69#14519765—21no—
70#14519753—41no—
71C4×Dic122—21no—
72C4×Dic122—21no—
73C4×D244—21no—
74#14519735—41no—
75#14519723—22no—
76#14519723—41no—
77#14519723—41no—
78#14519723—41no—
79#14519721—22no—
80#14519721—41no—
81#14519720—22no—
82#14519720—41no—
83#14519718—41no—
84C2×#11758185C2311yescharacteristic
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519702—21no—
90#14519702—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×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128Q16×C61—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131Q16×C61—21no—
132Q16×C61—81no—
133SD16×C61—81no—
134SD16×C61—81no—
135D8×C61—21no—
136D8×C61—21no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61D811yescharacteristic
147C4⋊C4×C61D811yescharacteristic
148C4⋊C4×C61D811yescharacteristic
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C42×C61—41no—
155C42×C61—41no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—21no—
165C4⋊Dic61—41no—
166C4⋊Dic61—41no—
167#11758196—41no—
168#11758196—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171Dic244—21no—
172Dic244—21no—
173Dic244—21no—
174Dic244—21no—
175Dic244—81no—
176D488—21no—
177D488—21no—
178#11758190—81no—
179#11758190—81no—
180#11758185—21no—
181#11758185—21no—
182#11758185—21no—
183#11758185#4112yes—
184C23×C61—41no—
185C22×D61—41no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—82no—
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207D244—41no—
208D244—41no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212Dic122—41no—
213Dic122—41no—
214C8×C61—21no—
215C8×C61—21no—
216C8×C61—21no—
217C8×C61C2×#4111yescharacteristic
218C8×C61H(5,𝔽2)11yescharacteristic
219#33969—611nomaximal, Sylow
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61#55011yescharacteristic
237C4×C61#57611yescharacteristic
238Dic61—81no—
239Dic61—81no—
240Dic61—81no—
241#3393—611no—
242#3393D6111yescharacteristic, complemented
243#2251—611no—
244#2251—611no—
245#2249—611no—
246#2211—611no—
247#2211—611no—
248#2210—611no—
249#2210—611no—
250#2209—611no—
251#2209—611no—
252#2183—611no—
253#2175—611no—
254#2175—611no—
255C4×Q32—611no—
256C2×C61—161no—
257C2×C61#339111yescharacteristic
258C2×C61#341411yescharacteristic
259C2×C61#341711yescharacteristic
260D61—161no—
261#556—21no—
262#556—1221no—
263C2×Q32—611no—
264C2×Q32—611no—
265C2×Q32—611no—
266C2×Q32—611no—
267C2×SD32—611no—
268C2×SD32—611no—
269C2×SD32—611no—
270C2×SD32—611no—
271C2×D16—611no—
272#495—611no—
273#495—611no—
274#495C2×D6111yescharacteristic
275#495C2×D6111yescharacteristic
276#494—611no—
277#494C2×D6111yescharacteristic
278#464—21no—
279#464—21no—
280#464—1221no—
281#464—1221no—
282#461—21no—
283#461—1221no—
284C8×Q8C2×D6111yescharacteristic
285C4×Q16—611no—
286C4×Q16—611no—
287C4×Q16C2×D6111yescharacteristic
288C4×Q16C2×D6111yescharacteristic
289C4×D8—611no—
290C4×D8C2×D6111yescharacteristic
291#366—611no—
292C4⋊C16—611no—
293C4⋊C16—611no—
294C4⋊C16—611no—
295#358—611no—
296#358—611no—
297#358—611no—
298#358—611no—
299#357—611no—
300#357—611no—
301#346—611no—
302#346—611no—
303C4×C16—611no—
304C61#3396911yescharacteristic, complemented, Sylow
305C2×Q16—21no—
306C2×Q16—611no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16C22×D6111yescharacteristic
310C2×Q16C22×D6111yescharacteristic
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—1221no—
314C2×SD16—1221no—
315C2×D8—611no—
316C2×D8C22×D6111yescharacteristic
317C4⋊Q8—21no—
318C4⋊Q8—1221no—
319#126—41no—
320#126—41no—
321#126—2441no—
322#126—2441no—
323#124—21no—
324#124—21no—
325#124—1221no—
326#124—1221no—
