Finite Group Databasebeta

← #29834980

Subgroups — #29834980

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
1C2×C4×D488C211yesmaximal, characteristic, complemented
2#21448019C211yesmaximal, characteristic, complemented
3#21448002C211yesmaximal, characteristic, complemented
4#21447973C212yesmaximal, complemented
5#21447871C212yesmaximal, complemented
6#21447747C211yesmaximal, characteristic, complemented
7#21447730C211yesmaximal, characteristic
8C4×#14519720C212yesmaximal, complemented
9C4×#14519718C212yesmaximal, complemented
10#21447618C211yesmaximal, characteristic, complemented
11C61×#3168C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13C2×C4×D244—21no—
14C2×#14519720C2212yes—
15C2×#14519718C2212yes—
16C4×D488—21no—
17C4×D488—21no—
18C4×D488C2212yes—
19C4×D488C2212yes—
20C2×#14519711C2211yescharacteristic
21C2×#14519699—21no—
22#16839589C2212yescomplemented
23#16839587C2212yescomplemented
24C4⋊#11758185C2212yescomplemented
25#16839566C2212yescomplemented
26#16839563C2212yescomplemented
27#16839562C2212yes—
28#16839562C2212yes—
29C22⋊#11758185C2211yescharacteristic
30C22⋊#11758185C2211yescharacteristic
31C4×#11758185C2212yescomplemented
32C61×#529—21no—
33C61×#499C2211yescharacteristic, complemented
34C61×#495C2211yescharacteristic, complemented
35C61×#488—21no—
36C61×#467C2211yescharacteristic
37C61×#466C2211yescharacteristic
38C61×#465—21no—
39C4×Q16×C61C2211yescharacteristic, complemented
40C4×SD16×C61—21no—
41C4×D8×C61C2211yescharacteristic, complemented
42C2×C4×C8×C61C2211yescharacteristic
43C22×D244—21no—
44C22×C4×D61—41no—
45C2×D488—21no—
46C2×D488—24no—
47C2×D488C2312yes—
48C4×D244—22no—
49C4×D244—22no—
50C4×D244—41no—
51C4×D244—41no—
52C2×C4⋊Dic61—21no—
53C2×#11758198—41no—
54#14519720—24no—
55#14519718—24no—
56C2×#11758185C2312yes—
57C2×#11758185C2312yes—
58#14519711—21no—
59#14519711C2312yes—
60#14519699—22no—
61#14519699—22no—
62C2×Q16×C61C2311yescharacteristic
63C2×SD16×C61—21no—
64C2×D8×C61C2311yescharacteristic
65C22×C8×C61D411yescharacteristic
66C22×C8×C61C2311yescharacteristic
67#126×C61—41no—
68#125×C61—21no—
69#124×C61—21no—
70#123×C61—41no—
71C22⋊Q8×C61—21no—
72#121×C61—21no—
73C4×Q8×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—41no—
76#117×C61—41no—
77C2×C42×C61D411yescharacteristic
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#103×C61—21no—
82#103×C61—21no—
83#102×C61—21no—
84#102×C61—21no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C23×D61—41no—
90C2×Q8×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61D811yescharacteristic
95C22×C4×C61D811yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×Dic61—41no—
98C2×D244—24no—
99C2×D244—22no—
100C2×D244—42no—
101C2×D244—41no—
102C2×C4×D61—42no—
103C2×C4×D61—42no—
104C2×C4×D61—84no—
105Q16×C61—22no—
106SD16×C61—42no—
107D8×C61—22no—
108C2×C8×C61—22no—
109C2×C8×C61—41no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C42×C61—41no—
126C42×C61D811yescharacteristic
127C42×C61D811yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130#11758198—42no—
131#11758198—42no—
132C4⋊Dic61—22no—
133C4⋊Dic61—41no—
134D488—24no—
135D488—22no—
136D488#4114yes—
137#11758185—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141C22×D61—44no—
142C22×D61—42no—
143C22×D61—82no—
144C22×D61—82no—
145Q8×C61—41no—
146Q8×C61—41no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—82no—
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×Dic61—42no—
165C2×Dic61—82no—
166D244—24no—
167D244—44no—
168D244—42no—
169C4×D61—84no—
170C4×D61—84no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61—21no—
174C8×C61C2×#4112yes—
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—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—81no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61#50912yes—
195C4×C61#52511yescharacteristic
196C4×C61#57511yescharacteristic
197Dic61—82no—
198#3168D6111yescharacteristic, complemented
199C2×C4×D8—611no—
200#2236—611no—
201#2233—611no—
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—21no—
217C2×C4×D4—1221no—
218C2×SD32—612no—
219C2×D16—612no—
220#499C2×D6111yescharacteristic, complemented
221#495C2×D6111yescharacteristic, complemented
222#488—21no—
223#467C2×D6111yescharacteristic
224#466C2×D6111yescharacteristic
225#465—21no—
226C4×Q16C2×D6111yescharacteristic, complemented
227C4×SD16—21no—
