Finite Group Databasebeta

← #29663585

Subgroups — #29663585

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic, complemented
2#21456757C211yesmaximal, characteristic, complemented
3#21456271C211yesmaximal, characteristic
4#21445303C212yesmaximal, complemented
5#21445291C212yesmaximal, complemented
6#21445179C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444469C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#2962C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14#16840312C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16#16840151—22no—
17#16840139C2212yes—
18#16840136—21no—
19#16840135C2212yes—
20#16840110—21no—
21#16840049—21no—
22#16840047C2212yescomplemented
23#16839993—22no—
24#16839964—21no—
25#16839962C2212yes—
26#16839962C2212yes—
27#16839962C2212yescomplemented
28#16839912—22no—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31#16839515C2212yescomplemented
32#16839513C2212yescomplemented
33#16839509C2211yescharacteristic, complemented
34#16839509C2211yescharacteristic, complemented
35#16839505C2212yes—
36#16839505C2212yes—
37#16839505C2211yescharacteristic, complemented
38#16839505C2211yescharacteristic, complemented
39C61×#533C2211yescharacteristic
40C61×#446—22no—
41C61×#446C2212yescomplemented
42C8×Q8×C61C2212yescomplemented
43C61×#424—21no—
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#405—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519805—44no—
52#14519805—44no—
53C2×#11758190—22no—
54C2×#11758190—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×#11758201—22no—
58C2×D488—22no—
59C2×D488—22no—
60C22×#92418—21no—
61#14519777C2311yescharacteristic
62#14519774—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774C2312yes—
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×D244—21no—
71C4×D244—21no—
72#14519708—22no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519697—22no—
79#14519697—22no—
80#14519697—22no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C2×M16×C61D411yescharacteristic
84C2×M16×C61D411yescharacteristic
85C22×C8×C61—21no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61D412yes—
91C4×Q8×C61—21no—
92C4×Q8×C61C2312yes—
93#117×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96C4⋊C8×C61—22no—
97C4⋊C8×C61—21no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100#97×C61—22no—
101#97×C61D412yes—
102C4×C8×C61—22no—
103C4×C8×C61C2312yes—
104C23×D61—41no—
105C2×Q8×C61—21no—
106C2×Q8×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—41no—
111#11758222—42no—
112#11758222—42no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—22no—
118C2×D244—21no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—41no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—81no—
125C2×D244—81no—
126C2×D244—82no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129M16×C61—22no—
130M16×C61SD1614yescomplemented
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—42no—
134C2×C8×C61C2×D412yes—
135C2×C8×C61C2×D412yes—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C42×C61—22no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148#11758201—82no—
149#11758201—82no—
150#11758201—82no—
151#11758198—41no—
152#11758198—41no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—81no—
159D488—44no—
160D488—44no—
161#11758190—44no—
162#11758190—44no—
163C23×C61#12111yescharacteristic
164C22×D61—41no—
165C22×D61—42no—
166C22×D61—82no—
167C22×D61—82no—
168C22×D61—82no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—81no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61C2×SD1612yes—
181C2×C4×C61AGL(1,ℤ/8)12yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C22⋊D61—82no—
186C2×Dic61—41no—
187D244—42no—
188D244—41no—
189D244—42no—
190D244—41no—
191D244—41no—
192D244—42no—
193D244—42no—
194D244—41no—
195D244—81no—
196D244—82no—
197D244—82no—
198D244—82no—
199D244—82no—
200C4×D61—82no—
201Dic122—41no—
202Dic122—41no—
203C8×C61—22no—
204C8×C61—41no—
205C8×C61—41no—
206C8×C61C2×SD1614yes—
207#92418—42no—
208#92418—41no—
209#92418—41no—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61#46012yes—
214C22×C61C2×#12111yescharacteristic
215C2×D61—81no—
216C2×D61—82no—
217C2×D61—82no—
218C2×D61—81no—
219C2×D61—82no—
220C2×D61—81no—
221C2×D61—164no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61#46012yes—
228C4×C61C22×SD1611yescharacteristic
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57611yescharacteristic
231C4×C61#57711yescharacteristic
232Dic61—81no—
233#3246—611no—
234#3208—611no—
235#3151—611no—
236#2962D6111yescharacteristic, complemented
237#1688—612no—
238#1686—612no—
239#1668—612no—
240#1666—612no—
241#1565—611no—
242#1540—611no—
243C2×#331—611no—
244C2×C61—22no—
245C2×C61C2×#46011yescharacteristic
246C2×C61#306211yescharacteristic
247C2×C61#306811yescharacteristic
248D61—161no—
249D61—162no—
250C2×AGL(1,ℤ/8)—1221no—
251C2×AGL(1,ℤ/8)—1221no—
252#533C2×D6111yescharacteristic
253#532—611no—
254C2×#127—611no—
255#496—1222no—
256#493—612no—
257#492—612no—
258#492—1221no—
259#486—1221no—
260#476—1221no—
261#475—612no—
262#465—1222no—
