Finite Group Databasebeta

← #30608461

Subgroups — #30608461

Order 15616.

336 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#21462100C211yesmaximal, characteristic
2#21462096C211yesmaximal, characteristic
3#21460960C212yesmaximal, complemented
4#21460953C212yesmaximal, complemented
5#21456893C211yesmaximal, characteristic
6#16840741—21no—
7C2×#14519836—21no—
8#16840669—21no—
9#16840667—21no—
10#16840667C2212yes—
11#16840667C2212yes—
12#16840374C411yescharacteristic, complemented
13#16840331C2211yescharacteristic
14#16840156C411yescharacteristic, complemented
15#16840135C2211yescharacteristic
16#16840127C2211yescharacteristic
17#16840117C411yescharacteristic, complemented
18#16839985—21no—
19#16839963—21no—
20#16839926—21no—
21#16839915—21no—
22C61×#433C411yescharacteristic, complemented
23#14519856—41no—
24C2×C22⋊#6923—41no—
25#14519836—22no—
26#14519836—22no—
27#14519836—41no—
28#14519835—41no—
29C2×#11758222D411yescharacteristic
30C22×D244D411yescharacteristic
31C2×Dic244—21no—
32C2×#11758190—21no—
33C2×#11758190—21no—
34C2×#11758190—21no—
35#14519777C2×C411yescharacteristic, complemented
36#14519774D411yescharacteristic
37#14519774C2311yescharacteristic
38#14519770—41no—
39#14519769C2×C411yescharacteristic, complemented
40#14519767—41no—
41#14519765C2×C411yescharacteristic
42#14519765D411yescharacteristic
43#14519756—21no—
44#14519752—41no—
45#14519747—41no—
46C4×Dic122—21no—
47#14519702—21no—
48#14519699—21no—
49#14519699—21no—
50#14519699—21no—
51#117×C61C2×C411yescharacteristic
52C4⋊C8×C61—21no—
53C22⋊C8×C61—21no—
54#97×C61C2×C411yescharacteristic, complemented
55C4×C8×C61C2×C411yescharacteristic, complemented
56C23×D61—21no—
57C22×#6923—81no—
58C22×C4×C61C22⋊C411yescharacteristic
59C2×C22⋊D61—21no—
60#11758222—81no—
61C2×D244—21no—
62C2×D244—41no—
63C2×D244—41no—
64C2×D244—81no—
65C2×D244C22⋊C411yescharacteristic
66C2×D244C2×D411yescharacteristic
67C2×D244C2×D411yescharacteristic
68C2×C4×D61—21no—
69C2×Dic122—21no—
70C2×Dic122C22⋊C411yescharacteristic
71C22⋊#6923—42no—
72C22⋊#6923—42no—
73C22⋊#6923—81no—
74C4⋊#6923—81no—
75C4×#6923—81no—
76C2×C8×C61—21no—
77C2×C8×C61—21no—
78C4⋊C4×C61—21no—
79C22⋊C4×C61—21no—
80C42×C61C22⋊C411yescharacteristic
81C42×C61C22×C411yescharacteristic
82C22⋊Dic61—41no—
83#11758198—41no—
84#11758198—41no—
85C4⋊Dic61—21no—
86C4⋊Dic61—41no—
87#11758196—41no—
88#11758196—41no—
89#11758196—41no—
90C4×Dic61—41no—
91Dic244—81no—
92#11758190—81no—
93#11758190—81no—
94#11758190—81no—
95C23×C61#9911yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—161no—
102C2×#6923—82no—
103C2×#6923—82no—
104C2×#6923—162no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—41no—
108C2×C4×C61#9911yescharacteristic
109C2×C4×C61C2×C22⋊C411yescharacteristic
110C22⋊D61—81no—
111C2×Dic61—21no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114D244—41no—
115D244—41no—
116D244—41no—
117D244—81no—
118D244—161no—
119D244—161no—
120C4×D61—41no—
121C4×D61—41no—
122Dic122—41no—
123Dic122—41no—
124Dic122—41no—
125C8×C61—41no—
126C8×C61—41no—
127#13610—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—41no—
130C22×C61C2×#9911yescharacteristic
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—82no—
135C2×D61—81no—
136C2×D61—81no—
137C2×D61—81no—
138C2×D61—161no—
139#6923—162no—
140#6923—162no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—41no—
144C4×C61—41no—
145Dic61—41no—
146Dic61—81no—
147Dic61—81no—
148#3222—611no—
149#2128—611no—
150#2127—611no—
151#1412—612no—
