Finite Group Databasebeta

← #30594376

Subgroups — #30594376

Order 15616.

292 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#16840657C211yesmaximal, characteristic
2#21461555C211yesmaximal, characteristic
3#21447428C211yesmaximal, characteristic
4C22×#11758187C412yescomplemented
5D61⋊(C22⋊C8)C411yescharacteristic
6C2×#14519831—21no—
7C4×C22⋊#6923C411yescharacteristic
8#16840657—21no—
9#16840657—21no—
10(C2×#6923)⋊C8—21no—
11C22×C8×D61C2211yescharacteristic
12C2×#14519716—21no—
13C22⋊#11758185C411yescharacteristic, complemented
14#16839541—21no—
15C61×C22⋊C16C411yescharacteristic, complemented
16C2×D61⋊C8—21no—
17C2×C22⋊#6923—21no—
18C2×C4×#6923—21no—
19C2×#11758187—22no—
20C2×#11758187—42no—
21#14519834C812yes—
22C22⋊#92420—21no—
23#14519831—42no—
24(C2×#6923)⋊C4C812yes—
25C22×C4×D61C2×C411yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C2×C8×D61—42no—
28C2×C8×D61D411yescharacteristic
29C2×C8×D61D411yescharacteristic
30C2×C8×D61D411yescharacteristic
31C2×C8×D61Q811yescharacteristic
32#14519716—42no—
33C2×#11758185—21no—
34C22×C8×C61C2×C411yescharacteristic
35C2×C16×C61—21no—
36C23×D61M1611yescharacteristic
37C22×#6923—22no—
38C22×C4×C61C4211yescharacteristic
39C22×Dic61M1611yescharacteristic
40C2×C4×D61—41no—
41C2×C4×D61—41no—
42C2×C4×D61C22⋊C411yescharacteristic
43C2×C4×D61C4⋊C411yescharacteristic
44C2×C4×D61C2×C811yescharacteristic
45C2×C4×D61C2×C811yescharacteristic
46C22⋊#6923—41no—
47C22⋊#6923—41no—
48C2×#92420—22no—
49C4×#6923—41no—
50C4×#6923—41no—
51D61⋊C8—41no—
52D61⋊C8—41no—
53C2×C8×C61—41no—
54C2×C8×C61C22⋊C411yescharacteristic
55C2×C8×C61C4⋊C411yescharacteristic
56C2×#92418—22no—
57C2×#92418C22⋊C411yescharacteristic
58C2×#92418C4⋊C411yescharacteristic
59C8×D61—22no—
60C8×D61—22no—
61C8×D61—41no—
62C8×D61—41no—
63#11758187—42no—
64C16×C61—41no—
65#11758185—41no—
66C23×C61#9711yescharacteristic
67C22×D61—21no—
68C22×D61—21no—
69C22×D61—22no—
70C22×D61—41no—
71C22×D61C22⋊C811yescharacteristic
72C22×D61C4⋊C811yescharacteristic
73C2×#6923—42no—
74C2×#6923—42no—
75C2×#6923—42no—
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61#9511yescharacteristic
79C2×C4×C61C4×C811yescharacteristic
80C2×Dic61—41no—
81C2×Dic61C22⋊C811yescharacteristic
82C2×Dic61C4⋊C811yescharacteristic
83C4×D61—21no—
84C4×D61—21no—
85C4×D61—21no—
86C4×D61—21no—
87C4×D61—41no—
88C4×D61—41no—
89#92420—42no—
90C8×C61—21no—
91C8×C61—21no—
92#92418—21no—
93#92418#9912yes—
94#7200—611nomaximal, Sylow
95C22×C61—21no—
96C22×C61—21no—
97C22×C61#33611yescharacteristic
98C22×C61#34711yescharacteristic
99C22×C61#34711yescharacteristic
100C2×D61—21no—
101C2×D61—21no—
102C2×D61—21no—
103C2×D61—21no—
104C2×D61—21no—
105C2×D61—42no—
106C2×D61—42no—
107C2×D61#35012yes—
108#6923—82no—
109C4×C61—21no—
110C4×C61#34211yescharacteristic
111C4×C61#34411yescharacteristic
112Dic61—21no—
113Dic61—21no—
114#2112—611no—
115C2×C22⋊C16—611no—
116#1752—611no—
117C2×C61—21no—
118C2×C61—21no—
