Finite Group Databasebeta

← #29834981

Subgroups — #29834981

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456384C211yesmaximal, characteristic, complemented
2C2×#16839844C211yesmaximal, characteristic, complemented
3#21448021C211yesmaximal, characteristic, complemented
4#21447975C212yesmaximal, complemented
5#21447873C212yesmaximal, complemented
6#21447750C211yesmaximal, characteristic, complemented
7#21447730C211yesmaximal, characteristic
8#21447670C212yesmaximal, complemented
9#21447663C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2233C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840293—21no—
14C2×#14519739—21no—
15C2×#14519722C2212yes—
16C2×#14519718C2212yes—
17#16840163C2211yescharacteristic, complemented
18#16840145C2211yescharacteristic, complemented
19#16840115—21no—
20#16840003C2211yescharacteristic
21#16839989—21no—
22#16839856C2211yescharacteristic, complemented
23#16839853—21no—
24D8×Dic61C2211yescharacteristic, complemented
25#16839844—21no—
26#16839844—21no—
27#16839844C2212yes—
28#16839844C2212yes—
29C2×#14519697—21no—
30C2×C8×Dic61C2211yescharacteristic
31#16839587C2212yescomplemented
32#16839580C2212yescomplemented
33#16839565C2212yescomplemented
34#16839564C2212yes—
35#16839562C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yescomplemented
38C61×#466C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yescomplemented
41C61×#357C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C22×D244—21no—
44C22×C4×D61—41no—
45C2×Dic244C2311yescharacteristic
46C2×#11758200—21no—
47C2×D488—21no—
48C2×D488—24no—
49C2×D488C2312yes—
50C22×#92418D411yescharacteristic
51#14519770—41no—
52#14519768—21no—
53#14519764—21no—
54#14519758—41no—
55#14519756—21no—
56#14519742—21no—
57D4×Dic61—21no—
58D4×Dic61—41no—
59#14519739—22no—
60#14519739—22no—
61#14519739—41no—
62#14519739—41no—
63#14519733—41no—
64C2×#11758198—41no—
65C2×C4×Dic61D411yescharacteristic
66#14519722—24no—
67#14519718—24no—
68C2×#11758185C2312yes—
69#14519711C2311yescharacteristic
70#14519708—21no—
71#14519702—21no—
72#14519700—21no—
73#14519698—21no—
74#14519697—22no—
75#14519697—22no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×D8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82#121×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C23×D61—41no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×Dic61—41no—
95C22×Dic61D811yescharacteristic
96C22×Dic61D811yescharacteristic
97C2×D244—24no—
98C2×D244—22no—
99C2×D244—42no—
100C2×D244—41no—
101C2×C4×D61—42no—
102C2×C4×D61—42no—
103C2×C4×D61—84no—
104C2×Dic122—21no—
105D8×C61—22no—
106C2×C8×C61—22no—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊Dic61—41no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116C22⋊Dic61—41no—
117#11758200—42no—
118#11758198—42no—
119#11758198—42no—
120C4⋊Dic61—21no—
121C4⋊Dic61—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418—41no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—22no—
134D488—24no—
135D488—22no—
136D488#4114yes—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141C22×D61—44no—
142C22×D61—42no—
143C22×D61—82no—
144C22×D61—82no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—82no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C22×D411yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—82no—
160C2×Dic61C2×D811yescharacteristic
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164D244—24no—
165D244—44no—
166D244—42no—
167C4×D61—84no—
168C4×D61—84no—
169Dic122—41no—
170Dic122—41no—
171C8×C61—21no—
172C8×C61C2×#4112yes—
173#92418—21no—
174#92418—21no—
175#33665—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61#50812yes—
182C22×C61C22×D811yescharacteristic
183C2×D61—44no—
184C2×D61—88no—
185C2×D61—82no—
186C2×D61—82no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61—82no—
190C4×C61#52511yescharacteristic
191C4×C61#57511yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—21no—
195Dic61—81no—
196Dic61—81no—
197Dic61#50912yes—
198#3168—611no—
199C2×C4×D8—611no—
200#2236—611no—
201#2233D6111yescharacteristic, complemented
202#2228—612no—
203#2210—612no—
204#2187—611no—
205#2184—611no—
206C4×SD32—612no—
207C4×D16—612no—
208#2162—611no—
209C2×C61—22no—
210C2×C61—161no—
211C2×C61#310011yescharacteristic
212C2×C61C2×#50811yescharacteristic
213C2×C61C2×#50911yescharacteristic
214D61—84no—
