Finite Group Databasebeta

← #30594359

Subgroups — #30594359

Order 15616.

310 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D61⋊#349C211yesmaximal, characteristic
2#21461487C211yesmaximal, characteristic
3#349×D61C211yesmaximal, characteristic
4C22×D61⋊C8C411yescharacteristic
5C2×#14519856C411yescharacteristic
6D61⋊M32—21no—
7#16840659—21no—
8#16840658C412yescomplemented
9(C2×#6923)⋊C8—21no—
10(C2×#6923)⋊C8—21no—
11C2×M16×D61C2211yescharacteristic
12M32×D61—21no—
13#16839544C411yescharacteristic, complemented
14#16839543—21no—
15C61×#349C411yescharacteristic, complemented
16C2×D61⋊C8—21no—
17C2×D61⋊C8—41no—
18C2×D61⋊C8—41no—
19C2×D61⋊C8C812yes—
20C22×#92420—21no—
21#14519856—41no—
22#14519856—41no—
23C2×C4⋊#6923C812yes—
24C2×C22⋊#6923—21no—
25C2×C4×#6923—21no—
26#14519852—41no—
27#14519851—22no—
28D61⋊C16—41no—
29C22×C4×D61C2×C411yescharacteristic
30C2×#11758194C2×C411yescharacteristic
31C2×#11758189D411yescharacteristic
32C2×#11758189Q811yescharacteristic
33M16×D61—41no—
34M16×D61—41no—
35C2×C8×D61D411yescharacteristic
36C2×C8×D61D411yescharacteristic
37#14519717—21no—
38#14519716—41no—
39C16×D61—41no—
40C2×M16×C61C2×C411yescharacteristic
41M32×C61—21no—
42C23×D61M1611yescharacteristic
43C22×#6923—22no—
44C22×C4×C61C4211yescharacteristic
45C22×Dic61M1611yescharacteristic
46C2×C4×D61—21no—
47C2×C4×D61—21no—
48C2×C4×D61—21no—
49C2×C4×D61—21no—
50C2×C4×D61C22⋊C411yescharacteristic
51C2×C4×D61C4⋊C411yescharacteristic
52C2×C4×D61C2×C811yescharacteristic
53C2×C4×D61C2×C811yescharacteristic
54C22⋊#6923—42no—
55C2×#92420—22no—
56C2×#92420—42no—
57C4⋊#6923—42no—
58C4×#6923—41no—
59C4×#6923—41no—
60D61⋊C8—41no—
61D61⋊C8—41no—
62D61⋊C8—42no—
63M16×C61—41no—
64C2×C8×C61C22⋊C411yescharacteristic
65C2×C8×C61C4⋊C411yescharacteristic
66#11758194—41no—
67C2×#92418C22⋊C411yescharacteristic
68C2×#92418C4⋊C411yescharacteristic
69#11758189—41no—
70#11758189—41no—
71C8×D61—21no—
72C8×D61—21no—
73C8×D61—21no—
74C8×D61—21no—
75#11758187—42no—
76C16×C61—41no—
77#11758185—41no—
78C23×C61#9711yescharacteristic
79C22×D61—21no—
80C22×D61—21no—
81C22×D61—42no—
82C22×D61C22⋊C811yescharacteristic
83C22×D61C4⋊C811yescharacteristic
84C2×#6923—44no—
85C2×#6923—42no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61#9511yescharacteristic
89C2×C4×C61C4×C811yescharacteristic
90C2×Dic61—21no—
91C2×Dic61—21no—
92C2×Dic61C22⋊C811yescharacteristic
93C2×Dic61C4⋊C811yescharacteristic
94C4×D61—21no—
95C4×D61—21no—
96C4×D61—21no—
97C4×D61—21no—
98C4×D61—21no—
99C4×D61—21no—
100C4×D61#9911yescharacteristic
101C4×D61#9911yescharacteristic
102C4×D61#10011yescharacteristic
103C4×D61#10111yescharacteristic
104#92420—42no—
105C8×C61—21no—
106C8×C61—21no—
107#92418—21no—
108#92418—21no—
109#7194—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—41no—
112C22×C61#33611yescharacteristic
113C2×D61—22no—
114C2×D61—21no—
115C2×D61—42no—
116C2×D61—42no—
117C2×D61—41no—
118C2×D61#32311yescharacteristic
119C2×D61#32411yescharacteristic
120#6923—82no—
121C4×C61—21no—
122C4×C61#34211yescharacteristic
123C4×C61#34311yescharacteristic
