Finite Group Databasebeta

← #30594058

Subgroups — #30594058

Order 15616.

338 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 42 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#21460740C211yesmaximal, characteristic, complemented
2#21460682C211yesmaximal, characteristic, complemented
3#21446818C211yesmaximal, characteristic
4#16840704—21no—
5C22⋊#11758187—21no—
6C22×D488C411yescharacteristic, complemented
7C2×#14519752—21no—
8C2×#14519697—21no—
9C2×#14519689C2211yescharacteristic
10C61×#340C411yescharacteristic, complemented
11C2×C4⋊#6923—41no—
12C2×C22⋊#6923—41no—
13C2×#11758187—41no—
14(C2×#6923)⋊C4—22no—
15C22×D244—21no—
16C22×D244—41no—
17C22×C4×D61D411yescharacteristic
18C2×D488—41no—
19C2×D488—42no—
20C22×#92418D411yescharacteristic
21#14519752—42no—
22#14519752—42no—
23C2×#11758198—41no—
24#14519697—42no—
25#14519689—41no—
26C22×C8×C61C2×C411yescharacteristic
27C2×C4⋊C4×C61—21no—
28C23×D61—41no—
29C23×D61D811yescharacteristic
30C22×#6923—42no—
31C22×C4×C61—41no—
32C22×C4×C61C22⋊C411yescharacteristic
33C22×Dic61SD1611yescharacteristic
34C2×D244—22no—
35C2×D244—41no—
36C2×D244—41no—
37C2×D244—41no—
38C2×D244—41no—
39C2×D244—41no—
40C2×D244—44no—
41C2×D244—82no—
42C2×D244—81no—
43C2×D244—81no—
44C2×C4×D61—21no—
45C2×C4×D61—21no—
46C2×C4×D61—41no—
47C2×C4×D61—41no—
48C22⋊#6923—81no—
49C22⋊#6923—81no—
50C4⋊#6923—81no—
51C4⋊#6923—81no—
52C2×C8×C61—21no—
53C2×C8×C61—41no—
54C4⋊C4×C61—22no—
55C4⋊C4×C61—41no—
56#11758198—44no—
57C2×#92418—21no—
58C2×#92418—41no—
59D488—42no—
60D488—42no—
61#11758187—81no—
62C23×C61#10211yescharacteristic
63C22×D61—21no—
64C22×D61—21no—
65C22×D61—41no—
66C22×D61—41no—
67C22×D61—42no—
68C22×D61—81no—
69C22×D61—81no—
70C22×D61—81no—
71C22×D61—81no—
72C22×D61—81no—
73C22×D61—81no—
74C22×D61D1611yescharacteristic
75C22×D61SD3211yescharacteristic
76C2×#6923—82no—
77C2×#6923—82no—
78C2×#6923—82no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—42no—
82C2×C4×C61—81no—
83C2×C4×C61—81no—
84C2×C4×C61#9911yescharacteristic
85C2×C4×C61#10411yescharacteristic
86C2×Dic61—21no—
87C2×Dic61—41no—
88D244—42no—
89D244—42no—
90D244—81no—
91D244—84no—
92D244—84no—
93D244—81no—
94C4×D61—41no—
95C4×D61—41no—
96C4×D61—81no—
97C8×C61—41no—
98#92418—41no—
99#7076—611nomaximal, Sylow
100C22×C61—21no—
101C22×C61—21no—
102C22×C61#32711yescharacteristic
103C22×C61#35711yescharacteristic
104C22×C61#36011yescharacteristic
105C2×D61—21no—
106C2×D61—21no—
107C2×D61—41no—
108C2×D61—41no—
109C2×D61—41no—
110C2×D61—41no—
111C2×D61—81no—
112C2×D61—82no—
113C2×D61—82no—
114C2×D61—82no—
115C2×D61—161no—
116C2×D61—161no—
117C2×D61—161no—
118C2×D61—161no—
119#6923—162no—
120C4×C61—21no—
121C4×C61—82no—
122C4×C61AΣL(1,𝔽16)11yescharacteristic
123C4×C61#35311yescharacteristic
124Dic61—41no—
125#1999—611no—
126#1314—611no—
127#1289—611no—
128C2×C61—21no—
129C2×C61—21no—
130C2×C61#133911yescharacteristic
131C2×C61#134311yescharacteristic
132C2×C61#134711yescharacteristic
133D61—41no—
134D61—41no—
