Finite Group Databasebeta

← #29577567

Subgroups — #29577567

Order 15616.

620 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456269C211yesmaximal, characteristic
2#21444533C211yesmaximal, characteristic, complemented
3#21444365C212yesmaximal, complemented
4#21443683C212yesmaximal, complemented
5C61×#2380C211yesmaximal, characteristic, complemented
6C2×#14519803—22no—
7#16840312C2211yescharacteristic
8#16840149—22no—
9#16840138C412yescomplemented
10#16840134C412yescomplemented
11#16839819C2212yes—
12C4×#11758194C2211yescharacteristic
13#16839506—22no—
14#16839495—24no—
15C2×#127×C61C2211yescharacteristic
16C61×#386—22no—
17C61×#386—24no—
18C61×#383C2212yes—
19C2×#11758222—21no—
20#14519803—44no—
21#14519803—44no—
22C2×#11758200—24no—
23C2×#11758199—24no—
24C2×#11758194—21no—
25#14519777C2×C411yescharacteristic, complemented
26#14519774C2×C411yescharacteristic, complemented
27#14519765—21no—
28#14519751—22no—
29C4×D244—22no—
30C4⋊#92418—22no—
31C22⋊#92418—22no—
32#14519688—21no—
33#14519688—22no—
34C4×#92418C2×C412yescomplemented
35C22×D4×C61—22no—
36C22×D4×C61D412yes—
37C23×C4×C61—22no—
38#127×C61—22no—
39#127×C61C2×C412yes—
40#127×C61D412yes—
41C2×C22⋊C4×C61—24no—
42C2×C22⋊C4×C61—22no—
43C2×C42×C61C2311yescharacteristic
44#95×C61—22no—
45#95×C61—22no—
46#95×C61D414yes—
47C24×C61—22no—
48C2×D4×C61—24no—
49C2×D4×C61—24no—
50C2×D4×C61—22no—
51C2×D4×C61—42no—
52C2×D4×C61—42no—
53C2×D4×C61—82no—
54C2×D4×C61—82no—
55C2×D4×C61C22⋊C414yes—
56C22×C4×C61—24no—
57C22×C4×C61—84no—
58C22×C4×C61—82no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C2×C22⋊D61—41no—
62#11758222—42no—
63#11758222—42no—
64C2×D244—21no—
65C2×D244—22no—
66C2×C4×D61—41no—
67C2×Dic122—21no—
68C22⋊C4×C61—84no—
69C22⋊C4×C61—82no—
70C42×C61C22×C411yescharacteristic
71C42×C61C2×D411yescharacteristic
72C42×C61#4112yes—
73#11758200—48no—
74#11758199—48no—
75#11758198—42no—
76C4⋊Dic61—22no—
77#11758194—24no—
78C2×#92418—22no—
79C2×#92418—42no—
80C23×C61—24no—
81C23×C61—44no—
82C23×C61—44no—
83C23×C61—42no—
84C23×C61—42no—
85C23×C61#12011yescharacteristic
86C22×D61—41no—
87D4×C61—44no—
88D4×C61—82no—
89D4×C61—84no—
90D4×C61—84no—
91C2×C4×C61—22no—
92C2×C4×C61—42no—
93C2×C4×C61—48no—
94C2×C4×C61—44no—
95C2×C4×C61—88no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C4×D412yes—
98C2×C4×C61#12112yes—
99C2×C4×C61#12311yescharacteristic
100C22⋊D61—82no—
101C2×Dic61—41no—
102D244—42no—
103D244—82no—
104C4×D61—81no—
105C4×D61—81no—
106Dic122—42no—
107#92418—24no—
108#92418—82no—
109#9428—611nomaximal, Sylow
110C22×C61—22no—
111C22×C61—44no—
112C22×C61—48no—
113C22×C61—82no—
114C22×C61—82no—
115C22×C61—84no—
116C22×C61#38611yescharacteristic
117C22×C61#45312yes—
118C2×D61—81no—
119C2×D61—82no—
120C4×C61—21no—
121C4×C61—21no—
122C4×C61—22no—
123C4×C61—22no—
124C4×C61—84no—
125Dic61—81no—
126#3151—611no—
127#2380D6111yescharacteristic, complemented
128#1555—611no—
129#1514—612no—
130#1288—612no—
