Finite Group Databasebeta

← #29667918

Subgroups — #29667918

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4#21456215C211yesmaximal, characteristic, complemented
5#21456193C211yesmaximal, characteristic, complemented
6#21445481C212yesmaximal, complemented
7#21445460C212yesmaximal, complemented
8#21445430C212yesmaximal, complemented
9#21445423C212yesmaximal, complemented
10C8×#11758194C211yesmaximal, characteristic, complemented
11C2×C61×#486C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840160C2211yescharacteristic, complemented
15#16840141C2212yescomplemented
16#16840138C2212yes—
17#16840135C2211yescharacteristic, complemented
18#16840126—22no—
19#16840124—22no—
20#16840114C2212yescomplemented
21#16840113C2211yescharacteristic, complemented
22#16840112C2212yescomplemented
23#16840110C2212yescomplemented
24#16840109C2211yescharacteristic, complemented
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2212yes—
28#16839991—22no—
29#16839969—22no—
30#16839954—22no—
31#16839918—22no—
32#16839896—22no—
33C4×#11758190—22no—
34#16839815C2211yescharacteristic
35#16839767—22no—
36#16839761—22no—
37C4×#11758194C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#486—22no—
43C61×#486C2212yes—
44C61×#486C2212yes—
45C2×#102×C61—22no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758202—24no—
49C2×#11758190—22no—
50C2×#11758199—24no—
51C2×#11758194D412yes—
52#14519777C2311yescharacteristic
53#14519777C2311yescharacteristic
54#14519777C2311yescharacteristic
55#14519774C2311yescharacteristic
56#14519769—21no—
57#14519765—21no—
58#14519756—21no—
59#14519756—21no—
60#14519751—21no—
61#14519751—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519736—21no—
67#14519709—22no—
68C4⋊#92418D412yes—
69#14519702—22no—
70#14519701—24no—
71#14519700—24no—
72#14519699—22no—
73#14519698—24no—
74#14519698—24no—
75#14519698—24no—
76#14519697—24no—
77C22⋊#92418D412yes—
78#14519688D412yes—
79C4×#92418C2312yes—
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×C8×C61D412yes—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61C2312yes—
90#127×C61—21no—
91#127×C61—21no—
92#127×C61—21no—
93#127×C61C2312yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#102×C61—24no—
98#102×C61—24no—
99C4×C8×C61D412yes—
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C24×C61—41no—
103C2×Q8×C61—22no—
104C2×Q8×C61—41no—
105C2×Q8×C61—82no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—44no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61—82no—
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—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×C8×C61—42no—
129C2×C8×C61C2×D412yes—
130C2×C8×C61C2×D412yes—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2411yescharacteristic
139C22⋊Dic61—41no—
140#11758202—84no—
141#11758199—84no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150#11758196—41no—
151#11758196—41no—
152C4×Dic61—41no—
153#11758194—24no—
154C2×#92418C2×D414yes—
155C2×#92418C2×D414yes—
156#11758190—44no—
157C23×C61—42no—
158C23×C61—82no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61#12711yescharacteristic
162C22×D61—41no—
163Q8×C61—44no—
164Q8×C61—82no—
165D4×C61—44no—
166D4×C61—44no—
167D4×C61—44no—
168D4×C61—82no—
169D4×C61—82no—
170D4×C61—82no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61#12711yescharacteristic
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#13514yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186D244—42no—
187D244—42no—
188D244—42no—
189C4×D61—82no—
190Dic122—42no—
191Dic122—42no—
192Dic122—42no—
193C8×C61—22no—
194C8×C61—22no—
195#92418—24no—
196#92418—24no—
197#19627—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—82no—
203C22×C61—164no—
204C22×C61#49512yes—
205C22×C61C2×#12711yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—82no—
212C4×C61#49512yes—
213C4×C61C2×#13512yes—
214C4×C61#57512yes—
215C4×C61#57612yes—
216Dic61—81no—
217Dic61—81no—
218Dic61—81no—
219#3209—611no—
220#3205—611no—
221#3167—611no—
222#3141—611no—
223#3138—611no—
224C2×#486D6111yescharacteristic, complemented
225#1720—612no—
226#1717—612no—
227#1711—612no—
228#1709—612no—
229C8×M16—611no—
230C2×C61—22no—
231C2×C61—162no—
232C2×C61C2×#49511yescharacteristic
233C2×C61#315311yescharacteristic
234C2×C61#315511yescharacteristic
235D61—161no—
236#533—611no—
237#532—611no—
238C2×C4⋊Q8C2×D6111yescharacteristic
