Finite Group Databasebeta

← #29662328

Subgroups — #29662328

Order 15616.

716 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#21456144C211yesmaximal, characteristic, complemented
2#21455823C211yesmaximal, characteristic, complemented
3#21455820C211yesmaximal, characteristic
4C2×#16839827C211yesmaximal, characteristic, complemented
5#21445273C212yesmaximal, complemented
6#21445259C212yesmaximal, complemented
7#21445245C212yesmaximal, complemented
8#21445231C212yesmaximal, complemented
9#21444565C211yesmaximal, characteristic, complemented
10#21444469C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840312C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840119—21no—
16#16840119—21no—
17#16840114C2211yescharacteristic, complemented
18#16840114C2211yescharacteristic, complemented
19#16840110C2211yescharacteristic, complemented
20#16840110C2211yescharacteristic, complemented
21#16840090—21no—
22#16840076—21no—
23#16840049C2212yes—
24#16840043C2212yes—
25#16839964C2212yes—
26#16839958C2212yes—
27#16839912—21no—
28#16839898—21no—
29C4×#11758201—22no—
30C4×D488—22no—
31#16839827—21no—
32#16839827—21no—
33#16839827C2212yes—
34#16839827C2212yes—
35#16839815C2211yescharacteristic
36#16839756—21no—
37#16839756—21no—
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758201—22no—
53C2×D488—22no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56C2×C8×D61—22no—
57C2×C8×D61—22no—
58C2×C8×D61—44no—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D412yes—
69C4×D244C2312yes—
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758198—21no—
72#14519711—22no—
73#14519708—22no—
74C4⋊#92418C2312yes—
75#14519701—22no—
76#14519700—21no—
77#14519700—21no—
78#14519700—21no—
79#14519700—21no—
80#14519699—22no—
81#14519699—22no—
82#14519697—22no—
83#14519697—21no—
84#14519697—21no—
85#14519697—21no—
86#14519697—21no—
87C22⋊#92418—21no—
88#14519689—22no—
89#14519689—22no—
90C4×#92418—21no—
91C4×#92418D411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418C2311yescharacteristic
94C4×#92418C2311yescharacteristic
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—41no—
117C22×Dic61—21no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—21no—
121C2×D244—24no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244C2×D412yes—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—42no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C42×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147#11758201—44no—
148#11758198—22no—
149#11758198—22no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C2×D412yes—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163D488—44no—
164C8×D61—44no—
165C8×D61—44no—
166C23×C61#12111yescharacteristic
167C22×D61—24no—
168C22×D61—22no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—41no—
172Q8×C61—42no—
173Q8×C61—81no—
174C2×C4×C61—22no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#13512yes—
182C2×C4×C61AGL(1,ℤ/8)12yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C22⋊D61—82no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190D244—24no—
191D244—22no—
192D244—42no—
193D244—41no—
194D244—41no—
195D244—81no—
196D244#13514yes—
197C4×D61—42no—
198C4×D61—42no—
199C4×D61—44no—
200C4×D61—82no—
201Dic122—41no—
202Dic122—41no—
203C8×C61—42no—
204C8×C61—42no—
205#92418—21no—
206#92418—21no—
207#92418—21no—
208#92418—21no—
209#92418—42no—
210#19110—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61#46312yes—
214C22×C61C2×#12111yescharacteristic
215C2×D61—24no—
216C2×D61—48no—
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—81no—
226C4×C61—81no—
227C4×C61#46312yes—
228C4×C61C2×#13512yes—
229C4×C61C2×#13511yescharacteristic
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231Dic61—42no—
232Dic61—81no—
233#3131—611no—
234#3101—611no—
235#3100—611no—
236C2×C8×D4—611no—
237#1683—612no—
238#1681—612no—
239#1679—612no—
240#1677—612no—
241#1564—611no—
242#1542D6111yescharacteristic, complemented
243#1540—611no—
244C2×C61—22no—
245C2×C61C2×#46311yescharacteristic
246C2×C61#305011yescharacteristic
247C2×C61#306711yescharacteristic
248D61—44no—
249D61—161no—
250C23×C8—1221no—
251#533C2×D6111yescharacteristic
252#532—611no—
253C2×C4×D4—611no—
254#488—1221no—
255#488—1221no—
256#487—611no—
257#487—611no—
258#486—611no—
259#486—611no—
260#483—1221no—
261#480—1221no—
262#476—612no—
263#474—612no—
264#461—612no—
265#459—612no—
266#450—1221no—
267#447—1221no—
268C4×SD16—1222no—
269C4×D8—1222no—
270C8×D4—612no—
271C8×D4—612no—
272C8×D4—1221no—
273C8×D4—1221no—
274#431—611no—