131C2×C61—22no—
132C2×C61—84no—
133C2×C61#188711yescharacteristic
134C2×C61#190412yes—
135D61—161no—
136C2×AGL(1,ℤ/8)—1222no—
137#532—611no—
138C2×#127C2×D6111yescharacteristic
139#496—1222no—
140#493—612no—
141#492—612no—
142#432—612no—
143C4×M16—611no—
144#386—22no—
145#386—24no—
146#383C2×D6112yes—
147#331—1222no—
148#327—1224no—
149C61#942811yescharacteristic, complemented, Sylow
150C2×#41—1221no—
151C22×D4—22no—
152C22×D4C22⋊D6112yes—
153C23×C4—22no—
154AGL(1,ℤ/8)—2444no—
155AGL(1,ℤ/8)—2444no—
156C2×SD16—1224no—
157C2×D8—1224no—
158C2×M16—1221no—
159C4⋊Q8—611no—
160#127—22no—
161#127—611no—
162#127C4×D6112yes—
163#127D24412yes—
164#124—1221no—
165#121—1222no—
166C4×D4—1222no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C42C22×D6111yescharacteristic
170C4⋊C8—1222no—
171C22⋊C8—1222no—
172#97—1221no—
173#97—1222no—
174C4×C8—612no—
175#95—22no—
176#95—22no—
177#95C22⋊D6114yes—
178C24—22no—
179#41—2442no—
180#41—2442no—
181C2×Q8—1221no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—1221no—
190C2×D4—1222no—
191C2×D4—2441no—
192C2×D4#1175819814yes—
193C22×C4—24no—
194C22×C4—84no—
195C22×C4—82no—
196C22×C4—2441no—
197C22×C4C2×C22⋊D6111yescharacteristic
198C22×C4C2×C22⋊D6112yes—
199SD16—2448no—
200D8—2448no—
201M16—1224no—
202C2×C8—1222no—
203C2×C8—2442no—
204C4⋊C4—1222no—
205C22⋊C4—84no—
206C22⋊C4—82no—
207C22⋊C4—2442no—
208C42C2×C4×D6111yescharacteristic
209C42C2×D24411yescharacteristic
210C42#1175822212yes—
211C23—24no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—2441no—
217C23#1451974611yescharacteristic
218Q8—2442no—
219D4—44no—
220D4—82no—
221D4—84no—
222D4—84no—
223D4—4882no—
224D4—4882no—
225D4—2442no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—88no—
231C2×C4—4881no—
232C2×C4—4881no—
233C2×C4—2441no—
234C2×C4C2×#1175819811yescharacteristic
235C2×C4C4×C22⋊D6112yes—
236C2×C4#1451975112yes—
237C2×C4#1451976011yescharacteristic
238C8—1224no—
239C8—4882no—
240C22—22no—
241C22—44no—
242C22—48no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—4881no—
247C22—4882no—
248C22#1683964111yescharacteristic
249C22#1683993012yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—84no—
255C4—4881no—
256C2—22no—
257C2—84no—
258C2—9761no—
259C2#2144621711yescharacteristic
260C2#2144630612yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2380C2
socleC22×C61#386
prime radical#21456269C2
cosocleC2×C42×C61C23
hypercenter#2380D61
hypocenterC61#9428
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
2. #21444533C2semidirect
3. #21444365C2semidirect
4. #21443683C2semidirect
5. C61×#2380C2semidirect
9. #16840138C4semidirect
10. #16840134C4semidirect
25. #14519777C2×C4semidirect
26. #14519774C2×C4semidirect
34. C4×#92418C2×C4semidirect
127. #2380D61semidirect
149. C61#9428semidirect

Maximal subgroups

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

subgroupindex
#214562692
#214445332
#214443652
#214436832
C61×#23802
#942861