Finite Group Databasebeta

← #29663821

Subgroups — #29663821

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#21456379C211yesmaximal, characteristic, complemented
2#21455910C211yesmaximal, characteristic
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21445268C212yesmaximal, complemented
5#21445258C212yesmaximal, complemented
6#21445192C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444580C211yesmaximal, characteristic, complemented
9#21444519C211yesmaximal, characteristic, complemented
10#21444515C211yesmaximal, characteristic, complemented
11C61×#3110C211yesmaximal, characteristic, complemented
12C23×#92418—21no—
13#16840312C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840136C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840110C2211yescharacteristic, complemented
18#16840080—21no—
19#16840052C2212yes—
20#16839993—21no—
21#16839993—21no—
22#16839957C2212yes—
23#16839896—21no—
24#16839872—22no—
25C4×#11758201—21no—
26C4×#11758201—21no—
27D4×#92418—21no—
28D4×#92418—21no—
29D4×#92418C2212yes—
30D4×#92418C2212yes—
31#16839819C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839510C2212yescomplemented
38#16839508C2212yescomplemented
39#16839508C2212yescomplemented
40#16839505C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#484—21no—
44C61×#474C2212yes—
45C61×#461C2212yes—
46C61×#450—21no—
47C61×#440—22no—
48C61×#432C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758201—21no—
51C2×#11758201—21no—
52C2×#11758199—22no—
53C22×#92418—21no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418—44no—
57C22×#92418D411yescharacteristic
58C22×#92418D411yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519708—22no—
67#14519708—21no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—21no—
73#14519701—21no—
74#14519698—22no—
75#14519697—22no—
76#14519697—21no—
77#14519697—21no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81#14519688—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109#97×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125#11758222—82no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×C4×D61—41no—
130C2×Dic122—21no—
131SD16×C61—44no—
132C2×C8×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61C2×D412yes—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—41no—
144C42×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149#11758201—42no—
150#11758201—42no—
151#11758199—44no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—41no—
160C2×#92418—44no—
161C2×#92418—42no—
162C2×#92418—44no—
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C23×C61—24no—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12111yescharacteristic
172C22×D61—41no—
173Q8×C61—42no—
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—24no—
177D4×C61—22no—
178D4×C61#13514yes—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—44no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—82no—
197C2×Dic61—41no—
198D244—42no—
199D244—41no—
200D244—41no—
201D244—81no—
202C4×D61—82no—
203Dic122—81no—
204C8×C61—42no—
205#92418—21no—
206#92418—21no—
207#92418—21no—
208#92418—42no—
209#92418—42no—
210#92418AGL(1,ℤ/8)12yes—
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×C61—81no—
229C4×C61—81no—
230C4×C61C2×#13512yes—
231C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C61#57511yescharacteristic
233Dic61—81no—
234#3167—611no—
235#3110D6111yescharacteristic, complemented
236#3109—611no—
237C2×C8×D4—611no—
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—1221no—
252#533C2×D6111yescharacteristic
253#532—611no—
254C2×C4×D4C2×D6111yescharacteristic
255#498—611no—
256#492—611no—
257#487—611no—
258#486—611no—
259#484—21no—
260#481—1221no—
261#476—612no—
262#474C2×D6112yes—
263#465—1221no—
264#465—1221no—
265#461C2×D6112yes—
266#459—612no—
267#450—21no—
268#447—1221no—
269#442—1222no—
270#440—22no—
271C4×SD16—1221no—
272C4×SD16—1221no—
273C8×D4—612no—
274C8×D4—612no—
275C8×D4—1221no—
276C8×D4—1221no—
