Finite Group Databasebeta

← #29834503

Subgroups — #29834503

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840202C211yesmaximal, characteristic, complemented
2C22×#14519717C211yesmaximal, characteristic, complemented
3C2×#16840003C211yesmaximal, characteristic, complemented
4#21447930C212yesmaximal, complemented
5#21447925C212yesmaximal, complemented
6#21447913C212yesmaximal, complemented
7#21447908C212yesmaximal, complemented
8#21447608C211yesmaximal, characteristic, complemented
9#21447538C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C61×#3050C211yesmaximal, characteristic
12C22×Dic244C2211yescharacteristic
13C2×#14519756—21no—
14#16840202—24no—
15C2×#14519723C2212yescomplemented
16C2×#14519721C2212yescomplemented
17C2×#14519717—24no—
18C2×#14519717C2211yescharacteristic
19C2×#14519717C2212yes—
20C2×#14519717C2211yescharacteristic
21C22×#11758185C2212yes—
22#16840003—21no—
23#16840003—21no—
24#16840003C2212yes—
25#16840003C2212yes—
26C2×#14519711C2211yescharacteristic
27C2×#14519702—21no—
28#16839589C2212yescomplemented
29#16839587C2212yescomplemented
30#16839566C2212yescomplemented
31#16839565C2212yescomplemented
32#16839565C2212yescomplemented
33#16839563C2212yescomplemented
34C2×#135×C61C2211yescharacteristic
35C23×C8×C61C2211yescharacteristic
36C61×#525—21no—
37C61×#467C2211yescharacteristic
38C61×#467C2211yescharacteristic
39C61×#466C2211yescharacteristic
40C61×#466C2211yescharacteristic
41C61×#465—21no—
42C61×#465—21no—
43C61×#427C2211yescharacteristic
44C61×#416—21no—
45C22×Dic122—21no—
46C2×Dic244—22no—
47C2×Dic244—22no—
48C2×Dic244—21no—
49C2×Dic244C2312yes—
50#14519756—22no—
51#14519756—22no—
52#14519756—41no—
53#14519756—41no—
54C2×C4⋊Dic61—21no—
55C2×#11758196—41no—
56C2×C22⋊Dic61—41no—
57#14519723—24no—
58#14519721—24no—
59#14519717—22no—
60#14519717—24no—
61#14519717D414yescomplemented
62C2×#11758185—24no—
63C2×#11758185C2312yes—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2312yes—
67#14519702—22no—
68#14519702—22no—
69C2×#41×C61—21no—
70C23×C4×C61D411yescharacteristic
71#135×C61—22no—
72#135×C61—22no—
73C2×Q16×C61C2311yescharacteristic
74C2×SD16×C61—21no—
75C2×D8×C61C2311yescharacteristic
76C22×C8×C61—22no—
77C22×C8×C61—22no—
78C22×C8×C61D411yescharacteristic
79C22×C8×C61D411yescharacteristic
80C22×C8×C61D411yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#123×C61—41no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87#121×C61—21no—
88#121×C61—21no—
89#120×C61—41no—
90C4×D4×C61—41no—
91C4×D4×C61—41no—
92#117×C61—21no—
93#107×C61C2311yescharacteristic
94#107×C61C2311yescharacteristic
95#106×C61—21no—
96#103×C61—21no—
97#103×C61—21no—
98#102×C61—21no—
99#102×C61—21no—
100C24×C61D811yescharacteristic
101#41×C61—42no—
102#41×C61—42no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—41no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—41no—
111C22×C4×C61D811yescharacteristic
112C22×C4×C61D811yescharacteristic
113C22×C4×C61D811yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—41no—
118C22×Dic61—41no—
119C2×Dic122—22no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—41no—
123Q16×C61—21no—
124Q16×C61—21no—
125SD16×C61—42no—
126D8×C61—21no—
127D8×C61—21no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D412yes—
139C2×C8×C61#4112yes—
140C2×C8×C61C2411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61—41no—
151C22⋊Dic61—42no—
152C22⋊Dic61—42no—
153C4⋊Dic61—22no—
154C4⋊Dic61—41no—
155#11758196—42no—
156#11758196—42no—
157Dic244—22no—
158Dic244—22no—
159Dic244—42no—
160#11758185—22no—
161#11758185C2×D414yes—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61C2×D811yescharacteristic
167C23×C61C2×D811yescharacteristic
168C23×C61C2×D811yescharacteristic
169Q8×C61—41no—
170Q8×C61—41no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—82no—
174D4×C61—82no—
175D4×C61—82no—
176C2×C4×C61—22no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—82no—
187C2×C4×C61#12112yes—
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C2×D811yescharacteristic
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61#13512yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196C2×Dic61—82no—
197Dic122—42no—
198Dic122—42no—
199Dic122—82no—
200C8×C61—21no—
201C8×C61—21no—
202C8×C61#12112yes—
203C8×C61C22×D411yescharacteristic
204C8×C61C2×#4111yescharacteristic
205#33606—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—44no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61#46612yes—
218C22×C61#51012yescomplemented
219C22×C61C22×D811yescharacteristic
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61#46612yes—
226C4×C61C2×#12111yescharacteristic
227C4×C61C2×#13511yescharacteristic
228Dic61—82no—
229Dic61—82no—
230C2×#510—611no—
231C22×M32—611no—
232#3050D6111yescharacteristic
233C2×#467—611no—
234#2220—612no—
235#2219—612no—
236#2217—612no—
237#2216—612no—
238#2159—611no—
239#2141—611no—
