Finite Group Databasebeta

← #29667961

Subgroups — #29667961

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic, complemented
2#21456734C211yesmaximal, characteristic, complemented
3#21456394C211yesmaximal, characteristic, complemented
4#21456271C211yesmaximal, characteristic, complemented
5C2×#16840110C211yesmaximal, characteristic, complemented
6#21445491C212yesmaximal, complemented
7#21445483C212yesmaximal, complemented
8#21445426C212yesmaximal, complemented
9#21445409C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3152C211yesmaximal, characteristic, complemented
12C2×#14519805—22no—
13#16840312C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840165C2211yescharacteristic, complemented
16#16840165C2212yescomplemented
17#16840151—22no—
18#16840139C2212yes—
19#16840138C2212yes—
20#16840138C2211yescharacteristic, complemented
21#16840135C2212yes—
22#16840110—22no—
23#16840110C2212yes—
24#16840110C2212yes—
25#16840110C2212yes—
26#16840110C2211yescharacteristic, complemented
27#16840110C2212yescomplemented
28#16840108C2211yescharacteristic, complemented
29#16840075—22no—
30#16840039—22no—
31#16839996—22no—
32#16839957—22no—
33#16839895—22no—
34C4×#11758199—22no—
35#16839815C2211yescharacteristic
36C2×#14519697—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C2×#137×C61—22no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#497—22no—
44C61×#494C2212yescomplemented
45C61×#492C2212yescomplemented
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519805—48no—
52C2×#11758190—24no—
53C2×D488—24no—
54C2×#11758199—22no—
55C2×#11758199—24no—
56C22×#92418D412yes—
57#14519777C2311yescharacteristic
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774C2312yes—
62#14519774C2311yescharacteristic
63#14519765—21no—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519710—22no—
69#14519710—24no—
70C4⋊#92418D412yes—
71#14519698—22no—
72#14519697—24no—
73#14519697—22no—
74#14519697—24no—
75#14519697—24no—
76#14519697—24no—
77C22⋊#92418D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C2×#41×C61—21no—
83C22×Q8×C61—21no—
84#137×C61—48no—
85C2×Q16×C61—24no—
86C2×SD16×C61—24no—
87C2×M16×C61D412yes—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61C2312yes—
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#97×C61D412yes—
103C4×C8×C61C2312yes—
104C23×D61—41no—
105#41×C61—44no—
106C2×Q8×C61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—41no—
109C2×Q8×C61—44no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C2×C22⋊D61—41no—
120#11758222—44no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—44no—
128C2×D244—44no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—82no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135Q16×C61—48no—
136SD16×C61—48no—
137M16×C61D818yescomplemented
138C2×C8×C61C2×D414yes—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2411yescharacteristic
151#11758199—44no—
152#11758199—84no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C2×#92418—42no—
158C2×#92418C2×D412yes—
159C2×#92418C2×D412yes—
160C2×#92418C2×D414yes—
161D488—48no—
162#11758190—48no—
163C23×C61—41no—
164C23×C61#12711yescharacteristic
165C22×D61—41no—
166C22×D61—42no—
167C22×D61—82no—
168C22×D61—82no—
169C22×D61—82no—
170Q8×C61—44no—
171Q8×C61—42no—
172Q8×C61—82no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—42no—
176D4×C61—82no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—82no—
182C2×C4×C61—82no—
183C2×C4×C61#12711yescharacteristic
184C2×C4×C61#12711yescharacteristic
185C2×C4×C61#12711yescharacteristic
186C2×C4×C61C2×D814yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C22⋊D61—82no—
191C2×Dic61—41no—
192D244—44no—
193D244—42no—
194D244—42no—
195D244—44no—
196D244—44no—
197D244—42no—
198D244—82no—
199D244—82no—
200D244—82no—
201C4×D61—82no—
202Dic122—42no—
203C8×C61C2×D818yes—
204#92418—22no—
205#92418—22no—
206#92418—24no—
207#19631—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#49312yes—
214C22×C61C2×#12711yescharacteristic
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—81no—
225C4×C61—82no—
226C4×C61#49312yes—
227C4×C61C22×D812yes—
228C4×C61#57512yes—
229C4×C61#57712yes—
230Dic61—81no—
231#3226—611no—
232#3205—611no—
233#3169—611no—
234#3152D6111yescharacteristic, complemented
235#3151—611no—
236C2×#486—611no—
237#1722—612no—
238#1721—612no—
239#1709—612no—
240#1706—612no—
241C8×M16—611no—
242C2×C61—22no—
243C2×C61—161no—
244C2×C61C2×#49311yescharacteristic
245C2×C61#315311yescharacteristic
246C2×C61#315411yescharacteristic
247D61—161no—
248D61—162no—
249C2×#137—22no—
250C2×AGL(1,ℤ/8)—1222no—
251#532—611no—
252#532C2×D6111yescharacteristic
253C2×C4⋊Q8C2×D6111yescharacteristic
254C2×#127—611no—
255#500—611no—
256#500—612no—