124Dic61—21no—
125Dic61#32311yescharacteristic
126Dic61#32411yescharacteristic
127C2×#349—611no—
128C2×#349—611no—
129#1729—611no—
130C2×C61—21no—
131C2×C61—41no—
132C2×C61#128011yescharacteristic
133D61—21no—
134D61—41no—
135D61#132112yes—
136C22×M16—611no—
137C23×C8—611no—
138C2×#117—611no—
139C2×M32—1221no—
140C2×M32—1221no—
141#349—612no—
142#349—611no—
143#349—1221no—
144#349—1221no—
145#349#692311yescharacteristic, complemented
146#336—1221no—
147#336—1221no—
148C61#719411yescharacteristic, complemented, Sylow
149C23×C4—611no—
150C2×M16—611no—
151C2×M16—611no—
152C2×M16—611no—
153C2×M16—2441no—
154C2×M16—2441no—
155C2×M16C2×#692311yescharacteristic
156C22×C8—611no—
157C22×C8—611no—
158C22×C8—612no—
159C22×C8—1221no—
160C22×C8—1221no—
161C22×C8—2441no—
162C22×C8—2441no—
163#117—2441no—
164#117—2441no—
165C2×C4⋊C4—612no—
166C2×C22⋊C4—1221no—
167C2×C42—1221no—
168M32—21no—
169M32—1222no—
170M32—1221no—
171M32—2441no—
172M32—2441no—
173C2×C16—2441no—
174C2×C16—2441no—
175C24—611no—
176C22×C4—611no—
177C22×C4—611no—
178C22×C4—611no—
179C22×C4—611no—
180C22×C4—611no—
181C22×C4—1221no—
182C22×C4—1221no—
183C22×C4—1221no—
184C22×C4—1221no—
185C22×C4—1222no—
186C22×C4C4×#692311yescharacteristic
187M16—41no—
188M16—2441no—
189M16—2441no—
190M16—2441no—
191C2×C8—611no—
192C2×C8—611no—
193C2×C8—1221no—
194C2×C8—1221no—
195C2×C8—1221no—
196C2×C8—1221no—
197C2×C8—1222no—
198C2×C8—2441no—
199C2×C8—2441no—
200C2×C8—2442no—
201C2×C8—2442no—
202C2×C8C4⋊#692311yescharacteristic
203C2×C8C22⋊#692311yescharacteristic
204C4⋊C4—2442no—
205C22⋊C4—2442no—
206C42—2441no—
207C42—2441no—
208C16—41no—
209C16—2441no—
210C16—2442no—
211C23—611no—
212C23—611no—
213C23—1221no—
214C23—1221no—
215C23—2442no—
216C23#1451983111yescharacteristic
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—611no—
220C2×C4—611no—
221C2×C4—611no—
222C2×C4—611no—
223C2×C4—611no—
224C2×C4—611no—
225C2×C4—1221no—
226C2×C4—1221no—
227C2×C4—1221no—
228C2×C4—1221no—
229C2×C4—1221no—
230C2×C4—1221no—
231C2×C4—1221no—
232C2×C4—1221no—
233C2×C4—2442no—
234C2×C4—2444no—
235C2×C4(C2×#6923)⋊C411yescharacteristic
236C2×C4C8×#692311yescharacteristic
237C8—21no—
238C8—21no—
239C8—1221no—
240C8—1221no—
241C8—2442no—
242C22—21no—
243C22—41no—
244C22—611no—
245C22—611no—
246C22—1222no—
247C22—1221no—
248C22—2441no—
249C22—2442no—
250C22—2442no—
251C22(C2×#6923)⋊C811yescharacteristic
252C4—21no—
253C4—611no—
254C4—611no—
255C4—1221no—
256C4—4882no—
257C4#1684063911yescharacteristic
258C4#1684064111yescharacteristic
259C2—21no—
260C2—41no—
261C2—612no—
262C2—1221no—
263C2—2441no—
264C2#2146065611yescharacteristic
subgroupG/H
centerC4#16840639
commutatorC2×C4×C61C4×C8
FrattiniC2×M16C2×#6923
FittingC61×#349C4
socleC2×C61#1280
prime radicalC23×D61M16
cosocleC2×M16×D61C22
hypercenter#349#6923
hypocenterC61#7194
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
8. #16840658C4semidirect
13. #16839544C4semidirect
15. C61×#349C4semidirect
145. #349#6923semidirect
148. C61#7194semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3492
#214614872
#349×D612
#719461