240#2139—611no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—41no—
244C2×C61—161no—
245C2×C61C2×#46611yescharacteristic
246C2×C61C2×#51011yescharacteristic
247C2×C61#341411yescharacteristic
248C2×#135C2×D6111yescharacteristic
249C22×Q16—611no—
250C23×C8C2×D6111yescharacteristic
251#525—21no—
252C2×C22⋊Q8—1221no—
253#510—1224no—
254C2×Q32—612no—
255C2×SD32—612no—
256C2×M32—611no—
257C2×M32—611no—
258C2×M32—612no—
259C2×M32—1224no—
260C22×C16—612no—
261#467—612no—
262#467—612no—
263#467—1221no—
264#467—1221no—
265#467C2×D6111yescharacteristic
266#467C2×D6111yescharacteristic
267#466C2×D6111yescharacteristic
268#466C2×D6111yescharacteristic
269#465—21no—
270#465—21no—
271#427C2×D6111yescharacteristic
272C2×#107—611no—
273#416—21no—
274C2×#103—1221no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358—612no—
279#358—612no—
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×Q8—1221no—
284C23×C4C22⋊D6111yescharacteristic
285#135—22no—
286#135—22no—
287C2×Q16—612no—
288C2×Q16—1222no—
289C2×Q16—1222no—
290C2×Q16—1221no—
291C2×Q16C22×D6111yescharacteristic
292C2×SD16—21no—
293C2×D8C22×D6111yescharacteristic
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D6111yescharacteristic
297C22×C8C22⋊D6111yescharacteristic
298C22×C8C22⋊D6111yescharacteristic
299C22×C8C22×D6111yescharacteristic
300C22×C8C22×D6111yescharacteristic
301C22×C8C22×D6111yescharacteristic
302#123—41no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—1222no—
306C22⋊Q8—1222no—
307C22⋊Q8—2441no—
308C22⋊Q8—2441no—
309#121—21no—
310#121—21no—
311#120—41no—
312C4×D4—41no—
313C4×D4—41no—
314#117—21no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—1224no—
319SD32—1224no—
320M32—614no—
321M32—1224no—
322M32—1222no—
323C2×C16—612no—
324C2×C16—612no—
325C2×C16—1224no—
326#107—612no—
327#107—1221no—
328#107C22×D6111yescharacteristic
329#107C22×D6111yescharacteristic
330#106—21no—
331#103—21no—
332#103—21no—
333#103—1222no—
334#103—1222no—
335#102—21no—
336#102—21no—
337C24#1175819911yescharacteristic
338#41—42no—
339#41—42no—
340C2×Q8—21no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×Q8—2441no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—2441no—
353C22×C4—2441no—
354C22×C4#1175819911yescharacteristic
355C22×C4#1175819911yescharacteristic
356C22×C4#1175819911yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—1222no—
363Q16—1222no—
364Q16—2442no—
365SD16—42no—
366D8—21no—
367D8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8D4×D6112yes—
376C2×C8#1175822412yes—
377C2×C8C2×C22⋊D6111yescharacteristic
378C2×C8C2×C22⋊D6111yescharacteristic
379C2×C8C2×C22⋊D6111yescharacteristic
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—1222no—
386C4⋊C4—2441no—
387C4⋊C4—2442no—
388C4⋊C4—2442no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—2442no—
395C22⋊C4—2442no—
396C42—41no—
397C16—614no—
398C16—1222no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23C2×#1175819911yescharacteristic
404C23C2×#1175819911yescharacteristic
405C23C2×#1175819911yescharacteristic
406Q8—41no—
407Q8—41no—
408Q8—4882no—
409Q8—2442no—
410Q8—2442no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415D4—82no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—4882no—
428C2×C4—4882no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4#1451974912yes—
432C2×C4C2×#1175819911yescharacteristic
433C2×C4C2×#1175819911yescharacteristic
434C2×C4C2×#1175819911yescharacteristic
435C2×C4#1451979712yes—
436C2×C4C22×C22⋊D6111yescharacteristic
437C8—21no—
438C8—21no—
439C8#1451974912yes—
440C8C2×D4×D6111yescharacteristic
441C8C2×#1175822411yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#1683999612yes—
454C22#1684020212yescomplemented
455C22C22×#1175819911yescharacteristic
456C4—21no—
457C4—21no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—4882no—
462C4—4882no—
463C4#1683999612yes—
464C4C2×#1451974911yescharacteristic
465C4C2×#1451979711yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—161no—
470C2C2×#1683999611yescharacteristic
471C2C2×#1684020211yescharacteristic
472C2#2145895311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3050C2
socleC22×C61C22×D8
hypercenter#3050D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C22×#14519717C2semidirect
3. C2×#16840003C2semidirect
4. #21447930C2semidirect
5. #21447925C2semidirect
6. #21447913C2semidirect
7. #21447908C2semidirect
8. #21447608C2semidirect
9. #21447538C2semidirect
10. #21447531C2semidirect
15. C2×#14519723C22semidirect
16. C2×#14519721C22semidirect
28. #16839589C22semidirect
29. #16839587C22semidirect
30. #16839566C22semidirect
31. #16839565C22semidirect
32. #16839565C22semidirect
33. #16839563C22semidirect
61. #14519717D4semidirect
218. C22×C61#510semidirect
281. C61#33606semidirect
454. C22#16840202semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C22×#145197172
C2×#168400032
#214479302
#214479252
#214479132
#214479082
#214476082
#214475382
#214475312
C61×#30502
#3360661