327C22⋊Q8—41no—
328C22⋊Q8—2441no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—1221no—
332C4×Q8—1221no—
333C4×Q8C22⋊D6111yescharacteristic
334C4×D4—21no—
335C4×D4—1221no—
336#117—41no—
337#117—2441no—
338Q32—1222no—
339Q32—2441no—
340Q32—2441no—
341Q32—2441no—
342SD32—1222no—
343SD32—1222no—
344SD32—2441no—
345SD32—2441no—
346D16—2441no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16—611no—
350C2×C16—611no—
351#107—611no—
352#107C22×D6111yescharacteristic
353C4⋊C8C22⋊D6111yescharacteristic
354C4⋊C8C22⋊D6111yescharacteristic
355C4⋊C8C22⋊D6111yescharacteristic
356#103—21no—
357#103—21no—
358#103—1221no—
359#103—1221no—
360#102—21no—
361#102—1221no—
362C4×C8C22×D6111yescharacteristic
363C4×C8C22×D6111yescharacteristic
364C4×C8C22×D6111yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8#1175819911yescharacteristic
370C2×D4—21no—
371C2×D4—1221no—
372C22×C4—41no—
373C22×C4—2441no—
374Q16—21no—
375Q16—21no—
376Q16—21no—
377Q16—21no—
378Q16—81no—
379Q16—1221no—
380Q16—1221no—
381Q16—1221no—
382Q16—1221no—
383Q16—4881no—
384SD16—81no—
385SD16—81no—
386SD16—4881no—
387SD16—4881no—
388D8—21no—
389D8—21no—
390D8—1221no—
391D8—1221no—
392C2×C8C2×C22⋊D6111yescharacteristic
393C2×C8C2×C22⋊D6111yescharacteristic
394C2×C8C2×C22⋊D6111yescharacteristic
395C2×C8C23×D6111yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—1221no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4#1175819911yescharacteristic
407C4⋊C4#1175819911yescharacteristic
408C4⋊C4#1175819911yescharacteristic
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C42—41no—
420C42—41no—
421C42—2441no—
422C42—2441no—
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C2×C22⋊D6111yescharacteristic
426C16—612no—
427C16—1221no—
428C16—1221no—
429C16—1221no—
430C23—41no—
431C23—2441no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—2441no—
438Q8—2441no—
439Q8—2441no—
440Q8—2441no—
441D4—41no—
442D4—41no—
443D4—2441no—
444D4—2441no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—4882no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4C2×#1175819911yescharacteristic
454C2×C4C2×#1175819911yescharacteristic
455C2×C4C2×#1175819911yescharacteristic
456C2×C4C2×#1175819911yescharacteristic
457C2×C4C2×#1175819911yescharacteristic
458C2×C4C2×#1175819911yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C8—21no—
461C8—21no—
462C8—21no—
463C8#41×D6111yescharacteristic
464C8#1451982311yescharacteristic
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22C22×#1175819911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—4881no—
482C4—4881no—
483C4—4881no—
484C4#1684043111yescharacteristic
485C4#1684056511yescharacteristic
486C2—161no—
487C2—9761no—
488C2#2145879111yescharacteristic
489C2#2145895311yescharacteristic
490C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3393C2
socleC22×C61C22×D8
prime radicalC2×#14519718C22
hypercenter#3393D61
hypocenterC61#33969
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. #21458808C2semidirect
2. #21448091C2semidirect
3. #21448088C2semidirect
5. #21447881C2semidirect
6. #21447878C2semidirect
9. #21447869C2semidirect
10. #21447866C2semidirect
11. #21447728C2semidirect
12. #21447683C2semidirect
13. #21447681C2semidirect
14. C4×#14519723C2semidirect
15. C61×#3393C2semidirect
242. #3393D61semidirect
304. C61#33969semidirect

Maximal subgroups

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

subgroupindex
#214588082
#214480912
#214480882
#214480782
#214478812
#214478782
#214478742
#214478712
#214478692
#214478662
#214477282
#214476832
#214476812
C4×#145197232
C61×#33932
#3396961