Finite Group Databasebeta

← #30599003

Subgroups — #30599003

Order 15616.

486 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 96 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#21461698C212yesmaximal, complemented
2#21461074C212yesmaximal
3#21460691C212yesmaximal, complemented
4C2×C8×C22⋊D61C211yesmaximal, characteristic
5C2×#14519847—22no—
6#16840733C2214yes—
7#16840638—22no—
8#16840608—22no—
9#16840605—22no—
10C22×C8×D61C2211yescharacteristic
11C2×C4×C22⋊D61C2211yescharacteristic
12C8×C22⋊D61—22no—
13C8×C22⋊D61—22no—
14C2×#14519706C411yescharacteristic
15C2×C22⋊#92418C411yescharacteristic
16C2×#14519689C2211yescharacteristic
17C2×C8×Dic61C411yescharacteristic
18C23×C8×C61C411yescharacteristic
19C2×#11758213—24no—
20#14519847—44no—
21#14519834—24no—
22C22⋊#92420—24no—
23C22×C22⋊D61Q811yescharacteristic
24C22×C4×D61C2311yescharacteristic
25C22×#92418C2×C411yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C2×C8×D61—42no—
28C2×C8×D61D412yes—
29C2×C8×D61D412yes—
30C4×C22⋊D61—22no—
31C4×C22⋊D61—22no—
32C2×#11758196D411yescharacteristic
33C2×C22⋊Dic61D411yescharacteristic
34C2×#11758198Q811yescharacteristic
35C2×C4×Dic61C2×C411yescharacteristic
36#14519706—22no—
37C22⋊#92418—22no—
38#14519689—22no—
39C8×Dic61—21no—
40C8×Dic61—21no—
41C23×C4×C61C2×C411yescharacteristic
42C22×C8×C61—21no—
43C22×C8×C61—21no—
44C22×C8×C61—42no—
45C22×C8×C61C2×C411yescharacteristic
46C22×C8×C61C2×C411yescharacteristic
47C24×C61C4⋊C411yescharacteristic
48C23×D61C2×Q811yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61C4⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C2×C22⋊D61—22no—
56C2×C22⋊D61—22no—
57C2×C22⋊D61—22no—
58C22×Dic61C4⋊C411yescharacteristic
59C22×Dic61C4⋊C411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C2×C4×D61—41no—
62C2×C4×D61—41no—
63C2×C4×D61C2×D412yes—
64C2×C4×D61#4112yes—
65C2×#92420—44no—
66#11758213—44no—
67#11758213—44no—
68C2×C8×C61—22no—
69C2×C8×C61—21no—
70C2×C8×C61—42no—
71C2×C8×C61—42no—
72C2×C8×C61—41no—
73C2×C8×C61C22⋊C411yescharacteristic
74C2×C8×C61C22⋊C411yescharacteristic
75C22⋊Dic61—22no—
76#11758198—22no—
77#11758196—22no—
78C4×Dic61—21no—
79C4×Dic61—21no—
80C2×#92418—21no—
81C2×#92418—41no—
82C2×#92418—41no—
83C2×#92418C22⋊C412yes—
84C8×D61—24no—
85C8×D61—42no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61#10812yescomplemented
91C23×C61C2×C4⋊C411yescharacteristic
92C22×D61—21no—
93C22×D61—21no—
94C22×D61—41no—
95C22×D61—41no—
96C22×D61C22⋊Q812yes—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—42no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61#11711yescharacteristic
106C22⋊D61—28no—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61#10812yescomplemented
113C2×Dic61#12112yes—
114C4×D61—22no—
115C4×D61—22no—
116C4×D61—42no—
117#92420—84no—
118C8×C61—21no—
119C8×C61—41no—
120C8×C61#9912yes—
121#92418—22no—
122#92418—41no—
123#8611—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61#38011yescharacteristic
132C22×C61C2×#10811yescharacteristic