239C2×#127C2×D6111yescharacteristic
240#498—611no—
241#494—612no—
242#493—612no—
243#492—611no—
244#490—1222no—
245#489—1222no—
246#487—612no—
247#487—612no—
248#487—611no—
249#486—22no—
250#486—612no—
251#486—612no—
252#486—612no—
253#486—612no—
254#486—611no—
255#486C2×D6112yes—
256#486C2×D6112yes—
257#465—1222no—
258#462—1222no—
259#459—1222no—
260#451—1222no—
261#447—1222no—
262C4×SD16—1222no—
263#431—611no—
264#418—1222no—
265#417—1222no—
266C2×#102—22no—
267C4×M16—611no—
268C2×C4×C8C2×D6111yescharacteristic
269#322—612no—
270C82—612no—
271C61#1962711yescharacteristic, complemented, Sylow
272C2×#41—1221no—
273C22×Q8—21no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4—21no—
277C2×Q16—1224no—
278C2×SD16—1222no—
279C2×D8—1224no—
280C2×M16—612no—
281C22×C8C22⋊D6112yes—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—611no—
286C4⋊Q8—611no—
287C4⋊Q8—611no—
288C4⋊Q8C22×D6112yes—
289#127—21no—
290#127—21no—
291#127—21no—
292#127—611no—
293#127C22×D6112yes—
294#125—1221no—
295#124—1221no—
296C22⋊Q8—1221no—
297C22⋊Q8—1221no—
298#121—1221no—
299#121—1221no—
300C4×Q8—1221no—
301C4×Q8—1221no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—1221no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C42C22×D6111yescharacteristic
308#106—1222no—
309C4⋊C8—612no—
310#103—1224no—
311#103—1224no—
312#103—1222no—
313#102—24no—
314#102—24no—
315#102—1224no—
316#102—1224no—
317#102—1224no—
318#102—1224no—
319#102—1222no—
320C22⋊C8—612no—
321#97—612no—
322C4×C8—612no—
323C4×C8—612no—
324C4×C8C22⋊D6112yes—
325C4×C8C22×D6111yescharacteristic
326C4×C8C22×D6111yescharacteristic
327C24—41no—
328#41—2444no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—82no—
332C2×Q8—1221no—
333C2×Q8—1221no—
334C2×Q8—1221no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—82no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—1221no—
347C2×D4—2441no—
348C22×C4—41no—
349C22×C4—2441no—
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353Q16—4884no—
354SD16—2444no—
355D8—4884no—
356M16—1224no—
357C2×C8—42no—
358C2×C8—614no—
359C2×C8—614no—
360C2×C8C2×C22⋊D6112yes—
361C2×C8C2×C22⋊D6112yes—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—2441no—
373C4⋊C4—2441no—
374C22⋊C4—2441no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C42—2441no—
378C42D4×D6111yescharacteristic
379C42D4×D6111yescharacteristic
380C42C2×C22⋊D6111yescharacteristic
381C42C23×D6111yescharacteristic
382C23—42no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—2441no—
387C23#1451977311yescharacteristic
388Q8—44no—
389Q8—82no—
390Q8—2442no—
391Q8—2442no—
392Q8—2442no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4—4882no—
400D4—2442no—
401D4—2442no—
402D4—2442no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—4882no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4#1451977311yescharacteristic
412C2×C4#1451977311yescharacteristic
413C2×C4#1451977311yescharacteristic
414C2×C4#1451979814yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C8—22no—
419C8—22no—
420C8—1224no—
421C8—1224no—
422C22—21no—
423C22—21no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—164no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22#1684014312yes—
432C22C2×#1451977311yescharacteristic
433C4—21no—
434C4—21no—
435C4—82no—
436C4—4881no—
437C4—4881no—
438C4—4881no—
439C4#1684014312yes—
440C4C2×#1451979812yes—
441C4#1684056012yes—
442C4#1684056312yes—
443C2—22no—
444C2—162no—
445C2—9761no—
446C2C2×#1684014311yescharacteristic
447C2#2145628311yescharacteristic
448C2#2145630011yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC22×C61C2×#127
prime radical#21456737C2
hypercenterC2×#486D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
3. #21456378C2semidirect
4. #21456215C2semidirect
5. #21456193C2semidirect
6. #21445481C2semidirect
7. #21445460C2semidirect
8. #21445430C2semidirect
9. #21445423C2semidirect
10. C8×#11758194C2semidirect
11. C2×C61×#486C2semidirect
14. #16840160C22semidirect
15. #16840141C22semidirect
17. #16840135C22semidirect
20. #16840114C22semidirect
21. #16840113C22semidirect
22. #16840112C22semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
224. C2×#486D61semidirect
271. C61#19627semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214567372
#214563782
#214562152
#214561932
#214454812
#214454602
#214454302
#214454232
C8×#117581942
C2×C61×#4862
#1962761