Finite Group Databasebeta

← #30608336

Subgroups — #30608336

Order 15616.

420 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840671C211yesmaximal, characteristic, complemented
2C2×#16840669C211yesmaximal, characteristic, complemented
3#21460949C212yesmaximal, complemented
4#21460947C212yesmaximal, complemented
5#21456285C211yesmaximal, characteristic
6C2×#14519838—21no—
7C2×#14519836—21no—
8#16840671—21no—
9#16840671C2212yes—
10#16840669—21no—
11#16840669C2212yes—
12C2×#14519810—21no—
13C2×#14519805—21no—
14#16840312C411yescharacteristic, complemented
15C2×#14519765C2211yescharacteristic
16#16840156C411yescharacteristic, complemented
17#16840156C2211yescharacteristic
18#16840151C411yescharacteristic, complemented
19#16840151C2211yescharacteristic
20#16840146—21no—
21#16840146—21no—
22C4×M16×C61C411yescharacteristic, complemented
23C2×#11758217—41no—
24C2×C22⋊#6923—41no—
25#14519838—22no—
26#14519838—41no—
27#14519836—22no—
28#14519836—41no—
29C2×#11758222—21no—
30C22×Dic122D411yescharacteristic
31C22×D244D411yescharacteristic
32#14519810—42no—
33#14519810—42no—
34#14519805—42no—
35#14519805—42no—
36C2×Dic244—21no—
37C2×Dic244—21no—
38C2×#11758190—21no—
39C2×#11758190—21no—
40C2×#11758190—21no—
41C2×#11758190—21no—
42C2×D488—21no—
43C2×D488—21no—
44#14519777C2×C411yescharacteristic, complemented
45#14519774C2×C411yescharacteristic, complemented
46#14519765—21no—
47#14519765—41no—
48#14519765C2×C411yescharacteristic
49#14519765D411yescharacteristic
50#14519765D411yescharacteristic
51#14519765C2311yescharacteristic
52#14519751—41no—
53C4×D244—41no—
54C2×#11758198—41no—
55C2×M16×C61—21no—
56C2×C42×C61C2×C411yescharacteristic
57#97×C61C2×C411yescharacteristic, complemented
58#97×C61C2×C411yescharacteristic, complemented
59C4×C8×C61—21no—
60C23×D61—21no—
61C22×#6923—81no—
62C22×C4×C61—21no—
63C22×C4×C61C22⋊C411yescharacteristic
64C2×C22⋊D61—41no—
65C22×Dic61—21no—
66#11758222—41no—
67#11758222—41no—
68#11758222—41no—
69#11758222—41no—
70C2×D244—21no—
71C2×D244—21no—
72C2×D244—41no—
73C2×D244—41no—
74C2×D244—41no—
75C2×D244C22⋊C411yescharacteristic
76C2×D244C2×D411yescharacteristic
77C2×C4×D61—41no—
78C2×Dic122—21no—
79C2×Dic122—21no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122C22⋊C411yescharacteristic
83C2×Dic122C2×D411yescharacteristic
84C22⋊#6923—42no—
85C22⋊#6923—81no—
86#11758217—42no—
87#11758217—81no—
88C2×#92420—81no—
89M16×C61—22no—
90M16×C61—22no—
91C2×C8×C61—21no—
92C2×C8×C61—21no—
93C42×C61—21no—
94C42×C61C22⋊C411yescharacteristic
95C42×C61C22×C411yescharacteristic
96#11758198—41no—
97#11758198—41no—
98#11758198—41no—
99#11758198—41no—
100#11758198—42no—
101C4⋊Dic61—41no—
102Dic244—42no—
103Dic244—42no—
104D488—42no—
105D488—42no—
106#11758190—42no—
107#11758190—42no—
108#11758190—42no—
109#11758190—42no—
110C23×C61#9911yescharacteristic
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—41no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—82no—
117C2×#6923—82no—
118C2×#6923—162no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—81no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61C2×C22⋊C411yescharacteristic
126C22⋊D61—81no—
127C22⋊D61—81no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133D244—41no—
134D244—41no—
135D244—41no—
136D244—41no—
137D244—81no—
138D244—82no—
139C4×D61—81no—
140C4×D61—81no—
141Dic122—41no—
142Dic122—41no—
143Dic122—41no—
144Dic122—41no—
145Dic122—81no—
146#92420—82no—
147C8×C61—22no—
148C8×C61—22no—
149#13578—611nomaximal, Sylow
150C22×C61—41no—
151C22×C61#35311yescharacteristic
152C22×C61#35311yescharacteristic
153C22×C61C2×#9911yescharacteristic
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—82no—
157C2×D61—81no—
158C2×D61—81no—
159C2×D61—82no—
160C2×D61—81no—
161C2×D61—161no—
162#6923—162no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—41no—
166C4×C61—41no—
167Dic61—41no—
168Dic61—41no—
169Dic61—81no—
170#3153—611no—
171C2×#355—611no—
172C2×#354—611no—
