Finite Group Databasebeta

← #30594748

Subgroups — #30594748

Order 15616.

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

normal subgroups only

This group has 342 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#21461679C211yesmaximal, characteristic
2#21461078C211yesmaximal, characteristic
3#21446116C211yesmaximal, characteristic
4#16840783—21no—
5C2×#14519835C412yes—
6#16840704—21no—
7#16840640—21no—
8#16840606—21no—
9#16840605C412yescomplemented
10C2×C4×C22⋊D61C2211yescharacteristic
11C2×C22⋊#92418C411yescharacteristic, complemented
12C2×#14519689—21no—
13#16839516—21no—
14C22×M16×C61C411yescharacteristic, complemented
15C2×C4⋊#6923—41no—
16C2×C22⋊#6923—22no—
17C4⋊#92420—41no—
18#14519835—42no—
19C22⋊#92420—22no—
20#14519832—41no—
21(C2×#6923)⋊C4—41no—
22C22×C22⋊D61C2×C411yescharacteristic
23C22×C4×D61D411yescharacteristic
24C22×#92418—21no—
25C4×C22⋊D61—41no—
26C4×C22⋊D61—41no—
27C2×#11758196Q811yescharacteristic
28C2×C22⋊Dic61C2×C411yescharacteristic
29C2×#11758198D411yescharacteristic
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×D61C22⋊C411yescharacteristic
39C22×#6923—42no—
40C22×C4×C61—21no—
41C22×C4×C61—21no—
42C22×C4×C61—41no—
43C22×C4×C61—41no—
44C22×C4×C61C22⋊C411yescharacteristic
45C22×C4×C61C4⋊C411yescharacteristic
46C2×C22⋊D61—21no—
47C2×C22⋊D61—21no—
48C2×C22⋊D61—41no—
49C2×C22⋊D61—41no—
50C22×Dic61C22⋊C411yescharacteristic
51C22×Dic61C4⋊C411yescharacteristic
52C22×Dic61C4⋊C411yescharacteristic
53C2×C4×D61—21no—
54C2×C4×D61—21no—
55C2×C4×D61—41no—
56C2×C4×D61—41no—
57C22⋊#6923—42no—
58C22⋊#6923—42no—
59C2×#92420—42no—
60C4⋊#6923—81no—
61C4⋊#6923—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—41no—
84C22×D61—41no—
85C22×D61#10011yescharacteristic
86C22×D61#10111yescharacteristic
87C2×#6923—82no—
88C2×#6923—82no—
89C2×#6923—82no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—42no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61#9911yescharacteristic
98C2×C4×C61#9911yescharacteristic
99C22⋊D61—42no—
100C22⋊D61—42no—
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—21no—
104C2×Dic61—21no—
105C2×Dic61—21no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C4×D61—41no—
109C4×D61—41no—
110C4×D61—81no—
111#92420—82no—
112C8×C61—42no—
113#92418—81no—
114#7320—611nomaximal, Sylow
115C22×C61—21no—
116C22×C61—21no—
117C22×C61—41no—
118C22×C61—41no—
119C22×C61—41no—
120C22×C61—81no—
121C22×C61—81no—
122C22×C61#34211yescharacteristic
123C22×C61#34311yescharacteristic
124C22×C61#34411yescharacteristic
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—41no—
128C2×D61—41no—
129C2×D61—41no—
130C2×D61—41no—
131C2×D61—81no—
132#6923—162no—
133C4×C61—21no—
134C4×C61—41no—
135C4×C61AΣL(1,𝔽16)11yescharacteristic
136C4×C61#35211yescharacteristic
137Dic61—41no—
138Dic61—41no—
139Dic61—41no—
140#1870—611no—
141#1799—611no—
142#1466—611no—
143C2×C61—21no—
144C2×C61—21no—
145C2×C61—81no—
146C2×C61#130511yescharacteristic
147C2×C61#139011yescharacteristic
148C2×C61#139111yescharacteristic
149D61—41no—
150D61—41no—
151C22×M16#692311yescharacteristic, complemented
152C2×C4×D4—611no—
153#432—1221no—
154C2×#99—612no—
155C2×C22⋊C8—611no—
156C2×C22⋊C8—1221no—
157#380—1221no—
158#342—1221no—
159#336—1221no—
160#323—612no—
161#323—1221no—
162C61#732011yescharacteristic, complemented, Sylow
163C22×D4—611no—
164C23×C4—611no—
165C23×C4C2×#692311yescharacteristic
166C2×M16—42no—
167C2×M16—41no—
168C22×C8—21no—
169C22×C8—1221no—
170C4×D4—2441no—
171C4×D4—2441no—
172C2×C4⋊C4—611no—
173C2×C4⋊C4—2441no—
174C2×C22⋊C4—611no—
175C2×C22⋊C4—611no—
176C2×C22⋊C4—1222no—
177C2×C42—611no—
178C4⋊C8—2441no—
179#99—2442no—
180C22⋊C8—1222no—
181C22⋊C8—2442no—
182C22⋊C8—2441no—
183#97—2441no—
184#95—2441no—
185C24—611no—
186C24C4×#692311yescharacteristic
187C2×D4—1221no—
188C2×D4—1221no—
189C2×D4—2441no—
190C2×D4—2441no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—611no—
196C22×C4—611no—
197C22×C4—611no—
198C22×C4—1221no—
199C22×C4—1221no—
200C22×C4—2441no—
201C22×C4—2441no—
202C22×C4—2442no—
203C22×C4C4⋊#692311yescharacteristic
204C22×C4C22⋊#692311yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—2441no—
211C2×C8—2441no—
212C2×C8—2442no—
213C2×C8—2441no—
214C4⋊C4—4881no—
215C4⋊C4—4881no—
216C4⋊C4—2441no—
217C22⋊C4—2441no—
218C22⋊C4—2441no—
219C22⋊C4—2442no—
220C22⋊C4—2442no—
221C42—1221no—
222C42—1221no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—611no—
229C23—611no—
230C23—1221no—
231C23—1221no—
232C23—2441no—
233C23—2441no—
234C23(C2×#6923)⋊C411yescharacteristic
235D4—2442no—
236D4—2442no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—1221no—
245C2×C4—1221no—
246C2×C4—1221no—
247C2×C4—1221no—
248C2×C4—1221no—
249C2×C4—4881no—
250C2×C4—4882no—
251C2×C4—4882no—
252C2×C4—4882no—
253C2×C4—2441no—
254C2×C4—2441no—
255C2×C4—2441no—
256C2×C4—2441no—
257C2×C4#1451983511yescharacteristic
258C2×C4#1451983611yescharacteristic
259C8—42no—
260C8—4881no—
261C8—4882no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—1221no—
270C22—1221no—
271C22—4881no—
272C22—2441no—
273C22—2441no—
274C22—2441no—
275C22—2441no—
276C22#1684064011yescharacteristic
277C22#1684064111yescharacteristic
278C22#1684064511yescharacteristic
279C4—21no—
280C4—41no—
281C4—9762no—
282C4—2441no—
283C4—2441no—
284C4—2441no—
285C4#1684066211yescharacteristic
286C4#1684066411yescharacteristic
287C2—21no—
288C2—21no—
289C2—81no—
290C2—2441no—
291C2—2441no—
292C2#2146071911yescharacteristic
293C2#2146090311yescharacteristic
294C2#2146090811yescharacteristic
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#7320
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
9. #16840605C4semidirect
11. C2×C22⋊#92418C4semidirect
14. C22×M16×C61C4semidirect
151. C22×M16#6923semidirect
162. C61#7320semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610782
#214461162
#732061