215C22×D8—611no—
216#529—1221no—
217C2×C4×D4—1221no—
218C2×SD32—612no—
219C2×D16—612no—
220#499—611no—
221#495—611no—
222#488—1221no—
223#467—611no—
224#466C2×D6111yescharacteristic
225#465—1221no—
226C4×Q16—611no—
227C4×SD16—1221no—
228C4×D8—611no—
229C4×D8—612no—
230C4×D8—612no—
231C4×D8—1221no—
232C4×D8—1221no—
233C2×#107C2×D6111yescharacteristic
234C2×#102—1221no—
235C2×C4×C8—611no—
236#367C2×D6112yescomplemented
237#366—612no—
238C4⋊C16—612no—
239#358—612no—
240#357—612no—
241#357—612no—
242#357C2×D6112yescomplemented
243C22⋊C16—611no—
244C22⋊C16C2×D6111yescharacteristic
245C4×C16—612no—
246C61#3366511yescharacteristic, complemented, Sylow
247C22×D4—1221no—
248C23×C4—2441no—
249C2×Q16—611no—
250C2×SD16—1221no—
251C2×D8—612no—
252C2×D8—1224no—
253C2×D8—1221no—
254C2×D8C22×D6111yescharacteristic
255C22×C8—611no—
256C22×C8C22×D6111yescharacteristic
257#126—2441no—
258#125—1221no—
259#124—1221no—
260#123—2441no—
261C22⋊Q8—1221no—
262#121—21no—
263C4×Q8—1221no—
264C4×D4—1222no—
265C4×D4—1222no—
266C4×D4—1221no—
267C4×D4—2441no—
268C4×D4—2441no—
269C4×D4—2441no—
270#117—2441no—
271C2×C4⋊C4—21no—
272C2×C22⋊C4—2441no—
273C2×C42—611no—
274SD32—1224no—
275D16—1224no—
276C2×C16—612no—
277C2×C16C22×D6112yes—
278#107—21no—
279#107—611no—
280#107C22×D6111yescharacteristic
281#107C22×D6112yes—
282#106—1221no—
283#103—1221no—
284#103—1221no—
285#102—21no—
286#102—1222no—
287#102—1222no—
288#102—1221no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293C24—2441no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1224no—
298C2×D4—1222no—
299C2×D4—2442no—
300C2×D4—2441no—
301C22×C4—41no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—4884no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4—2441no—
308C22×C4D4×D6111yescharacteristic
309Q16—1222no—
310SD16—2442no—
311D8—22no—
312D8—614no—
313D8—1224no—
314D8—1222no—
315C2×C8—22no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—2441no—
319C2×C8D4×D6111yescharacteristic
320C2×C8C23×D6111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—1221no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C22⋊C4—41no—
330C22⋊C4—2442no—
331C22⋊C4—2442no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C22⋊C4—2441no—
336C42—611no—
337C42—611no—
338C42—611no—
339C42—611no—
340C42—2441no—
341C16—22no—
342C16—1222no—
343C23—41no—
344C23—4882no—
345C23—4882no—
346C23—2444no—
347C23—2442no—
348C23D8×D6111yescharacteristic
349Q8—2441no—
350Q8—2441no—
351D4—41no—
352D4—41no—
353D4—82no—
354D4—1224no—
355D4—2444no—
356D4—2442no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—1222no—
367C2×C4—1222no—
368C2×C4—4884no—
369C2×C4—4884no—
370C2×C4—4882no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4D8×D6111yescharacteristic
374C2×C4C2×D4×D6111yescharacteristic
375C8—21no—
376C8—1221no—
377C8—1221no—
378C8#41×D6112yes—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—4888no—
385C22—4882no—
386C22—4882no—
387C22—2444no—
388C22#1684019312yes—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—81no—
392C4—82no—
393C4—612no—
394C4—1221no—
395C4—1221no—
396C4—1221no—
397C4—4881no—
398C4—4881no—
399C4#1684027011yescharacteristic
400C4#1684056111yescharacteristic
401C2—22no—
402C2—161no—
403C2—4884no—
404C2#2145581611yescharacteristic
405C2C2×#1684019311yescharacteristic
406C2#509×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2233C2
socleC22×C61C22×D8
prime radical#21447730C2
hypercenter#2233D61
hypocenterC61#33665
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
1. #21456384C2semidirect
2. C2×#16839844C2semidirect
3. #21448021C2semidirect
4. #21447975C2semidirect
5. #21447873C2semidirect
6. #21447750C2semidirect
8. #21447670C2semidirect
9. #21447663C2semidirect
10. #21447615C2semidirect
11. C61×#2233C2semidirect
17. #16840163C22semidirect
18. #16840145C22semidirect
22. #16839856C22semidirect
24. D8×Dic61C22semidirect
31. #16839587C22semidirect
32. #16839580C22semidirect
33. #16839565C22semidirect
37. C16×Dic61C22semidirect
40. C61×#367C22semidirect
41. C61×#357C22semidirect
201. #2233D61semidirect
236. #367C2×D61semidirect
242. #357C2×D61semidirect
246. C61#33665semidirect

Maximal subgroups

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

subgroupindex
#214563842
C2×#168398442
#214480212
#214479752
#214478732
#214477502
#214477302
#214476702
#214476632
#214476152
C61×#22332
#3366561