Finite Group Databasebeta

← #30599056

Subgroups — #30599056

Order 15616.

388 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461381C212yesmaximal, complemented
2#21461307C212yesmaximal
3#21460725C212yesmaximal, complemented
4#21454527C211yesmaximal, characteristic
5#16840783C2214yes—
6#16840625—22no—
7#16840624—22no—
8#16840618—22no—
9#16840617—22no—
10C2×#14519785C411yescharacteristic
11C2×#14519784C411yescharacteristic
12#16840308C2211yescharacteristic
13#16839837—22no—
14#16839825—22no—
15#16839785C2211yescharacteristic
16#16839728C411yescharacteristic
17C2×C8×Dic61C2211yescharacteristic
18C61×#408C411yescharacteristic
19C4⋊#92420—24no—
20C22⋊#92420—24no—
21#14519832—24no—
22C2×#11758222C2×C411yescharacteristic
23#14519785—42no—
24#14519784—42no—
25C2×#11758194—21no—
26C2×#11758194C2×C411yescharacteristic
27C2×#11758189—21no—
28C22×#92418—21no—
29C22×#92418C2×C411yescharacteristic
30C2×C8×D61—21no—
31#14519775Q811yescharacteristic
32#14519773Q811yescharacteristic
33#14519766D411yescharacteristic
34#14519764D411yescharacteristic
35#14519749—22no—
36C4×C22⋊D61—22no—
37C2×C4×Dic61C2311yescharacteristic
38#14519706—22no—
39#14519689—22no—
40C8×Dic61D412yes—
41C8×Dic61D412yes—
42C2×#41×C61C2×C411yescharacteristic
43C2×M16×C61C2×C411yescharacteristic
44C22×C8×C61C2×C411yescharacteristic
45C22⋊C8×C61—22no—
46#41×C61—41no—
47#41×C61—41no—
48C2×Q8×C61C4⋊C411yescharacteristic
49C2×D4×C61—21no—
50C2×D4×C61C4⋊C411yescharacteristic
51C22×C4×C61—21no—
52C22×C4×C61C22×C411yescharacteristic
53C2×C22⋊D61—21no—
54C2×C22⋊D61—22no—
55C22×Dic61#4112yes—
56#11758222—41no—
57#11758222—41no—
58C2×D244C4⋊C411yescharacteristic
59C2×C4×D61—21no—
60C2×Dic122C4⋊C411yescharacteristic
61C2×#92420—44no—
62M16×C61—41no—
63C2×C8×C61—21no—
64C2×C8×C61—41no—
65C2×C8×C61C22⋊C412yes—
66C22⋊Dic61—22no—
67#11758198—22no—
68#11758196—22no—
69C4×Dic61C2×D411yescharacteristic
70C4×Dic61C2×D412yes—
71C4×Dic61C2×Q811yescharacteristic
72#11758194—42no—
73#11758194—41no—
74C2×#92418—22no—
75C2×#92418—21no—
76C2×#92418—42no—
77C2×#92418C22⋊C411yescharacteristic
78C2×#92418C22⋊C411yescharacteristic
79#11758189—42no—
80C8×D61—42no—
81C23×C61—21no—
82C23×C61#11711yescharacteristic
83C22×D61—21no—
84Q8×C61—41no—
85D4×C61—41no—
86D4×C61—42no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61C2×C4⋊C411yescharacteristic
94C22⋊D61—44no—
95C22⋊D61—42no—
96C2×Dic61—21no—
97C2×Dic61—42no—
98C2×Dic61#12112yes—
99C2×Dic61C22⋊Q812yes—
100D244—41no—
101C4×D61—41no—
102C4×D61—41no—
103Dic122—41no—
104#92420—84no—
105C8×C61—22no—
106C8×C61—41no—
107#92418—21no—
108#92418—41no—
109#92418#10012yes—
110#8630—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—22no—
113C22×C61—41no—
114C22×C61—42no—
115C22×C61#38011yescharacteristic
116C2×D61—41no—
117C2×D61—42no—
118C4×C61—21no—
119C4×C61—41no—
120C4×C61C2×#10011yescharacteristic
121C4×C61#41311yescharacteristic
