Finite Group Databasebeta

← #30601078

Subgroups — #30601078

Order 15616.

508 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 74 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#21462364C211yesmaximal, characteristic, complemented
2#21461353C211yesmaximal, characteristic, complemented
3#21461239C212yesmaximal
4#21460679C212yesmaximal, complemented
5#21456273C211yesmaximal, characteristic
6#16840785C2212yes—
7(C2×#6923)⋊Q8—22no—
8#16840711—24no—
9#16840707C2212yes—
10#16840622—22no—
11#16840613—24no—
12C2×#14519806—22no—
13#16840310C2211yescharacteristic
14C4⋊Q8×D61C2211yescharacteristic
15#16840153—22no—
16#16840142C411yescharacteristic, complemented
17#16840141C411yescharacteristic, complemented
18#16840134C411yescharacteristic, complemented
19C4×#11758189C2211yescharacteristic
20C61×#492C411yescharacteristic, complemented
21C2×C4⋊#6923—22no—
22C2×C4×#6923—22no—
23#14519844—22no—
24#14519834—22no—
25#14519832—22no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—22no—
28(C2×#6923)⋊C4—22no—
29(C2×#6923)⋊C4D414yes—
30C2×#11758224—21no—
31C2×Q8×D61D412yes—
32#14519806—44no—
33#14519806—44no—
34C2×Dic244—22no—
35C2×#11758202—22no—
36C2×#11758200—22no—
37C2×#11758189—21no—
38#14519777C2×C411yescharacteristic
39#14519777C2×C411yescharacteristic
40#14519776—22no—
41#14519775D412yes—
42#14519764—21no—
43#14519748—22no—
44D4×Dic61—22no—
45C4⋊C4×D61—22no—
46C42×D61C2311yescharacteristic
47#14519687—21no—
48C4×#92418C2×C411yescharacteristic
49C2×SD16×C61—22no—
50C4⋊Q8×C61C2×C411yescharacteristic
51#127×C61C2×C411yescharacteristic
52C4×C8×C61C2×C411yescharacteristic
53C22×#6923—24no—
54C2×Q8×C61C22⋊C412yes—
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C2×C22⋊D61—41no—
58C22×Dic61—41no—
59Q8×D61—42no—
60Q8×D61—42no—
61#11758224—42no—
62#11758224—42no—
63C2×C4×D61—22no—
64C2×C4×D61C2×D411yescharacteristic
65C2×C4×D61C2×D412yes—
66C2×Dic122—21no—
67C2×Dic122—22no—
68C2×Dic122C22⋊C412yes—
69C2×#92420—42no—
70C4⋊#6923—82no—
71C4×#6923—82no—
72SD16×C61—44no—
73C2×C8×C61—21no—
74C4⋊C4×C61—22no—
75C42×C61C22×C411yescharacteristic
76C22⋊Dic61—42no—
77#11758202—44no—
78#11758200—44no—
79C4⋊Dic61—22no—
80C4⋊Dic61—22no—
81#11758196—24no—
82C4×Dic61C2×D411yescharacteristic
83C4×Dic61#4112yes—
84C2×#92418—21no—
85Dic244—44no—
86#11758189—24no—
87C23×C61—41no—
88C22×D61#12011yescharacteristic
89C2×#6923—48no—
90C2×#6923—44no—
91Q8×C61—42no—
92D4×C61—42no—
93D4×C61—82no—
94C2×C4×C61—22no—
95C2×C4×C61C2×C22⋊C411yescharacteristic
96C2×C4×C61C4×D412yes—
97C22⋊D61—82no—
98C2×Dic61—22no—
99C2×Dic61—41no—
100C2×Dic61—81no—
101C2×Dic61—81no—
102C2×Dic61#12112yes—
103C2×Dic61#12311yescharacteristic
104C4×D61—22no—
105C4×D61—42no—
106C4×D61—42no—
107C4×D61—42no—
108Dic122—42no—
109Dic122—42no—
110Dic122—82no—
111#92420—82no—
112C8×C61—22no—
113#92418—22no—
114#9429—611nomaximal, Sylow
115C22×C61—81no—
116C22×C61—82no—
117C22×C61#38611yescharacteristic
118C2×D61—22no—
119C2×D61#45312yes—
120#6923—84no—
121C4×C61—21no—
122C4×C61—22no—
123C4×C61—42no—
124Dic61—21no—