228C4×D8—612no—
229C4×D8—612no—
230C4×D8—1221no—
231C4×D8—1221no—
232C4×D8C2×D6111yescharacteristic, complemented
233C2×#107—611no—
234C2×#102—1221no—
235C2×C4×C8C2×D6111yescharacteristic
236#367—612no—
237#366—612no—
238C4⋊C16—612no—
239#358—612no—
240#357—612no—
241#357—612no—
242#357—612no—
243C22⋊C16—611no—
244C22⋊C16—611no—
245C4×C16—612no—
246C61#3366511yescharacteristic, complemented, Sylow
247C22×D4—1221no—
248C23×C4—2441no—
249C2×Q16C22×D6111yescharacteristic
250C2×SD16—21no—
251C2×D8—612no—
252C2×D8—1224no—
253C2×D8—1221no—
254C2×D8C22×D6111yescharacteristic
255C22×C8C22⋊D6111yescharacteristic
256C22×C8C22×D6111yescharacteristic
257#126—41no—
258#125—21no—
259#124—21no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—41no—
266C4×D4—1222no—
267C4×D4—1222no—
268C4×D4—2441no—
269C4×D4—2441no—
270#117—41no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—2441no—
273C2×C42C22⋊D6111yescharacteristic
274SD32—1224no—
275D16—1224no—
276C2×C16—612no—
277C2×C16—612no—
278#107—612no—
279#107—1221no—
280#107C22×D6111yescharacteristic
281#107C22×D6111yescharacteristic
282#106—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—1222no—
288#102—1222no—
289C4×C8C22⋊D6111yescharacteristic
290C4×C8C22⋊D6111yescharacteristic
291C4×C8C22×D6111yescharacteristic
292C4×C8C22×D6111yescharacteristic
293C24—2441no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1224no—
298C2×D4—1222no—
299C2×D4—2442no—
300C2×D4—2441no—
301C22×C4—41no—
302C22×C4—4884no—
303C22×C4—2441no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4#1175819911yescharacteristic
307C22×C4#1175819911yescharacteristic
308C22×C4C2×C22⋊D6111yescharacteristic
309Q16—22no—
310SD16—42no—
311D8—22no—
312D8—614no—
313D8—1224no—
314D8—1222no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8C2×C22⋊D6111yescharacteristic
318C2×C8C2×C22⋊D6111yescharacteristic
319C2×C8C2×C22⋊D6111yescharacteristic
320C2×C8C23×D6111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—1222no—
328C4⋊C4—2441no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C42—41no—
337C42#1175819911yescharacteristic
338C42#1175819911yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C2×C22⋊D6111yescharacteristic
341C16—1222no—
342C16—1222no—
343C23—41no—
344C23—4882no—
345C23—4882no—
346C23—2444no—
347C23—2442no—
348C23C2×#1175819911yescharacteristic
349Q8—41no—
350Q8—41no—
351D4—41no—
352D4—41no—
353D4—82no—
354D4—1224no—
355D4—2444no—
356D4—2442no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—4884no—
366C2×C4—4882no—
367C2×C4—4884no—
368C2×C4—2442no—
369C2×C4C2×#1175819911yescharacteristic
370C2×C4C2×#1175819911yescharacteristic
371C2×C4C2×#1175819911yescharacteristic
372C2×C4C2×#1175819911yescharacteristic
373C2×C4C2×#1175819911yescharacteristic
374C2×C4C22×C22⋊D6111yescharacteristic
375C8—21no—
376C8—21no—
377C8—21no—
378C8#41×D6112yes—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—4888no—
385C22—4882no—
386C22—4882no—
387C22—2444no—
388C22#1684019212yes—
389C22C22×#1175819911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—4882no—
398C4#1684019712yes—
399C4#1684026911yescharacteristic
400C4#1684056011yescharacteristic
401C2—22no—
402C2—161no—
403C2—4884no—
404C2#2145581511yescharacteristic
405C2C2×#1684019211yescharacteristic
406C2C2×#1684019711yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3168C2
socleC22×C61C22×D8
prime radical#21447730C2
hypercenter#3168D61
hypocenterC61#33665
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C2×C4×D488C2semidirect
2. #21448019C2semidirect
3. #21448002C2semidirect
4. #21447973C2semidirect
5. #21447871C2semidirect
6. #21447747C2semidirect
8. C4×#14519720C2semidirect
9. C4×#14519718C2semidirect
10. #21447618C2semidirect
11. C61×#3168C2semidirect
22. #16839589C22semidirect
23. #16839587C22semidirect
24. C4⋊#11758185C22semidirect
25. #16839566C22semidirect
26. #16839563C22semidirect
31. C4×#11758185C22semidirect
33. C61×#499C22semidirect
34. C61×#495C22semidirect
39. C4×Q16×C61C22semidirect
41. C4×D8×C61C22semidirect
198. #3168D61semidirect
220. #499C2×D61semidirect
221. #495C2×D61semidirect
226. C4×Q16C2×D61semidirect
232. C4×D8C2×D61semidirect
246. C61#33665semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4882
#214480192
#214480022
#214479732
#214478712
#214477472
#214477302
C4×#145197202
C4×#145197182
#214476182
C61×#31682
#3366561