Finite Group Databasebeta

← #29667958

Subgroups — #29667958

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#21456942C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21456279C211yesmaximal, characteristic, complemented
4#21456269C211yesmaximal, characteristic, complemented
5C2×#16840108C211yesmaximal, characteristic, complemented
6#21445492C212yesmaximal, complemented
7#21445484C212yesmaximal, complemented
8#21445423C212yesmaximal, complemented
9#21445408C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3205C211yesmaximal, characteristic, complemented
12C2×#14519807—22no—
13C2×#14519803—22no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840167C2212yescomplemented
17#16840165C2211yescharacteristic, complemented
18#16840157—22no—
19#16840149—22no—
20#16840141C2212yescomplemented
21#16840138C2212yes—
22#16840138C2211yescharacteristic, complemented
23#16840134C2212yes—
24#16840134C2212yescomplemented
25#16840110C2211yescharacteristic, complemented
26#16840109C2212yescomplemented
27#16840108—22no—
28#16840108C2212yes—
29#16840108C2212yes—
30#16840108C2211yescharacteristic, complemented
31#16840076—22no—
32#16840043—22no—
33#16839996—22no—
34C4×#11758199—22no—
35C2×#14519698—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#532C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#493C2212yes—
43C61×#486C2212yes—
44C61×#459—22no—
45C61×#447—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519807—48no—
50#14519803—48no—
51C2×#11758202—24no—
52C2×#11758200—24no—
53C2×#11758200—24no—
54C2×#11758199—22no—
55C2×#11758199—24no—
56C2×#11758194D412yes—
57C22×#92418D412yes—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777C2312yes—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71#14519711—24no—
72#14519710—22no—
73#14519699—24no—
74#14519698—22no—
75#14519698—24no—
76#14519698—24no—
77#14519697—22no—
78#14519688D412yes—
79C4×#92418D412yes—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C4×#92418C2312yes—
83C2×#41×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C2×D8×C61—24no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61—21no—
90#127×C61—21no—
91#127×C61—21no—
92#127×C61C2312yes—
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61D412yes—
101#102×C61—24no—
102C22⋊C8×C61D412yes—
103C4×C8×C61C2312yes—
104C24×C61—41no—
105#41×C61—44no—
106C2×Q8×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—44no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—44no—
119C2×D4×C61—82no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C2×C22⋊D61—41no—
125C22×Dic61—41no—
126#11758222—44no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122—44no—
134C2×Dic122—41no—
135D8×C61—84no—
136C2×C8×C61C2×D414yes—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61C2×D411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2411yescharacteristic
145#11758202—48no—
146#11758200—48no—
147#11758200—48no—
148#11758199—44no—
149#11758199—48no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158#11758194D818yescomplemented
159C2×#92418—42no—
160C2×#92418C2×D412yes—
161C2×#92418C2×D412yes—
162C2×#92418C2×D414yes—
163C23×C61—41no—
164C23×C61—42no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61—82no—
168C23×C61#12711yescharacteristic
169C22×D61—41no—
170Q8×C61—42no—
171D4×C61—44no—
172D4×C61—42no—
173D4×C61—42no—
174D4×C61—44no—
175D4×C61—42no—
176D4×C61—44no—
177D4×C61—82no—
178D4×C61—82no—
179D4×C61—82no—
180D4×C61—82no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—82no—
185C2×C4×C61#12711yescharacteristic
186C2×C4×C61#12711yescharacteristic
187C2×C4×C61#12711yescharacteristic
188C2×C4×C61C2×D814yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C22⋊D61—82no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—82no—
196D244—42no—
197D244—42no—
198D244—42no—
199C4×D61—82no—
200Dic122—42no—
201Dic122—44no—
202Dic122—82no—
203C8×C61—24no—
204#92418—22no—
205#92418—22no—
206#92418C2×D818yes—
207#19631—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—81no—
211C22×C61—82no—
212C22×C61—82no—
213C22×C61—81no—
214C22×C61—82no—
215C22×C61—81no—
216C22×C61—164no—
217C22×C61#49312yes—
218C22×C61C2×#12711yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—81no—
225C4×C61#49312yes—
226C4×C61C22×D812yes—
227C4×C61#57512yes—
228C4×C61#57712yes—
229Dic61—82no—
230Dic61—81no—
231#3226—611no—
232#3205D6111yescharacteristic, complemented
233#3169—611no—
234#3152—611no—
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×C61—162no—
245C2×C61C2×#49311yescharacteristic
246C2×C61#315311yescharacteristic
247C2×C61#315411yescharacteristic
248D61—161no—
249C2×#137—1222no—
250C2×AGL(1,ℤ/8)—1222no—