257#497—22no—
258#496—1222no—
259#494C2×D6112yescomplemented
260#493—611no—
261#493—612no—
262#493—612no—
263#492—612no—
264#492C2×D6112yescomplemented
265#486—612no—
266#486—612no—
267#486—611no—
268#486—612no—
269#486—612no—
270#486—611no—
271#486—1222no—
272#480—1222no—
273#474—1222no—
274#466—1222no—
275#459—1222no—
276#447—1222no—
277C4×D8—1222no—
278#431—611no—
279C2×#102—1222no—
280C4×M16C2×D6111yescharacteristic
281C2×C4×C8—611no—
282#322—612no—
283C82—612no—
284C61#1963111yescharacteristic, complemented, Sylow
285C2×#41—21no—
286C2×#41—1221no—
287C22×Q8—21no—
288C22×D4—1221no—
289C22×D4—1221no—
290C22×D4—1221no—
291#137—48no—
292AGL(1,ℤ/8)—2448no—
293C2×Q16—24no—
294C2×SD16—24no—
295C2×SD16—1224no—
296C2×D8—1224no—
297C2×D8—1222no—
298C2×D8—1224no—
299C2×M16C22⋊D6112yes—
300C22×C8—612no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—611no—
305C4⋊Q8C22×D6112yes—
306C4⋊Q8C22×D6111yescharacteristic
307#127—612no—
308#127—611no—
309#127—1221no—
310#127—1221no—
311#127—1221no—
312#127C22×D6111yescharacteristic
313#124—21no—
314#124—1221no—
315#121—21no—
316#121—21no—
317#121—1221no—
318#121—1221no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—1221no—
322C4×D4—1221no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C42C22×D6111yescharacteristic
326#107—1222no—
327#107—1224no—
328C4⋊C8—612no—
329#102—1224no—
330#102—1224no—
331#102—1222no—
332#102—1224no—
333#102—1224no—
334#102—1222no—
335C22⋊C8—612no—
336#97C22⋊D6112yes—
337C4×C8—612no—
338C4×C8—611no—
339C4×C8—612no—
340C4×C8—611no—
341C4×C8C22×D6112yes—
342C24—2441no—
343#41—44no—
344#41—2444no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—44no—
349C2×Q8—1221no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1221no—
355C2×D4—1222no—
356C2×D4—1221no—
357C2×D4—1222no—
358C2×D4—1221no—
359C2×D4—1222no—
360C2×D4—4882no—
361C2×D4—2444no—
362C2×D4—2441no—
363C2×D4—2441no—
364C2×D4—2441no—
365C2×D4—2441no—
366C2×D4—2444no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—2441no—
370C22×C4D4×D6111yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373Q16—48no—
374SD16—48no—
375SD16—2448no—
376D8—4884no—
377D8—2444no—
378D8—2448no—
379M16#1175819918yescomplemented
380C2×C8—612no—
381C2×C8—612no—
382C2×C8—614no—
383C2×C8—2442no—
384C2×C8C2×C22⋊D6114yes—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C42D4×D6111yescharacteristic
398C42D4×D6111yescharacteristic
399C42C2×C22⋊D6111yescharacteristic
400C42C23×D6111yescharacteristic
401C23—41no—
402C23—4882no—
403C23—4882no—
404C23—4882no—
405C23—2441no—
406C23—2442no—
407C23#1451977311yescharacteristic
408Q8—44no—
409Q8—42no—
410Q8—82no—
411Q8—2442no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—82no—
416D4—4882no—
417D4—4882no—
418D4—4882no—
419D4—4882no—
420D4—2442no—
421D4—2444no—
422D4—2442no—
423D4—2444no—
424D4—2444no—
425D4—2442no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—4882no—
433C2×C4—2441no—
434C2×C4#1451977311yescharacteristic
435C2×C4#1451977311yescharacteristic
436C2×C4#1451977311yescharacteristic
437C2×C4C2×#1175819914yes—
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C8—1222no—
442C8—1222no—
443C8—1224no—
444C8C2×#1175819918yes—
445C22—21no—
446C22—21no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—9764no—
451C22—4881no—
452C22—4882no—
453C22—4882no—
454C22—4881no—
455C22—4882no—
456C22—4881no—
457C22#1684013812yes—
458C22C2×#1451977311yescharacteristic
459C4—21no—
460C4—21no—
461C4—81no—
462C4—82no—
463C4—4881no—
464C4#1684013812yes—
465C4C22×#1175819912yes—
466C4#1684056112yes—
467C4#1684057112yes—
468C2—22no—
469C2—161no—
470C2—9762no—
471C2—9761no—
472C2C2×#1684013811yescharacteristic
473C2#2145628611yescharacteristic
474C2#2145629811yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3152C2
socleC22×C61C2×#127
hypercenter#3152D61
hypocenterC61#19631
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
2. #21456734C2semidirect
3. #21456394C2semidirect
4. #21456271C2semidirect
5. C2×#16840110C2semidirect
6. #21445491C2semidirect
7. #21445483C2semidirect
8. #21445426C2semidirect
9. #21445409C2semidirect
10. M16×#92418C2semidirect
11. C61×#3152C2semidirect
15. #16840165C22semidirect
16. #16840165C22semidirect
20. #16840138C22semidirect
26. #16840110C22semidirect
27. #16840110C22semidirect
28. #16840108C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
44. C61×#494C22semidirect
45. C61×#492C22semidirect
137. M16×C61D8semidirect
234. #3152D61semidirect
259. #494C2×D61semidirect
264. #492C2×D61semidirect
284. C61#19631semidirect
379. M16#11758199semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214567342
#214563942
#214562712
C2×#168401102
#214454912
#214454832
#214454262
#214454092
M16×#924182
C61×#31522
#1963161