Finite Group Databasebeta

← #29665237

Subgroups — #29665237

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic
3#21454843C211yesmaximal, characteristic, complemented
4#21445230C214yesmaximal, complemented
5#21445196C214yesmaximal, complemented
6#21444589C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C2×C61×#476C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840158C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—22no—
15#16840049—22no—
16#16839977—22no—
17#16839967C2212yes—
18#16839957C2212yes—
19D4×#92418—22no—
20D4×#92418C2214yes—
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839761—21no—
25C2×C22⋊#92418—22no—
26#16839512C2214yescomplemented
27#16839511C2214yescomplemented
28#16839509C2212yescomplemented
29#16839505C2212yescomplemented
30C2×C4⋊Q8×C61C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C61×#476—22no—
33C61×#476C2214yes—
34C2×#106×C61—21no—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#102×C61—21no—
37C2×#11758222—21no—
38#14519803—44no—
39C2×#11758200—22no—
40C2×#11758199—22no—
41C2×#11758194—21no—
42C22×#92418—24no—
43#14519777C2311yescharacteristic
44#14519774C2311yescharacteristic
45#14519765—21no—
46#14519751—21no—
47#14519751—21no—
48C4×D244—21no—
49C4×D244—21no—
50#14519708—24no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—22no—
56#14519697—22no—
57#14519697—24no—
58C22⋊#92418—21no—
59C22⋊#92418—24no—
60C22⋊#92418D412yes—
61C4×#92418—21no—
62C4×#92418C2312yes—
63C22×Q8×C61—21no—
64C22×D4×C61D411yescharacteristic
65C23×C4×C61—21no—
66C22×C8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61C2312yes—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61D412yes—
74C4×D4×C61C2312yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80#106×C61—24no—
81C4⋊C8×C61—21no—
82C4⋊C8×C61C2312yes—
83#102×C61—24no—
84C24×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—41no—
87C2×Q8×C61—82no—
88C2×D4×C61—21no—
89C2×D4×C61—22no—
90C2×D4×C61SD1614yes—
91C2×D4×C61C2×D412yes—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—41no—
96C22×C4×C61—82no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×Dic122—21no—
108C2×C8×C61—22no—
109C2×C8×C61—42no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61C2×D412yes—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—22no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758200—44no—
123#11758199—44no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—84no—
132C2×#92418C2×D414yes—
133C23×C61—22no—
134C23×C61—24no—
135C23×C61—81no—
136C23×C61—81no—
137C23×C61#12111yescharacteristic
138C22×D61—41no—
139Q8×C61—82no—
140Q8×C61—82no—
141D4×C61—22no—
142D4×C61—24no—
143D4×C61C2×SD1614yes—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—44no—
148C2×C4×C61—42no—
149C2×C4×C61—82no—
150C2×C4×C61—81no—
151C2×C4×C61—82no—
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61C2×SD1612yes—
156C2×C4×C61#13512yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C22⋊D61—82no—
161C2×Dic61—41no—
162D244—41no—
163D244—41no—
164D244—42no—
165D244—81no—
166C4×D61—82no—
167Dic122—41no—
168Dic122—41no—
169C8×C61—42no—
170#92418—24no—
171#92418—42no—
172#92418—42no—
173#19391—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—24no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—84no—
180C22×C61#46512yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—42no—
189C4×C61—82no—
190C4×C61#46512yes—
191C4×C61C22×SD1611yescharacteristic
192C4×C61C2×#13511yescharacteristic
193C4×C61#57712yes—
194Dic61—81no—
195#3227—611no—
196C2×#476D6111yescharacteristic, complemented
197#3036—611no—
198#2972—611no—
199#1677—614no—
200#1671—614no—
201#1570—611no—
202#1548—611no—
203#1540—611no—
204C2×C61—22no—
205C2×C61—44no—
206C2×C61#304211yescharacteristic
207C2×C61C2×#46511yescharacteristic
208C2×C61#306811yescharacteristic
209D61—161no—
210C2×AGL(1,ℤ/8)—1221no—
211#532—611no—
212C2×C4⋊Q8C2×D6111yescharacteristic
213C2×C4×D4C2×D6111yescharacteristic