251#532—611no—
252#532C2×D6111yescharacteristic
253C2×C4⋊Q8—611no—
254C2×#127C2×D6111yescharacteristic
255#500—611no—
256#500—612no—
257#497—1222no—
258#496—1222no—
259#494—612no—
260#493—612no—
261#493—611no—
262#493C2×D6112yes—
263#492—612no—
264#492—612no—
265#486—611no—
266#486—612no—
267#486—612no—
268#486—611no—
269#486—612no—
270#486—1222no—
271#486C2×D6112yes—
272#480—1222no—
273#474—1222no—
274#466—1222no—
275#459—22no—
276#447—22no—
277C4×D8—1222no—
278#431C2×D6111yescharacteristic
279C2×#102—1222no—
280C4×M16—611no—
281C2×C4×C8—611no—
282#322—612no—
283C82—612no—
284C61#1963111yescharacteristic, complemented, Sylow
285C2×#41—21no—
286C2×#41—1221no—
287C22×Q8—1221no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—21no—
291#137—2448no—
292AGL(1,ℤ/8)—2448no—
293C2×Q16—1224no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C2×D8—24no—
297C2×D8—1224no—
298C2×D8—1222no—
299C2×M16—612no—
300C22×C8—612no—
301C4⋊Q8—612no—
302C4⋊Q8—611no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8C22×D6111yescharacteristic
307#127—21no—
308#127—21no—
309#127—21no—
310#127—611no—
311#127C22×D6112yes—
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—1221no—
324C2×C4⋊C4—1221no—
325C2×C42C22×D6111yescharacteristic
326#107—1222no—
327#107—1224no—
328C4⋊C8C22⋊D6112yes—
329#102—24no—
330#102—1222no—
331#102—1224no—
332#102—1224no—
333#102—1222no—
334#102—1224no—
335C22⋊C8C22⋊D6112yes—
336#97—612no—
337C4×C8—612no—
338C4×C8—611no—
339C4×C8—612no—
340C4×C8—611no—
341C4×C8C22×D6112yes—
342C24—41no—
343#41—44no—
344#41—2444no—
345C2×Q8—21no—
346C2×Q8—1221no—
347C2×Q8—1222no—
348C2×Q8—2444no—
349C2×Q8—2441no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—21no—
356C2×D4—44no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—44no—
362C2×D4—82no—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—2441no—
367C22×C4—41no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4D4×D6111yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373Q16—2448no—
374SD16—2448no—
375SD16—2448no—
376D8—84no—
377D8—2448no—
378D8—2444no—
379M16—618no—
380C2×C8—614no—
381C2×C8—612no—
382C2×C8—612no—
383C2×C8—2442no—
384C2×C8C2×C22⋊D6114yes—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—1222no—
388C4⋊C4—1221no—
389C4⋊C4—1222no—
390C4⋊C4—1221no—
391C4⋊C4—2441no—
392C4⋊C4—2441no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C42D4×D6111yescharacteristic
398C42D4×D6111yescharacteristic
399C42C2×C22⋊D6111yescharacteristic
400C42C23×D6111yescharacteristic
401C23—41no—
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—2441no—
407C23#1451977311yescharacteristic
408Q8—42no—
409Q8—4882no—
410Q8—2442no—
411Q8—2444no—
412D4—44no—
413D4—42no—
414D4—42no—
415D4—44no—
416D4—42no—
417D4—44no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—82no—
422D4—4882no—
423D4—2442no—
424D4—2442no—
425D4—2442no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—4882no—
431C2×C4—4882no—
432C2×C4—2441no—
433C2×C4—2442no—
434C2×C4#1451977311yescharacteristic
435C2×C4#1451977311yescharacteristic
436C2×C4#1451977311yescharacteristic
437C2×C4D8×D6114yes—
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C8—24no—
442C8—618no—
443C8—1222no—
444C8—1222no—
445C22—21no—
446C22—21no—
447C22—81no—
448C22—82no—
449C22—82no—
450C22—81no—
451C22—82no—
452C22—81no—
453C22—164no—
454C22—4881no—
455C22—4881no—
456C22—4881no—
457C22#1684013712yes—
458C22C2×#1451977311yescharacteristic
459C4—21no—
460C4—21no—
461C4—81no—
462C4—4881no—
463C4—4882no—
464C4#1684013712yes—
465C4C2×D8×D6112yes—
466C4#1684055812yes—
467C4#1684057312yes—
468C2—22no—
469C2—162no—
470C2—161no—
471C2—9761no—
472C2C2×#1684013711yescharacteristic
473C2#2145628111yescharacteristic
474C2#2145629111yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3205C2
socleC22×C61C2×#127
hypercenter#3205D61
hypocenterC61#19631
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
2. #21456394C2semidirect
3. #21456279C2semidirect
4. #21456269C2semidirect
5. C2×#16840108C2semidirect
6. #21445492C2semidirect
7. #21445484C2semidirect
8. #21445423C2semidirect
9. #21445408C2semidirect
10. #21444100C2semidirect
11. C61×#3205C2semidirect
16. #16840167C22semidirect
17. #16840165C22semidirect
20. #16840141C22semidirect
22. #16840138C22semidirect
24. #16840134C22semidirect
25. #16840110C22semidirect
26. #16840109C22semidirect
30. #16840108C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
158. #11758194D8semidirect
232. #3205D61semidirect
284. C61#19631semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214563942
#214562792
#214562692
C2×#168401082
#214454922
#214454842
#214454232
#214454082
#214441002
C61×#32052
#1963161