Finite Group Databasebeta

← #29663818

Subgroups — #29663818

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic
4#21445270C212yesmaximal, complemented
5#21445256C212yesmaximal, complemented
6#21445194C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444582C211yesmaximal, characteristic, complemented
9#21444517C211yesmaximal, characteristic, complemented
10#21444512C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840160C2211yescharacteristic, complemented
15#16840135C2211yescharacteristic, complemented
16#16840113C2211yescharacteristic, complemented
17#16840109C2211yescharacteristic, complemented
18#16840097—21no—
19#16840080—21no—
20#16840052C2212yes—
21#16840042C2212yes—
22#16839991—21no—
23#16839991—21no—
24#16839967C2212yes—
25#16839957C2212yes—
26#16839913—21no—
27#16839896—21no—
28#16839872—22no—
29#16839862—22no—
30C4×#11758190—21no—
31C4×#11758190—21no—
32#16839819C2211yescharacteristic
33#16839819C2211yescharacteristic
34#16839515C2212yescomplemented
35#16839513C2212yescomplemented
36#16839510C2212yescomplemented
37#16839509C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yescomplemented
40C23×C8×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C8×D4×C61—21no—
43C8×D4×C61—21no—
44C8×D4×C61C2212yes—
45C8×D4×C61C2212yes—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×C22⋊C8×C61—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758190—21no—
51C2×#11758190—21no—
52C2×#11758200—22no—
53C2×#11758199—22no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519769—21no—
59#14519765—21no—
60#14519756—21no—
61#14519756—21no—
62#14519751—21no—
63#14519751—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519736—21no—
69#14519710—22no—
70#14519709—21no—
71#14519709—21no—
72#14519708—22no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519702—21no—
76#14519702—21no—
77#14519700—22no—
78#14519699—21no—
79#14519699—21no—
80#14519698—22no—
81#14519698—22no—
82#14519697—22no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85#14519688—21no—
86#14519688—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C22×C8×C61—21no—
90C22×C8×C61—22no—
91C22×C8×C61—22no—
92C22×C8×C61—44no—
93C22×C8×C61D411yescharacteristic
94C22×C8×C61D411yescharacteristic
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61C2312yes—
104C22⋊C8×C61—22no—
105C22⋊C8×C61—22no—
106C4×C8×C61D411yescharacteristic
107C4×C8×C61D411yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C24×C61—21no—
111C2×D4×C61—24no—
112C2×D4×C61—21no—
113C2×D4×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C2×C22⋊D61—41no—
123#11758222—82no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—41no—
134C2×C8×C61—44no—
135C2×C8×C61—42no—
136C2×C8×C61—44no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149C22⋊Dic61—41no—
150#11758200—44no—
151#11758199—44no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163C2×#92418—22no—
164C2×#92418—22no—
165#11758190—42no—
166#11758190—42no—
167C23×C61—24no—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12111yescharacteristic
172C22×D61—41no—
173D4×C61—24no—
174D4×C61—22no—
175D4×C61#13514yes—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—44no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C22⋊D61—82no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195D244—42no—
196D244—41no—
197D244—41no—
198D244—81no—
199C4×D61—82no—
200Dic122—42no—
201Dic122—41no—
202Dic122—41no—
203Dic122—81no—
204C8×C61—21no—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61—42no—
208C8×C61AGL(1,ℤ/8)12yes—
209#92418—42no—
210#92418—42no—
211#19257—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—24no—
215C22×C61—48no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61#47112yes—
219C22×C61C2×#12111yescharacteristic
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—42no—
228C4×C61C2×#13512yes—
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57511yescharacteristic
231Dic61—81no—
232Dic61—81no—
233Dic61—81no—
234#3167—611no—
235#3110—611no—
236#3109—611no—
237C2×C8×D4D6111yescharacteristic, complemented
238#1683—612no—
239#1681—612no—
240#1671—612no—
241#1665—612no—
242#1568—611no—
243#1552—611no—
244#1551—611no—
245C2×C61—22no—
246C2×C61—44no—
247C2×C61#303611yescharacteristic
248C2×C61#305011yescharacteristic
249C2×C61C2×#47111yescharacteristic
250D61—161no—
251C23×C8—21no—
252#533—611no—
253#532—611no—
254C2×C4×D4C2×D6111yescharacteristic
255#498—611no—
256#492—611no—
257#487—611no—
258#486—611no—
259#484—1221no—
260#481—1221no—
261#476—612no—
262#474—612no—
263#465—1221no—
