Finite Group Databasebeta

← #30608458

Subgroups — #30608458

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#21460959C212yesmaximal, complemented
4#21460952C212yesmaximal, complemented
5#21456885C211yesmaximal, characteristic
6#16840741—21no—
7C2×#14519836—21no—
8#16840669—21no—
9#16840667—21no—
10#16840667C2212yes—
11#16840667C2212yes—
12#16840331C2211yescharacteristic
13#16840154C411yescharacteristic, complemented
14#16840136C2211yescharacteristic
15#16840128C2211yescharacteristic
16#16840119C411yescharacteristic, complemented
17#16839984—21no—
18#16839962—21no—
19#16839922—21no—
20#16839912—21no—
21#16839823C411yescharacteristic, complemented
22C61×#541C411yescharacteristic, 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×#11758202—21no—
32C2×#11758201—21no—
33C2×#11758201—21no—
34C2×#11758201—21no—
35#14519774D411yescharacteristic
36#14519774C2311yescharacteristic
37#14519765C2×C411yescharacteristic
38#14519765D411yescharacteristic
39#14519752—41no—
40#14519747—41no—
41C4⋊#92418—21no—
42#14519700—21no—
43#14519697—21no—
44#14519697—21no—
45#14519697—21no—
46C22⋊#92418—21no—
47#14519688C2×C411yescharacteristic, complemented
48C4×#92418C2×C411yescharacteristic, complemented
49C4⋊Q8×C61C2×C411yescharacteristic, complemented
50#126×C61—41no—
51#125×C61—41no—
52#125×C61C2×C411yescharacteristic, complemented
53C22⋊Q8×C61—21no—
54C4×Q8×C61—21no—
55#117×C61C2×C411yescharacteristic
56C23×D61—21no—
57C22×#6923—81no—
58C2×Q8×C61—21no—
59C22×C4×C61C22⋊C411yescharacteristic
60C2×C22⋊D61—21no—
61#11758222—81no—
62C2×D244—21no—
63C2×D244—41no—
64C2×D244—41no—
65C2×D244—81no—
66C2×D244C22⋊C411yescharacteristic
67C2×D244C2×D411yescharacteristic
68C2×D244C2×D411yescharacteristic
69C2×C4×D61—21no—
70C2×Dic122C22⋊C411yescharacteristic
71C22⋊#6923—42no—
72C22⋊#6923—42no—
73C22⋊#6923—81no—
74C4⋊#6923—81no—
75C4×#6923—81no—
76C4⋊C4×C61—21no—
77C4⋊C4×C61—21no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C22⋊C4×C61—21no—
83C22⋊C4×C61—41no—
84C42×C61—41no—
85C42×C61C22⋊C411yescharacteristic
86C42×C61C22×C411yescharacteristic
87#11758202—81no—
88#11758201—81no—
89#11758201—81no—
90#11758201—81no—
91#11758198—41no—
92#11758198—41no—
93C2×#92418—21no—
94C2×#92418—21no—
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—
105Q8×C61—41no—
106Q8×C61—41no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61#9911yescharacteristic
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C22⋊D61—81no—
115C2×Dic61—21no—
116D244—41no—
117D244—41no—
118D244—41no—
119D244—81no—
120D244—161no—
121D244—161no—
122C4×D61—41no—
123C4×D61—41no—
124Dic122—41no—
125#92418—41no—
126#92418—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—
145C4×C61—81no—
146C4×C61—81no—
147Dic61—41no—
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#692311yescharacteristic, complemented
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—611no—
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⋊Q8C2×#692311yescharacteristic, complemented
184#127—611no—
185#127—611no—
186#126—41no—
187#125—41no—
188#125C2×#692311yescharacteristic, complemented
189#124—611no—
190#124—611no—
191C22⋊Q8—21no—
192#121—2441no—
193#120—2441no—
194C4×Q8—21no—
195#117—2441no—
196#117C2×#692311yescharacteristic
197C2×C22⋊C4—2441no—
198C4⋊C8—1221no—
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—1221no—
208#97—611no—
209C4×C8—611no—
210C24—1221no—
211#41—4881no—
212C2×Q8—21no—
213C2×Q8—611no—
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—1221no—
230C2×C8—1221no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—4881no—
238C22⋊C4—21no—
239C22⋊C4—41no—
240C22⋊C4—4881no—
241C22⋊C4—2441no—
242C22⋊C4—2441no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C42—41no—
246C42—4881no—
247C42C22⋊#692311yescharacteristic
248C42C22×#692311yescharacteristic
249C23—1221no—
250C23—1221no—
251C23—1221no—
252C23—9761no—
253C23—2441no—
254C23—2441no—
255C23#1451983511yescharacteristic
256Q8—41no—
257Q8—41no—
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—41no—
270C2×C4—41no—
271C2×C4—1221no—
272C2×C4—9762no—
273C2×C4—4882no—
274C2×C4—4882no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4#1451983511yescharacteristic
278C2×C4C2×C22⋊#692311yescharacteristic
279C8—2441no—
280C8—2441no—
281C22—21no—
282C22—41no—
283C22—9761no—
284C22—4881no—
285C22—4881no—
286C22—4881no—
287C22—4882no—
288C22—2441no—
289C22—2441no—
290C22—2441no—
291C22C2×#1451983511yescharacteristic
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—9762no—
299C4—9762no—
300C4—2441no—
301C2—21no—
302C2—41no—
303C2—4881no—
304C2—4881no—
305C2—4881no—
306C2#2146209411yescharacteristic
subgroupG/H
centerC2#21462094
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#541C4
socleC2×C61#2127
prime radical#16840331C22
cosocle#14519774C23
hypercenter#541#6923
hypocenterC61#13610
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
3. #21460959C2semidirect
4. #21460952C2semidirect
13. #16840154C4semidirect
16. #16840119C4semidirect
21. #16839823C4semidirect
22. C61×#541C4semidirect
47. #14519688C2×C4semidirect
48. C4×#92418C2×C4semidirect
49. C4⋊Q8×C61C2×C4semidirect
52. #125×C61C2×C4semidirect
159. #541#6923semidirect
176. C61#13610semidirect
183. C4⋊Q8C2×#6923semidirect
188. #125C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214621002
#214620962
#214609592
#214609522
#214568852
#1361061