119C2×C61#128211yescharacteristic
120C2×C61#137511yescharacteristic
121C2×C61#137911yescharacteristic
122D61—22no—
123D61—41no—
124C23×C8—611no—
125C2×M32—1221no—
126C22×C16—612no—
127C2×C22⋊C8—611no—
128C2×#97—1221no—
129C4×C22⋊C4—611no—
130C22⋊C16—611no—
131C22⋊C16—1221no—
132C22⋊C16—1221no—
133C22⋊C16—1221no—
134C22⋊C16#692311yescharacteristic, complemented
135#336—1221no—
136C61#720011yescharacteristic, complemented, Sylow
137C23×C4—611no—
138C22×C8—611no—
139C22×C8—611no—
140C22×C8—611no—
141C22×C8—611no—
142C22×C8—611no—
143C22×C8—1221no—
144C22×C8—2442no—
145C22×C8C2×#692311yescharacteristic
146C2×C22⋊C4—1221no—
147C2×C42—1221no—
148M32—2442no—
149C2×C16—21no—
150C2×C16—1222no—
151C2×C16—1221no—
152C2×C16—2442no—
153C22⋊C8—612no—
154C22⋊C8—1221no—
155#97—2442no—
156#95—612no—
157C24—611no—
158C22×C4—611no—
159C22×C4—611no—
160C22×C4—611no—
161C22×C4—611no—
162C22×C4—611no—
163C22×C4—1222no—
164C22×C4—2441no—
165C22×C4—2441no—
166C22×C4C4×#692311yescharacteristic
167C2×C8—41no—
168C2×C8—611no—
169C2×C8—611no—
170C2×C8—1222no—
171C2×C8—1222no—
172C2×C8—1222no—
173C2×C8—1222no—
174C2×C8—2441no—
175C2×C8—2441no—
176C2×C8—2441no—
177C2×C8—2441no—
178C2×C8C4⋊#692311yescharacteristic
179C2×C8C22⋊#692311yescharacteristic
180C22⋊C4—2441no—
181C22⋊C4—2441no—
182C42—2441no—
183C42—2441no—
184C16—41no—
185C16—2441no—
186C16—2442no—
187C23—611no—
188C23—611no—
189C23—1222no—
190C23—1221no—
191C23—1221no—
192C23—2441no—
193C23#1451983111yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—611no—
197C2×C4—611no—
198C2×C4—1221no—
199C2×C4—1221no—
200C2×C4—1221no—
201C2×C4—1221no—
202C2×C4—2441no—
203C2×C4—2441no—
204C2×C4—2441no—
205C2×C4—2442no—
206C2×C4—2442no—
207C2×C4—2442no—
208C2×C4(C2×#6923)⋊C411yescharacteristic
209C2×C4C8×#692311yescharacteristic
210C8—21no—
211C8—21no—
212C8—612no—
213C8—1221no—
214C8—2442no—
215C22—21no—
216C22—21no—
217C22—612no—
218C22—1221no—
219C22—1221no—
220C22—1221no—
221C22—1221no—
222C22—1221no—
223C22—2442no—
224C22—2442no—
225C22(C2×#6923)⋊C811yescharacteristic
226C22#1684065211yescharacteristic
227C22#1684065311yescharacteristic
228C4—21no—
229C4—1221no—
230C4—1221no—
231C4—4882no—
232C4#1684063911yescharacteristic
233C4#1684064511yescharacteristic
234C2—21no—
235C2—21no—
236C2—1222no—
237C2—2441no—
238C2#2146066011yescharacteristic
239C2#2146085911yescharacteristic
240C2#2146087511yescharacteristic
subgroupG/H
centerC2×C4(C2×#6923)⋊C4
commutatorC2×C4×C61C4×C8
FrattiniC22×C8C2×#6923
FittingC61×C22⋊C16C4
socleC22×C61#336
prime radicalC23×D61M16
cosocleC22×C8×D61C22
hypercenterC22⋊C16#6923
hypocenterC61#7200
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
4. C22×#11758187C4semidirect
13. C22⋊#11758185C4semidirect
15. C61×C22⋊C16C4semidirect
134. C22⋊C16#6923semidirect
136. C61#7200semidirect

Maximal subgroups

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

subgroupindex
C2×#168406572
#214615552
#214474282
#720061