122Dic61—22no—
123Dic61—22no—
124Dic61—41no—
125#2931—611no—
126#1568—612no—
127#1536—612no—
128#1307—612no—
129C2×C61—41no—
130C2×C61—81no—
131C2×C61#180011yescharacteristic
132C2×C61#181412yes—
133D61—81no—
134C2×#131—611no—
135C2×#131—611no—
136#532—611no—
137#436—1222no—
138C8×D4—1222no—
139#432—614no—
140#423—611no—
141#408—611no—
142#408#692311yescharacteristic
143C2×C4×C8—611no—
144#335—1222no—
145#334—1222no—
146#329—1222no—
147#329—1222no—
148C61#863011yescharacteristic, complemented, Sylow
149C2×#41—611no—
150C2×#41C2×#692311yescharacteristic
151#131—2442no—
152#131—2442no—
153C2×M16—611no—
154C2×M16—1221no—
155C2×M16—1221no—
156C2×M16C2×#692311yescharacteristic
157C22×C8—611no—
158C22×C8—1221no—
159C22×C8—1221no—
160C22×C8C2×#692311yescharacteristic
161C4⋊Q8—611no—
162#127—611no—
163#124—611no—
164#124—611no—
165#121—1222no—
166C4×D4—1222no—
167C2×C42—611no—
168C4⋊C8—1222no—
169C4⋊C8—1224no—
170C22⋊C8—22no—
171C22⋊C8—1222no—
172C22⋊C8—1224no—
173#97—1224no—
174C4×C8—612no—
175C4×C8—612no—
176#41—41no—
177#41—41no—
178#41—2441no—
179#41—2441no—
180C2×Q8—611no—
181C2×Q8C4⋊#692311yescharacteristic
182C2×D4—21no—
183C2×D4—611no—
184C2×D4—1222no—
185C2×D4—1221no—
186C2×D4C4⋊#692311yescharacteristic
187C22×C4—21no—
188C22×C4—612no—
189C22×C4—1221no—
190C22×C4C22×#692311yescharacteristic
191M16—41no—
192M16—2442no—
193M16—2442no—
194M16—2441no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—611no—
198C2×C8—611no—
199C2×C8—1222no—
200C2×C8—1221no—
201C2×C8—2442no—
202C2×C8—2442no—
203C2×C8—2444no—
204C2×C8C22⋊#692312yes—
205C4⋊C4—1222no—
206C22⋊C4—1222no—
207C22⋊C4—1222no—
208C42—612no—
209C42—611no—
210C42—611no—
211C23—21no—
212C23—1221no—
213C23#1451985611yescharacteristic
214Q8—41no—
215Q8—2441no—
216D4—41no—
217D4—42no—
218D4—2441no—
219D4—2444no—
220D4—2442no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—612no—
227C2×C4—612no—
228C2×C4—1221no—
229C2×C4—2442no—
230C2×C4—2441no—
231C2×C4—2441no—
232C2×C4C2×C22⋊#692311yescharacteristic
233C2×C4C2×C4⋊#692311yescharacteristic
234C8—22no—
235C8—41no—
236C8—612no—
237C8—1221no—
238C8—4884no—
239C8—2441no—
240C22—21no—
241C22—22no—
242C22—41no—
243C22—42no—
244C22—2441no—
245C22—2442no—
246C22#1684070411yescharacteristic
247C4—21no—
248C4—41no—
249C4—1222no—
250C4—1222no—
251C4—2441no—
252C4D61⋊#10011yescharacteristic
253C4#1684074911yescharacteristic
254C2—41no—
255C2—81no—
256C2—4881no—
257C2#2146168411yescharacteristic
258C2#2146170612yes—
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#408C4
socleC22×C61#380
prime radical#16840308C22
cosocleC2×C4×Dic61C23
hypercenter#408#6923
hypocenterC61#8630
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #21461381C2semidirect
3. #21460725C2semidirect
148. C61#8630semidirect

Maximal subgroups

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

subgroupindex
#214613812
#214613072
#214607252
#214545272
#863061