275#431C2×D6111yescharacteristic
276C2×C4⋊C8C2×D6111yescharacteristic
277#416—1221no—
278#416—1221no—
279C2×C22⋊C8—1221no—
280C2×C4×C8—611no—
281#335—612no—
282#334—612no—
283#333C2×D6112yescomplemented
284#332—612no—
285#332C2×D6112yescomplemented
286#331—612no—
287C61#1911011yescharacteristic, complemented, Sylow
288C2×#41—1221no—
289C22×D4—611no—
290C23×C4—1221no—
291C2×SD16—1222no—
292C2×D8—1222no—
293C22×C8—21no—
294C22×C8—611no—
295C22×C8—611no—
296C22×C8—1222no—
297C22×C8—1222no—
298C22×C8—2444no—
299C4⋊Q8—611no—
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6111yescharacteristic
302#127—611no—
303#125—21no—
304#124—1221no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—1221no—
308#121—1221no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—612no—
312C4×D4—612no—
313C4×D4—1221no—
314C4×D4—1221no—
315C4×D4—1221no—
316C4×D4—1221no—
317#117—21no—
318C2×C4⋊C4—611no—
319C2×C22⋊C4—1221no—
320C2×C42C22×D6111yescharacteristic
321#107—1222no—
322#106—1222no—
323C4⋊C8—21no—
324C4⋊C8—612no—
325C4⋊C8C22×D6112yes—
326C4⋊C8C22×D6112yes—
327#103—1221no—
328#103—1221no—
329#103—1221no—
330#103—1221no—
331#103—1222no—
332#102—1221no—
333#102—1221no—
334#102—1221no—
335#102—1221no—
336#102—1222no—
337#102—1222no—
338#102—1222no—
339C22⋊C8—21no—
340C22⋊C8—1222no—
341C22⋊C8—1222no—
342C22⋊C8—1221no—
343C4×C8—21no—
344C4×C8—611no—
345C4×C8—611no—
346C4×C8—611no—
347C4×C8—611no—
348C4×C8—1221no—
349C24—1221no—
350#41—2442no—
351#41—2442no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1221no—
355C2×D4—612no—
356C2×D4—1224no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—2441no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—2441no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4#1175822211yescharacteristic
369C22×C4D4×D6111yescharacteristic
370C22×C4D4×D6111yescharacteristic
371SD16—2444no—
372D8—2444no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—42no—
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—2444no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—612no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C22⋊C4—41no—
396C22⋊C4—1222no—
397C22⋊C4—1222no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C42—41no—
401C42#1175822211yescharacteristic
402C42D4×D6111yescharacteristic
403C42D4×D6111yescharacteristic
404C42C23×D6111yescharacteristic
405C23—1224no—
406C23—1222no—
407C23—2442no—
408C23—2441no—
409C23—2442no—
410C23#1451975011yescharacteristic
411Q8—42no—
412Q8—81no—
413Q8—2441no—
414Q8—2441no—
415D4—614no—
416D4—1224no—
417D4—1222no—
418D4—4881no—
419D4—4882no—
420D4—2442no—
421D4—2441no—
422D4—2441no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—1222no—
428C2×C4—4882no—
429C2×C4—2441no—
430C2×C4—2442no—
431C2×C4—2444no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4#1451975011yescharacteristic
435C2×C4#1451975011yescharacteristic
436C2×C4#1451975011yescharacteristic
437C2×C4#1451980012yes—
438C2×C4#1451980412yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C2×D4×D6111yescharacteristic
441C2×C4C2×#1175822211yescharacteristic
442C8—42no—
443C8—42no—
444C8—1221no—
445C8—1221no—
446C8—1221no—
447C8—1221no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—1224no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22—2448no—
456C22—2442no—
457C22—2442no—
458C22#1683998012yes—
459C22C2×#1451975011yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—4881no—
466C4—2442no—
467C4#1683998012yes—
468C4#135×D6112yes—
469C4C2×#1451980011yescharacteristic
470C4C2×#1451980411yescharacteristic
471C2—22no—
472C2—9761no—
473C2—2444no—
474C2C2×#1683998011yescharacteristic
475C2#2145539911yescharacteristic
476C2#2145553811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radical#21455820C2
hypercenter#1542D61
hypocenterC61#19110
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456144C2semidirect
2. #21455823C2semidirect
4. C2×#16839827C2semidirect
5. #21445273C2semidirect
6. #21445259C2semidirect
7. #21445245C2semidirect
8. #21445231C2semidirect
9. #21444565C2semidirect
10. #21444469C2semidirect
11. C61×#1542C2semidirect
17. #16840114C22semidirect
18. #16840114C22semidirect
19. #16840110C22semidirect
20. #16840110C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839505C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
242. #1542D61semidirect
283. #333C2×D61semidirect
285. #332C2×D61semidirect
287. C61#19110semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214558232
#214558202
C2×#168398272
#214452732
#214452592
#214452452
#214452312
#214445652
#214444692
C61×#15422
#1911061