263#461—1221no—
264#460—612no—
265#460—612no—
266#460—612no—
267#450—1222no—
268#446—22no—
269#446C2×D6112yescomplemented
270C8×Q8C2×D6112yescomplemented
271#431—611no—
272#424—21no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8C2×D6111yescharacteristic
275#405—21no—
276C4×M16C2×D6111yescharacteristic
277#335—612no—
278#334—612no—
279#332—611no—
280#332—611no—
281#331—612no—
282#331—612no—
283#331—611no—
284#331—611no—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—1221no—
287C22×D4—1221no—
288C22×D4—1221no—
289C22×D4—1221no—
290AGL(1,ℤ/8)—2444no—
291AGL(1,ℤ/8)—2444no—
292C2×SD16—1222no—
293C2×SD16—1222no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×SD16—1222no—
297C2×D8—1222no—
298C2×D8—1222no—
299C2×M16C22⋊D6111yescharacteristic
300C2×M16C22⋊D6111yescharacteristic
301C22×C8—21no—
302C22×C8—1221no—
303C4⋊Q8—611no—
304C4⋊Q8D24411yescharacteristic
305C4⋊Q8D24411yescharacteristic
306#127—612no—
307#127—611no—
308#127—1221no—
309#127—1221no—
310#127—1221no—
311#125—21no—
312#124—1221no—
313C22⋊Q8—21no—
314C22⋊Q8D24412yes—
315#121—1221no—
316#121—1221no—
317C4×Q8—21no—
318C4×Q8C22×D6112yes—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—21no—
322C2×C42C22×D6111yescharacteristic
323#106—1222no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—21no—
327C4⋊C8—612no—
328C4⋊C8—611no—
329C4⋊C8—611no—
330C4⋊C8—1221no—
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6111yescharacteristic
333#103—1222no—
334#102—1222no—
335#102—1222no—
336#102—1222no—
337C22⋊C8—1221no—
338#97—22no—
339#97C22⋊D6112yes—
340C4×C8—22no—
341C4×C8—1221no—
342C4×C8C22×D6112yes—
343C24—2441no—
344#41—2442no—
345#41—2442no—
346C2×Q8—21no—
347C2×Q8—1221no—
348C2×Q8C2×D24412yes—
349C2×D4—1221no—
350C2×D4—1222no—
351C2×D4—1221no—
352C2×D4—1222no—
353C2×D4—1221no—
354C2×D4—1222no—
355C2×D4—4881no—
356C2×D4—4881no—
357C2×D4—4882no—
358C2×D4—2441no—
359C2×D4—2441no—
360C2×D4—2441no—
361C2×D4—2441no—
362C2×D4—2442no—
363C2×D4—2442no—
364C22×C4—2441no—
365C22×C4C2×D24411yescharacteristic
366C22×C4#1175822211yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368SD16—4882no—
369SD16—4882no—
370SD16—4882no—
371SD16—2444no—
372SD16—2444no—
373D8—2444no—
374D8—2444no—
375M16—22no—
376M16#1175820114yescomplemented
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—42no—
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—1222no—
383C2×C8—4881no—
384C2×C8C2×C22⋊D6112yes—
385C2×C8C2×C22⋊D6112yes—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4C2×D24412yes—
394C22⋊C4—22no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C42—22no—
398C42C2×D24411yescharacteristic
399C42#1175822211yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—4882no—
403C23—4882no—
404C23—4882no—
405C23—2441no—
406C23—2442no—
407C23#1451975111yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—81no—
411Q8—2441no—
412Q8—2441no—
413D4—4881no—
414D4—4882no—
415D4—4882no—
416D4—4882no—
417D4—4882no—
418D4—4882no—
419D4—2441no—
420D4—2442no—
421D4—2441no—
422D4—2442no—
423D4—2441no—
424D4—2442no—
425D4—2442no—
426D4—2441no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—4882no—
433C2×C4—2441no—
434C2×C4#1451975111yescharacteristic
435C2×C4#1451975111yescharacteristic
436C2×C4#1451975111yescharacteristic
437C2×C4C2×#1175820112yes—
438C2×C4#1451980212yes—
439C2×C4C22×C22⋊D6111yescharacteristic
440C2×C4C22×D24411yescharacteristic
441C2×C4C2×#1175822211yescharacteristic
442C8—22no—
443C8—41no—
444C8—41no—
445C8—2442no—
446C8—2441no—
447C8—2441no—
448C8C2×#1175820114yes—
449C22—21no—
450C22—21no—
451C22—9764no—
452C22—4882no—
453C22—4881no—
454C22—4881no—
455C22—4882no—
456C22—4882no—
457C22—4881no—
458C22#1683996212yes—
459C22C2×#1451975111yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—4881no—
466C4#1683996212yes—
467C4C22×#1175820111yescharacteristic
468C4C2×#1451980211yescharacteristic
469C4#1684056711yescharacteristic
470C4#1684057411yescharacteristic
471C2—22no—
472C2—9762no—
473C2—9761no—
474C2C2×#1683996211yescharacteristic
475C2#2145549911yescharacteristic
476C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2962C2
socleC22×C61C2×#121
prime radical#21456271C2
hypercenter#2962D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #21457131C2semidirect
2. #21456757C2semidirect
4. #21445303C2semidirect
5. #21445291C2semidirect
6. #21445179C2semidirect
7. #21445167C2semidirect
8. #21444571C2semidirect
9. #21444469C2semidirect
10. C2×#16839505C2semidirect
11. C61×#2962C2semidirect
22. #16840047C22semidirect
27. #16839962C22semidirect
31. #16839515C22semidirect
32. #16839513C22semidirect
33. #16839509C22semidirect
34. #16839509C22semidirect
37. #16839505C22semidirect
38. #16839505C22semidirect
41. C61×#446C22semidirect
42. C8×Q8×C61C22semidirect
130. M16×C61SD16semidirect
236. #2962D61semidirect
269. #446C2×D61semidirect
270. C8×Q8C2×D61semidirect
285. C61#19238semidirect
376. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214567572
#214562712
#214453032
#214452912
#214451792
#214451672
#214445712
#214444692
C2×#168395052
C61×#29622
#1923861