152#1409—612no—
153C2×C61—21no—
154C2×C61—41no—
155C2×C61#212711yescharacteristic
156D61—81no—
157D61—81no—
158D61—81no—
159#541—611no—
160#535—611no—
161#497—611no—
162#492—611no—
163#490—611no—
164#488—611no—
165#464—1221no—
166#460—1221no—
167#452—1221no—
168#450—1221no—
169#433#692311yescharacteristic, complemented
170#410—1221no—
171C2×#99—1221no—
172#354—1221no—
173#353—612no—
174#353—612no—
175#353—1221no—
176C61#1361011yescharacteristic, complemented, Sylow
177C2×#41—611no—
178C22×D4—611no—
179C2×Q16—1221no—
180C2×SD16—1221no—
181C2×SD16—1221no—
182C2×SD16—1221no—
183C4⋊Q8—611no—
184#127—611no—
185#127—611no—
186#126—2441no—
187#125—611no—
188#125—2441no—
189#124—611no—
190#124—611no—
191C22⋊Q8—1221no—
192#121—2441no—
193#120—2441no—
194C4×Q8—1221no—
195#117—2441no—
196#117C2×#692311yescharacteristic
197C2×C22⋊C4—2441no—
198C4⋊C8—21no—
199#103—1221no—
200#102—1221no—
201#102—1221no—
202#102—1221no—
203#99—1222no—
204#99—1222no—
205#99—2441no—
206#99—2441no—
207C22⋊C8—21no—
208#97C2×#692311yescharacteristic, complemented
209C4×C8C2×#692311yescharacteristic, complemented
210C24—1221no—
211#41—4881no—
212C2×Q8—611no—
213C2×Q8—1221no—
214C2×D4—611no—
215C2×D4—611no—
216C2×D4—611no—
217C2×D4—1221no—
218C2×D4—1221no—
219C2×D4—4881no—
220C2×D4—2441no—
221C2×D4—2441no—
222C22×C4—1221no—
223C22×C4—4881no—
224C22×C4C22⋊#692311yescharacteristic
225Q16—4881no—
226SD16—4881no—
227SD16—4881no—
228SD16—4881no—
229C2×C8—21no—
230C2×C8—21no—
231C4⋊C4—21no—
232C4⋊C4—1221no—
233C4⋊C4—4881no—
234C4⋊C4—2441no—
235C4⋊C4—2441no—
236C4⋊C4—2441no—
237C4⋊C4—2441no—
238C22⋊C4—21no—
239C22⋊C4—4881no—
240C22⋊C4—2441no—
241C22⋊C4—2441no—
242C22⋊C4—2442no—
243C22⋊C4—2442no—
244C22⋊C4—2441no—
245C42—4881no—
246C42—2441no—
247C42C22⋊#692311yescharacteristic
248C42C22×#692311yescharacteristic
249C23—1221no—
250C23—1221no—
251C23—1221no—
252C23—9761no—
253C23—2441no—
254C23—2441no—
255C23#1451983611yescharacteristic
256Q8—2441no—
257Q8—2441no—
258Q8—2441no—
259D4—9761no—
260D4—9761no—
261D4—4881no—
262D4—4881no—
263D4—2441no—
264D4—2441no—
265D4—2441no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—1221no—
270C2×C4—9762no—
271C2×C4—4882no—
272C2×C4—4882no—
273C2×C4—2441no—
274C2×C4—2441no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4#1451983611yescharacteristic
278C2×C4C2×C22⋊#692311yescharacteristic
279C8—41no—
280C8—41no—
281C22—21no—
282C22—41no—
283C22—9761no—
284C22—4881no—
285C22—4881no—
286C22—4881no—
287C22—4882no—
288C22—2441no—
289C22—2441no—
290C22—2441no—
291C22C2×#1451983611yescharacteristic
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—9762no—
297C4—9762no—
298C4—4881no—
299C4—4881no—
300C4—2441no—
301C2—21no—
302C2—41no—
303C2—4881no—
304C2—4881no—
305C2—4881no—
306C2#2146209611yescharacteristic
subgroupG/H
centerC2#21462096
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#433C4
socleC2×C61#2127
prime radical#16840331C22
cosocle#14519774C23
hypercenter#433#6923
hypocenterC61#13610
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
3. #21460960C2semidirect
4. #21460953C2semidirect
12. #16840374C4semidirect
14. #16840156C4semidirect
17. #16840117C4semidirect
22. C61×#433C4semidirect
35. #14519777C2×C4semidirect
39. #14519769C2×C4semidirect
54. #97×C61C2×C4semidirect
55. C4×C8×C61C2×C4semidirect
169. #433#6923semidirect
176. C61#13610semidirect
208. #97C2×#6923semidirect
209. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214621002
#214620962
#214609602
#214609532
#214568932
#1361061