214#498—612no—
215#486—612no—
216#481—1222no—
217#476—22no—
218#476—1222no—
219#476C2×D6114yes—
220#463—1222no—
221#461—612no—
222#459—612no—
223C8×D4—614no—
224C8×D4—1222no—
225#431—611no—
226#431—611no—
227C2×#106—21no—
228#423—611no—
229C2×C4⋊C8C2×D6111yescharacteristic
230#417—1221no—
231C2×#102—21no—
232C2×C22⋊C8—1222no—
233#334—614no—
234#333—614no—
235#332—612no—
236#331—612no—
237C61#1939111yescharacteristic, complemented, Sylow
238C2×#41—1221no—
239C22×Q8—21no—
240C22×D4C22⋊D6111yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2444no—
243C2×SD16—1222no—
244C2×D8—1222no—
245C2×M16—1221no—
246C22×C8—21no—
247C22×C8—1224no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—611no—
252C4⋊Q8C22×D6112yes—
253#127—611no—
254#124—1221no—
255#121—1221no—
256#121—1221no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1221no—
260C4×D4—1221no—
261C4×D4C22⋊D6112yes—
262C4×D4C22×D6112yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D6111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D6111yescharacteristic
268#106—24no—
269#106—1224no—
270C4⋊C8—21no—
271C4⋊C8—612no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8—1221no—
275C4⋊C8C22×D6112yes—
276#103—1222no—
277#102—24no—
278#102—1224no—
279#102—1222no—
280C22⋊C8—612no—
281C22⋊C8—1224no—
282C22⋊C8—1221no—
283C4×C8—612no—
284C4×C8—1221no—
285C24—21no—
286#41—2442no—
287#41—2442no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1221no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—2441no—
298C2×D4#1175820114yes—
299C2×D4C2×C22⋊D6112yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—2441no—
306C22×C4D4×D6111yescharacteristic
307C22×C4#1175822411yescharacteristic
308C22×C4C2×C22⋊D6111yescharacteristic
309SD16—2444no—
310D8—2444no—
311M16—2442no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—614no—
315C2×C8—1222no—
316C2×C8—1222no—
317C2×C8—4884no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4C2×C22⋊D6112yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C42D4×D6111yescharacteristic
331C42#1175822411yescharacteristic
332C42C2×C22⋊D6111yescharacteristic
333C42C23×D6111yescharacteristic
334C23—22no—
335C23—24no—
336C23—81no—
337C23—81no—
338C23—2441no—
339C23#1451974911yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—2441no—
343Q8—2441no—
344D4—24no—
345D4—22no—
346D4—4881no—
347D4—4882no—
348D4—2441no—
349D4—2441no—
350D4—2442no—
351D4C2×#1175820114yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—4882no—
361C2×C4—2441no—
362C2×C4#1451974911yescharacteristic
363C2×C4#1451974911yescharacteristic
364C2×C4#1451974911yescharacteristic
365C2×C4C2×#1175820112yes—
366C2×C4#1451979712yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C22×C22⋊D6111yescharacteristic
369C2×C4C2×#1175822411yescharacteristic
370C8—42no—
371C8—1224no—
372C8—2442no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—4881no—
381C22—4881no—
382C22—4881no—
383C22#1683999312yes—
384C22C2×#1451974911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4—4881no—
391C4#1683999312yes—
392C4C22×#1175820111yescharacteristic
393C4C2×#1451979711yescharacteristic
394C4#1684056812yes—
395C2—22no—
396C2—44no—
397C2—9761no—
398C2#2145532811yescharacteristic
399C2C2×#1683999311yescharacteristic
400C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#476C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC2×#476D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
3. #21454843C2semidirect
4. #21445230C2semidirect
5. #21445196C2semidirect
6. #21444589C2semidirect
7. #21444500C2semidirect
8. #21444469C2semidirect
9. C2×C61×#476C2semidirect
12. #16840158C22semidirect
13. #16840110C22semidirect
26. #16839512C22semidirect
27. #16839511C22semidirect
28. #16839509C22semidirect
29. #16839505C22semidirect
196. C2×#476D61semidirect
237. C61#19391semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214552832
#214548432
#214452302
#214451962
#214445892
#214445002
#214444692
C2×C61×#4762
#1939161