173#1407—612no—
174#1406—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61C2×#35311yescharacteristic
179D61—81no—
180D61—81no—
181D61—161no—
182C2×#137—1221no—
183C2×AGL(1,ℤ/8)—1221no—
184#532—611no—
185C2×#124—611no—
186#497—611no—
187#497—611no—
188#496—611no—
189#496—611no—
190#495—1221no—
191#495—1221no—
192C2×#101—1221no—
193C2×#99—1221no—
194C4×M16#692311yescharacteristic, complemented
195#355—612no—
196#355—1221no—
197#354—612no—
198#354—1221no—
199C61#1357811yescharacteristic, complemented, Sylow
200C2×#41—1221no—
201C22×Q8—611no—
202C22×D4—611no—
203#137—2442no—
204#137—2442no—
205AGL(1,ℤ/8)—2442no—
206AGL(1,ℤ/8)—2442no—
207C2×Q16—1221no—
208C2×Q16—1221no—
209C2×SD16—1221no—
210C2×SD16—1221no—
211C2×SD16—1221no—
212C2×SD16—1221no—
213C2×D8—1221no—
214C2×D8—1221no—
215C2×M16—21no—
216C2×M16—2441no—
217C4⋊Q8—611no—
218#127—611no—
219#124—611no—
220#124—611no—
221#124—611no—
222#124—611no—
223#124—1221no—
224#124—2441no—
225#121—2441no—
226C4×D4—2441no—
227C2×C22⋊C4—2441no—
228C2×C22⋊C4—2441no—
229C2×C42C2×#692311yescharacteristic
230#101—1222no—
231#101—2441no—
232#99—1222no—
233#99—2441no—
234#97C2×#692311yescharacteristic, complemented
235#97C2×#692311yescharacteristic, complemented
236C4×C8—21no—
237C24—1221no—
238#41—2441no—
239#41—2441no—
240#41—2441no—
241#41—2441no—
242C2×Q8—611no—
243C2×Q8—611no—
244C2×Q8—1221no—
245C2×Q8—1221no—
246C2×Q8—2441no—
247C2×Q8—2441no—
248C2×D4—611no—
249C2×D4—611no—
250C2×D4—1221no—
251C2×D4—1221no—
252C2×D4—2441no—
253C2×D4—2441no—
254C2×D4—2441no—
255C2×D4—2441no—
256C22×C4—21no—
257C22×C4—1221no—
258C22×C4—4881no—
259C22×C4—2441no—
260C22×C4C22⋊#692311yescharacteristic
261Q16—2442no—
262Q16—2442no—
263SD16—2442no—
264SD16—2442no—
265SD16—2442no—
266SD16—2442no—
267D8—2442no—
268D8—2442no—
269M16—22no—
270M16—22no—
271M16—4881no—
272M16—2442no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—4881no—
276C4⋊C4—2441no—
277C22⋊C4—4881no—
278C22⋊C4—2441no—
279C22⋊C4—2441no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C22⋊C4—2442no—
283C22⋊C4—2442no—
284C42—21no—
285C42C22⋊#692311yescharacteristic
286C42C22×#692311yescharacteristic
287C23—1221no—
288C23—1221no—
289C23—4882no—
290C23—2441no—
291C23—2441no—
292C23—2441no—
293C23#1451983611yescharacteristic
294Q8—4881no—
295Q8—2441no—
296Q8—2441no—
297Q8—2441no—
298Q8—2441no—
299D4—4881no—
300D4—4882no—
301D4—4881no—
302D4—4881no—
303D4—2441no—
304D4—2441no—
305D4—2441no—
306D4—2441no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—9762no—
315C2×C4—4881no—
316C2×C4—4881no—
317C2×C4—4882no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4#1451983611yescharacteristic
322C2×C4C2×C22⋊#692311yescharacteristic
323C8—22no—
324C8—22no—
325C8—4882no—
326C22—41no—
327C22—9761no—
328C22—4882no—
329C22—4881no—
330C22—4881no—
331C22—4882no—
332C22—4881no—
333C22—2441no—
334C22—2441no—
335C22#1684066711yescharacteristic
336C22#1684066711yescharacteristic
337C22C2×#1451983611yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—9762no—
343C4—4881no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—9761no—
350C2—4881no—
351C2—4881no—
352C2C2×#1684066711yescharacteristic
subgroupG/H
centerC2C2×#16840667
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC4×M16×C61C4
socleC2×C61C2×#353
prime radical#21456285C2
cosocle#14519765C23
hypercenterC4×M16#6923
hypocenterC61#13578
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16840671C2semidirect
2. C2×#16840669C2semidirect
3. #21460949C2semidirect
4. #21460947C2semidirect
14. #16840312C4semidirect
16. #16840156C4semidirect
18. #16840151C4semidirect
22. C4×M16×C61C4semidirect
44. #14519777C2×C4semidirect
45. #14519774C2×C4semidirect
57. #97×C61C2×C4semidirect
58. #97×C61C2×C4semidirect
194. C4×M16#6923semidirect
199. C61#13578semidirect
234. #97C2×#6923semidirect
235. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406712
C2×#168406692
#214609492
#214609472
#214562852
#1357861