135D61—162no—
136C22×D8—611no—
137C2×#121—1221no—
138C2×#102—1221no—
139C2×C22⋊C8—611no—
140#380—1221no—
141C22⋊C16—1221no—
142#340#692311yescharacteristic, complemented
143C61#707611yescharacteristic, complemented, Sylow
144C22×D4—1221no—
145C22×D4—2441no—
146C23×C4—611no—
147C2×D8—2441no—
148C2×D8—2442no—
149C22×C8—611no—
150C22×C8C2×#692311yescharacteristic
151#121—2442no—
152#121—2442no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—2441no—
155C2×C22⋊C4—2441no—
156C2×C22⋊C4—2441no—
157C2×C16—2441no—
158#102—2442no—
159C22⋊C8—2441no—
160#95—1222no—
161C24—611no—
162C24—2441no—
163C2×D4—1222no—
164C2×D4—4881no—
165C2×D4—4881no—
166C2×D4—4882no—
167C2×D4—2441no—
168C2×D4—2441no—
169C2×D4—2441no—
170C2×D4—2444no—
171C2×D4—2441no—
172C2×D4—2441no—
173C22×C4—41no—
174C22×C4—611no—
175C22×C4—1221no—
176C22×C4—1221no—
177C22×C4—2441no—
178C22×C4—2441no—
179C22×C4—2442no—
180C22×C4C22⋊#692311yescharacteristic
181D8—2442no—
182D8—2442no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—1221no—
186C2×C8—2441no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—4881no—
190C4⋊C4—4881no—
191C22⋊C4—4881no—
192C22⋊C4—4881no—
193C22⋊C4—2444no—
194C16—4881no—
195C23—611no—
196C23—611no—
197C23—1221no—
198C23—1221no—
199C23—4881no—
200C23—4881no—
201C23—4881no—
202C23—4881no—
203C23—4881no—
204C23—4881no—
205C23—2442no—
206C23—2441no—
207C23—2441no—
208C23#1451983911yescharacteristic
209D4—4881no—
210D4—4884no—
211D4—4884no—
212D4—4881no—
213D4—2442no—
214D4—2442no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—1221no—
221C2×C4—4881no—
222C2×C4—4882no—
223C2×C4—4882no—
224C2×C4—4882no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4—2441no—
228C2×C4#1451983611yescharacteristic
229C2×C4#1451984211yescharacteristic
230C8—41no—
231C8—2441no—
232C22—21no—
233C22—21no—
234C22—1221no—
235C22—1221no—
236C22—9761no—
237C22—9761no—
238C22—9761no—
239C22—9761no—
240C22—4882no—
241C22—4882no—
242C22—4881no—
243C22—4882no—
244C22—2441no—
245C22—2441no—
246C22—2441no—
247C22—2441no—
248C22#1684061411yescharacteristic
249C22#1684067411yescharacteristic
250C22#1684067911yescharacteristic
251C4—21no—
252C4—82no—
253C4—9762no—
254C4—2441no—
255C4#1684066311yescharacteristic
256C4#1684066711yescharacteristic
257C2—21no—
258C2—21no—
259C2—9762no—
260C2—2441no—
261C2—2441no—
262C2#2146079011yescharacteristic
263C2#2146079811yescharacteristic
264C2#2146080611yescharacteristic
subgroupG/H
centerC22#16840614
commutatorC22×C8×C61C2×C4
FrattiniC22×C8C2×#6923
FittingC61×#340C4
socleC22×C61#327
prime radical#21446818C2
cosocleC2×#14519689C22
hypercenter#340#6923
hypocenterC61#7076
LCSG ▷ C22×C8×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
1. #21460740C2semidirect
2. #21460682C2semidirect
6. C22×D488C4semidirect
10. C61×#340C4semidirect
142. #340#6923semidirect
143. C61#7076semidirect

Maximal subgroups

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

subgroupindex
#214607402
#214606822
#214468182
#707661