125Dic61—22no—
126Dic61—42no—
127Dic61—81no—
128#3152—611no—
129#2381—611no—
130#1558—611no—
131#1514—612no—
132#1288—612no—
133C2×C61—161no—
134C2×C61#188711yescharacteristic
135C2×C61#190512yes—
136D61—22no—
137C2×#137—1222no—
138#532—611no—
139C2×C4⋊Q8—611no—
140#497—1222no—
141#494—611no—
142#494—611no—
143#492—611no—
144#492#692311yescharacteristic, complemented
145#432—612no—
146C4×M16—611no—
147#391—1222no—
148#387—1224no—
149#383—612no—
150#333—1222no—
151#327—1224no—
152C61#942911yescharacteristic, complemented, Sylow
153C2×#41—1221no—
154C22×Q8—612no—
155#137—2444no—
156#137—2444no—
157C2×Q16—1222no—
158C2×Q16—1222no—
159C2×SD16—22no—
160C2×SD16—1222no—
161C2×M16—1221no—
162C4⋊Q8—612no—
163C4⋊Q8—611no—
164C4⋊Q8—611no—
165C4⋊Q8—1222no—
166C4⋊Q8C2×#692311yescharacteristic
167#127C2×#692311yescharacteristic
168#124—1221no—
169#121—1222no—
170C4×D4—1222no—
171C2×C4⋊C4—1222no—
172C2×C4⋊C4—1222no—
173C2×C42—611no—
174C2×C42—1222no—
175C4⋊C8—1222no—
176C22⋊C8—1222no—
177#97—1221no—
178#97—1222no—
179C4×C8—611no—
180C4×C8C2×#692311yescharacteristic
181#95—614no—
182#95—1222no—
183#95—1222no—
184#95—1224no—
185#41—2442no—
186#41—2442no—
187C2×Q8—612no—
188C2×Q8—1222no—
189C2×Q8—1221no—
190C2×Q8—2442no—
191C2×Q8—2442no—
192C2×Q8C22⋊#692312yes—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—2441no—
196C22×C4—611no—
197C22×C4—612no—
198C22×C4—1222no—
199C22×C4—1224no—
200C22×C4—2441no—
201Q16—2444no—
202Q16—2444no—
203SD16—44no—
204SD16—2444no—
205M16—1224no—
206C2×C8—21no—
207C2×C8—1221no—
208C2×C8—2442no—
209C4⋊C4—22no—
210C4⋊C4—1224no—
211C4⋊C4—1222no—
212C4⋊C4—1222no—
213C4⋊C4—4882no—
214C22⋊C4—2442no—
215C42—611no—
216C42—612no—
217C42—4882no—
218C42C22×#692311yescharacteristic
219C23—41no—
220C23—611no—
221Q8—42no—
222Q8—4882no—
223Q8—2442no—
224Q8—2442no—
225D4—42no—
226D4—82no—
227D4—4882no—
228C2×C4—22no—
229C2×C4—612no—
230C2×C4—611no—
231C2×C4—1222no—
232C2×C4—1222no—
233C2×C4—4881no—
234C2×C4—4881no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4—2444no—
239C2×C4—2448no—
240C2×C4—2441no—
241C2×C4C2×C22⋊#692311yescharacteristic
242C2×C4D4×#692312yes—
243C8—22no—
244C8—1222no—
245C8—4882no—
246C22—81no—
247C22—82no—
248C22—612no—
249C22—1222no—
250C22#1684071011yescharacteristic
251C4—21no—
252C4—22no—
253C4—42no—
254C4—1221no—
255C4—1222no—
256C4—4884no—
257C4—4881no—
258C4—2442no—
259C2—161no—
260C2—1222no—
261C2#2146182611yescharacteristic
262C2#2146185512yes—
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#492C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#492#6923
hypocenterC61#9429
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #21462364C2semidirect
2. #21461353C2semidirect
4. #21460679C2semidirect
16. #16840142C4semidirect
17. #16840141C4semidirect
18. #16840134C4semidirect
20. C61×#492C4semidirect
144. #492#6923semidirect
152. C61#9429semidirect

Maximal subgroups

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

subgroupindex
#214623642
#214613532
#214612392
#214606792
#214562732
#942961