Finite Group Databasebeta

← #30594752

Subgroups — #30594752

Order 15616.

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

normal subgroups only

This group has 328 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#21461683C211yesmaximal, characteristic
2#21461074C211yesmaximal, characteristic
3#21446116C211yesmaximal, characteristic
4#16840765—21no—
5C2×#14519837C412yes—
6#16840733—21no—
7#16840642—21no—
8#16840608—21no—
9#16840605C412yes—
10C2×C4×C22⋊D61C2211yescharacteristic
11C2×C22⋊#92418C411yescharacteristic
12C2×#14519689—21no—
13#16839516—21no—
14C22×M16×C61C411yescharacteristic
15C2×#11758213—41no—
16C2×#11758217—22no—
17#14519844—41no—
18C4⋊#92420—41no—
19#14519837—42no—
20#14519834—41no—
21C22⋊#92420—22no—
22C22×C22⋊D61C2×C411yescharacteristic
23C22×C4×D61D411yescharacteristic
24C22×#92418—21no—
25C4×C22⋊D61—41no—
26C4×C22⋊D61—41no—
27C2×#11758196D411yescharacteristic
28C2×C22⋊Dic61C2×C411yescharacteristic
29C2×#11758198Q811yescharacteristic
30C2×C4×Dic61D411yescharacteristic
31C22⋊#92418—41no—
32#14519689—42no—
33C23×C4×C61C2×C411yescharacteristic
34C2×M16×C61—42no—
35C2×M16×C61—41no—
36C22×C8×C61—21no—
37C24×C61C4211yescharacteristic
38C23×D61C4⋊C411yescharacteristic
39C22×C4×C61—21no—
40C22×C4×C61—21no—
41C22×C4×C61—41no—
42C22×C4×C61—41no—
43C22×C4×C61C22⋊C411yescharacteristic
44C22×C4×C61C4⋊C411yescharacteristic
45C2×C22⋊D61—21no—
46C2×C22⋊D61—21no—
47C2×C22⋊D61—41no—
48C2×C22⋊D61—41no—
49C22×Dic61C22⋊C411yescharacteristic
50C22×Dic61C22⋊C411yescharacteristic
51C22×Dic61C4⋊C411yescharacteristic
52C2×C4×D61—21no—
53C2×C4×D61—21no—
54C2×C4×D61—41no—
55C2×C4×D61—41no—
56#11758217—42no—
57#11758217—42no—
58C2×#92420—42no—
59C2×#92420—42no—
60#11758213—81no—
61#11758213—81no—
62M16×C61—42no—
63M16×C61—42no—
64C2×C8×C61—22no—
65C2×C8×C61—41no—
66C2×C8×C61—41no—
67C22⋊Dic61—41no—
68#11758198—41no—
69#11758196—41no—
70C4×Dic61—21no—
71C4×Dic61—21no—
72C2×#92418—41no—
73C2×#92418—41no—
74C2×#92418—41no—
75C23×C61—21no—
76C23×C61—21no—
77C23×C61—21no—
78C23×C61—41no—
79C23×C61—41no—
80C23×C61#9511yescharacteristic
81C22×D61—21no—
82C22×D61—21no—
83C22×D61—21no—
84C22×D61—41no—
85C22×D61—41no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—42no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61#9911yescharacteristic
94C2×C4×C61#9911yescharacteristic
95C22⋊D61—42no—
96C22⋊D61—42no—
97C2×Dic61—21no—
98C2×Dic61—21no—
99C2×Dic61—21no—
100C2×Dic61—21no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61#10011yescharacteristic
104C2×Dic61#10111yescharacteristic
105C4×D61—41no—
106C4×D61—41no—
107C4×D61—81no—
108#92420—82no—
109#92420—82no—
110C8×C61—42no—
111#92418—81no—
112#7321—611nomaximal, Sylow
113C22×C61—21no—
114C22×C61—21no—
115C22×C61—41no—
116C22×C61—41no—
117C22×C61—41no—
118C22×C61—81no—
119C22×C61—81no—
120C22×C61#34211yescharacteristic
121C22×C61#34311yescharacteristic
122C22×C61#34411yescharacteristic
123C2×D61—42no—
124C2×D61—41no—
125C2×D61—41no—
126C2×D61—41no—
127C2×D61—81no—
128C4×C61—21no—
129C4×C61—41no—
130C4×C61AΣL(1,𝔽16)11yescharacteristic
131C4×C61#35211yescharacteristic