264#465—1221no—
265#461—612no—
266#459—612no—
267#450—1221no—
268#447—1221no—
269#442—1222no—
270#440—1222no—
271C4×SD16—1221no—
272C4×SD16—1221no—
273C8×D4—21no—
274C8×D4—21no—
275C8×D4C2×D6112yes—
276C8×D4C2×D6112yes—
277#432—611no—
278#432—611no—
279C2×C4⋊C8C2×D6111yescharacteristic
280C2×C22⋊C8—21no—
281C2×C4×C8C2×D6111yescharacteristic
282#335—612no—
283#334—612no—
284#333—612no—
285#332—612no—
286#332—612no—
287#331—612no—
288C61#1925711yescharacteristic, complemented, Sylow
289C2×#41—1221no—
290C22×D4D24411yescharacteristic
291C23×C4—21no—
292C2×SD16—1222no—
293C2×SD16—1221no—
294C2×SD16—1221no—
295C2×D8—1222no—
296C22×C8—21no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—44no—
300C22×C8C22⋊D6111yescharacteristic
301C22×C8C22⋊D6111yescharacteristic
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305#127—611no—
306#125—1221no—
307#124—1221no—
308C22⋊Q8—1221no—
309C22⋊Q8—1221no—
310#121—1221no—
311#121—1221no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4D24412yes—
319C4×D4C22×D6112yes—
320#117—1221no—
321C2×C4⋊C4D24411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D6111yescharacteristic
324#107—1222no—
325#106—1221no—
326#106—1221no—
327#106—1222no—
328C4⋊C8—21no—
329C4⋊C8—612no—
330C4⋊C8—612no—
331C4⋊C8C22×D6112yes—
332#103—1221no—
333#103—1221no—
334#103—1222no—
335#102—1222no—
336#102—1222no—
337#102—1221no—
338#102—1221no—
339#102—1222no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—1221no—
343C22⋊C8—1221no—
344#97—1221no—
345#97—1221no—
346C4×C8C22⋊D6111yescharacteristic
347C4×C8C22⋊D6111yescharacteristic
348C4×C8C22×D6111yescharacteristic
349C4×C8C22×D6111yescharacteristic
350C24—21no—
351#41—4882no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—2441no—
361C2×D4C2×D24412yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2441no—
368C22×C4C2×D24411yescharacteristic
369C22×C4#1175822211yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371SD16—2442no—
372SD16—2444no—
373SD16—2442no—
374D8—2444no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—41no—
378C2×C8—44no—
379C2×C8—42no—
380C2×C8—44no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8C2×C22⋊D6111yescharacteristic
384C2×C8C2×C22⋊D6111yescharacteristic
385C2×C8C2×C22⋊D6111yescharacteristic
386C2×C8C2×C22⋊D6111yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4C2×D24412yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—2441no—
403C42C2×D24411yescharacteristic
404C42#1175822211yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—2441no—
412C23#1451975111yescharacteristic
413Q8—4881no—
414Q8—2442no—
415Q8—2441no—
416Q8—2441no—
417D4—24no—
418D4—22no—
419D4—4881no—
420D4—4882no—
421D4—2442no—
422D4—2441no—
423D4—2441no—
424D4#1451979914yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—4882no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4#1451975111yescharacteristic
439C2×C4#1451975111yescharacteristic
440C2×C4#1451975111yescharacteristic
441C2×C4C22×C22⋊D6111yescharacteristic
442C2×C4C22×D24411yescharacteristic
443C2×C4C2×#1175822211yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—42no—
448C8—2442no—
449C8—2442no—
450C8#1451980312yes—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—4881no—
458C22—4881no—
459C22—4881no—
460C22#1684002512yes—
461C22C2×#1451975111yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—4881no—
468C4—4881no—
469C4—4881no—
470C4C2×#1451979912yes—
471C4C2×#1451980311yescharacteristic
472C4#1684056011yescharacteristic
473C2—22no—
474C2—44no—
475C2—9761no—
476C2#2145528311yescharacteristic
477C2#2145540011yescharacteristic
478C2C2×#1684002511yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C8×D4×C61C2
socleC22×C61C2×#121
prime radical#21455910C2
hypercenterC2×C8×D4D61
hypocenterC61#19257
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #21456378C2semidirect
2. #21455918C2semidirect
4. #21445270C2semidirect
5. #21445256C2semidirect
6. #21445194C2semidirect
7. #21445160C2semidirect
8. #21444582C2semidirect
9. #21444517C2semidirect
10. #21444512C2semidirect
11. C2×C8×D4×C61C2semidirect
14. #16840160C22semidirect
15. #16840135C22semidirect
16. #16840113C22semidirect
17. #16840109C22semidirect
34. #16839515C22semidirect
35. #16839513C22semidirect
36. #16839510C22semidirect
37. #16839509C22semidirect
38. #16839507C22semidirect
39. #16839505C22semidirect
237. C2×C8×D4D61semidirect
288. C61#19257semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214559182
#214559102
#214452702
#214452562
#214451942
#214451602
#214445822
#214445172
#214445122
C2×C8×D4×C612
#1925761