277#432—611no—
278#432C2×D6111yescharacteristic
279C2×C4⋊C8—611no—
280C2×C22⋊C8—1221no—
281C2×C4×C8—611no—
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×D4C22⋊D6111yescharacteristic
291C23×C4—21no—
292C2×SD16—22no—
293C2×SD16—1221no—
294C2×SD16—1221no—
295C2×D8—1222no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—1222no—
299C22×C8—1222no—
300C22×C8—1221no—
301C22×C8—2444no—
302C4⋊Q8—611no—
303C4⋊Q8C22×D6111yescharacteristic
304C4⋊Q8C22×D6111yescharacteristic
305#127—611no—
306#125—21no—
307#124—1221no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—1221no—
311#121—1221no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4C22⋊D6112yes—
319C4×D4C22×D6112yes—
320#117—21no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D6111yescharacteristic
324#107—22no—
325#106—1221no—
326#106—1221no—
327#106—1222no—
328C4⋊C8—612no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8C22×D6112yes—
332#103—22no—
333#103—1221no—
334#103—1221no—
335#102—22no—
336#102—1221no—
337#102—1221no—
338#102—1222no—
339#102—1222no—
340C22⋊C8—21no—
341C22⋊C8—1222no—
342C22⋊C8—1222no—
343C22⋊C8—1221no—
344#97—21no—
345#97—1221no—
346C4×C8—611no—
347C4×C8—611no—
348C4×C8—611no—
349C4×C8—611no—
350C24—21no—
351#41—4882no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1221no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—2441no—
361C2×D4C2×C22⋊D6112yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2441no—
368C22×C4D4×D6111yescharacteristic
369C22×C4#1175822411yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371SD16—44no—
372SD16—2442no—
373SD16—2442no—
374D8—2444no—
375C2×C8—22no—
376C2×C8—611no—
377C2×C8—611no—
378C2×C8—611no—
379C2×C8—611no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—2444no—
384C2×C8—2441no—
385C2×C8—2444no—
386C2×C8—2442no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4C2×C22⋊D6112yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—41no—
403C42D4×D6111yescharacteristic
404C42#1175822411yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—2441no—
412C23#1451974911yescharacteristic
413Q8—42no—
414Q8—41no—
415Q8—41no—
416Q8—4881no—
417D4—24no—
418D4—22no—
419D4—4881no—
420D4—4882no—
421D4—2442no—
422D4—2441no—
423D4—2441no—
424D4#1451979714yes—
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—41no—
435C2×C4—41no—
436C2×C4—4882no—
437C2×C4—2441no—
438C2×C4#1451974911yescharacteristic
439C2×C4#1451974911yescharacteristic
440C2×C4#1451974911yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C22×C22⋊D6111yescharacteristic
443C2×C4C2×#1175822411yescharacteristic
444C8—42no—
445C8—612no—
446C8—1221no—
447C8—1221no—
448C8—1221no—
449C8—2442no—
450C8—2442no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—4881no—
458C22—4881no—
459C22—4881no—
460C22#1684002712yes—
461C22C2×#1451974911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—4881no—
470C4C2×#1451979712yes—
471C4AGL(1,ℤ/8)×D6111yescharacteristic
472C4#1684055811yescharacteristic
473C2—22no—
474C2—44no—
475C2—9761no—
476C2#2145528011yescharacteristic
477C2#2145539711yescharacteristic
478C2C2×#1684002711yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#121
prime radical#21455910C2
hypercenter#3110D61
hypocenterC61#19257
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
3. C2×D4×#92418C2semidirect
4. #21445268C2semidirect
5. #21445258C2semidirect
6. #21445192C2semidirect
7. #21445162C2semidirect
8. #21444580C2semidirect
9. #21444519C2semidirect
10. #21444515C2semidirect
11. C61×#3110C2semidirect
14. #16840158C22semidirect
15. #16840136C22semidirect
16. #16840112C22semidirect
17. #16840110C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839510C22semidirect
38. #16839508C22semidirect
39. #16839508C22semidirect
40. #16839505C22semidirect
235. #3110D61semidirect
288. C61#19257semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214559102
C2×D4×#924182
#214452682
#214452582
#214451922
#214451622
#214445802
#214445192
#214445152
C61×#31102
#1925761