132Dic61—21no—
133Dic61—21no—
134Dic61—41no—
135Dic61—41no—
136#1870—611no—
137#1800—611no—
138#1465—611no—
139C2×C61—21no—
140C2×C61—21no—
141C2×C61—81no—
142C2×C61#130511yescharacteristic
143C2×C61#139011yescharacteristic
144C2×C61#139211yescharacteristic
145D61—81no—
146C22×M16#692311yescharacteristic
147C2×C4×D4—611no—
148#423—1221no—
149C2×#100—612no—
150#407—1221no—
151C2×C22⋊C8—611no—
152C2×C22⋊C8—1221no—
153#343—1221no—
154#336—1221no—
155#324—1221no—
156#323—612no—
157C61#732111yescharacteristic, complemented, Sylow
158C22×D4—611no—
159C23×C4—611no—
160C23×C4C2×#692311yescharacteristic
161C2×M16—42no—
162C2×M16—41no—
163C2×M16—1222no—
164C2×M16—2441no—
165C22×C8—21no—
166C22×C8—1221no—
167C4×D4—2441no—
168C4×D4—2441no—
169C2×C4⋊C4—611no—
170C2×C22⋊C4—611no—
171C2×C22⋊C4—611no—
172C2×C42—611no—
173C4⋊C8—2441no—
174C4⋊C8—2441no—
175#100—2442no—
176C22⋊C8—1222no—
177C22⋊C8—2442no—
178C22⋊C8—2441no—
179C22⋊C8—2441no—
180C24—611no—
181C24C4×#692311yescharacteristic
182C2×D4—1221no—
183C2×D4—1221no—
184C2×D4—2441no—
185C2×D4—2441no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—611no—
191C22×C4—611no—
192C22×C4—611no—
193C22×C4—1221no—
194C22×C4—1221no—
195C22×C4—2441no—
196C22×C4—2441no—
197C22×C4C4⋊#692311yescharacteristic
198C22×C4C22⋊#692311yescharacteristic
199M16—42no—
200M16—42no—
201M16—4881no—
202M16—4881no—
203M16—2442no—
204M16—2442no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—2441no—
209C2×C8—2441no—
210C2×C8—2442no—
211C2×C8—2442no—
212C2×C8—2441no—
213C4⋊C4—2441no—
214C22⋊C4—2441no—
215C22⋊C4—2441no—
216C42—1221no—
217C42—1221no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—1221no—
224C23—1221no—
225C23—1221no—
226C23—2441no—
227C23—2441no—
228C23(C2×#6923)⋊C411yescharacteristic
229D4—2442no—
230D4—2442no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—611no—
239C2×C4—611no—
240C2×C4—1221no—
241C2×C4—1221no—
242C2×C4—1221no—
243C2×C4—1221no—
244C2×C4—4881no—
245C2×C4—2441no—
246C2×C4—2441no—
247C2×C4—2441no—
248C2×C4—2441no—
249C2×C4#1451983511yescharacteristic
250C2×C4#1451983611yescharacteristic
251C8—42no—
252C8—4881no—
253C8—4882no—
254C8—4882no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22—81no—
262C22—4881no—
263C22—2442no—
264C22—2441no—
265C22—2441no—
266C22—2441no—
267C22#1684064011yescharacteristic
268C22#1684064111yescharacteristic
269C22#1684064511yescharacteristic
270C4—21no—
271C4—41no—
272C4—1221no—
273C4—1221no—
274C4—2441no—
275C4—2441no—
276C4#1684066211yescharacteristic
277C4#1684066411yescharacteristic
278C2—21no—
279C2—21no—
280C2—81no—
281C2—4881no—
282C2#2146071911yescharacteristic
283C2#2146090311yescharacteristic
284C2#2146091111yescharacteristic
subgroupG/H
centerC22#16840640
commutatorC24×C61C42
FrattiniC23×C4C2×#6923
FittingC22×M16×C61C4
socleC22×C61#342
prime radicalC22×C22⋊D61C2×C4
cosocleC2×C4×C22⋊D61C22
hypercenterC22×M16#6923
hypocenterC61#7321
LCSG ▷ C24×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
157. C61#7321semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214610742
#214461162
#732161