133C22×C61C2×#10811yescharacteristic
134C2×D61—22no—
135C2×D61—22no—
136C2×D61—22no—
137C2×D61—42no—
138C2×D61—42no—
139C4×C61—21no—
140C4×C61—41no—
141C4×C61C2×#9911yescharacteristic
142C4×C61#41011yescharacteristic
143Dic61—22no—
144Dic61—22no—
145Dic61—41no—
146C2×C8×D4—611no—
147#1810—612no—
148#1465—612no—
149#1293—612no—
150C2×C61—21no—
151C2×C61—21no—
152C2×C61—42no—
153C2×C61#179911yescharacteristic
154C2×C61#181011yescharacteristic
155C2×C61#181311yescharacteristic
156D61—42no—
157C23×C8—611no—
158C23×C8#692311yescharacteristic
159C2×C4×D4—611no—
160C8×D4—1222no—
161C8×D4—1222no—
162C2×#108—1222no—
163C2×C4⋊C8—611no—
164#407—614no—
165C2×C22⋊C8—611no—
166C2×C22⋊C8—611no—
167C2×C4×C8—611no—
168#341—1222no—
169#324—1222no—
170#323—1222no—
171C61#861111yescharacteristic, complemented, Sylow
172C22×D4—611no—
173C23×C4—611no—
174C23×C4C2×#692311yescharacteristic
175C2×M16—1224no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—611no—
180C22×C8—612no—
181C22×C8—612no—
182C22×C8—611no—
183C22×C8—2442no—
184C22×C8C2×#692311yescharacteristic
185C22×C8C2×#692311yescharacteristic
186C4×D4—1222no—
187C4×D4—1222no—
188C2×C4⋊C4—611no—
189C2×C22⋊C4—611no—
190C2×C22⋊C4—611no—
191C2×C42—611no—
192#108—2444no—
193C4⋊C8—1222no—
194C22⋊C8—1222no—
195C22⋊C8—1222no—
196C22⋊C8—1224no—
197C22⋊C8—1224no—
198C4×C8—1221no—
199C4×C8—1221no—
200C24—611no—
201C24C4⋊#692311yescharacteristic
202C2×D4—1222no—
203C2×D4—1222no—
204C2×D4—1222no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—611no—
210C22×C4—611no—
211C22×C4—612no—
212C22×C4—612no—
213C22×C4—611no—
214C22×C4—2441no—
215C22×C4—2441no—
216C22×C4C4⋊#692311yescharacteristic
217C22×C4C22×#692311yescharacteristic
218M16—2444no—
219M16—2444no—
220C2×C8—22no—
221C2×C8—21no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—612no—
226C2×C8—1224no—
227C2×C8—1221no—
228C2×C8—2442no—
229C2×C8—2441no—
230C2×C8—2444no—
231C2×C8—2441no—
232C2×C8C22⋊#692311yescharacteristic
233C2×C8C22⋊#692311yescharacteristic
234C4⋊C4—1222no—
235C22⋊C4—1222no—
236C22⋊C4—1222no—
237C42—1221no—
238C42—1221no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—612no—
244C23—1221no—
245C23—1221no—
246C23—2441no—
247C23—2441no—
248C23#1451984712yescomplemented
249C23C2×C4⋊#692311yescharacteristic
250D4—1228no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—612no—
259C2×C4—612no—
260C2×C4—1221no—
261C2×C4—1221no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1221no—
265C2×C4—2442no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4C2×C22⋊#692311yescharacteristic
269C2×C4#1451985611yescharacteristic
270C8—21no—
271C8—41no—
272C8—1222no—
273C8—4884no—
274C8—2441no—
275C8#1451983512yes—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—1222no—
284C22—1222no—
285C22—1222no—
286C22—2442no—
287C22—2442no—
288C22#1684070411yescharacteristic
289C22C2×#1451984711yescharacteristic
290C22D61⋊#10811yescharacteristic
291C4—21no—
292C4—41no—
293C4—1222no—
294C4—1222no—
295C4—2441no—
296C4C2×#1451983511yescharacteristic
297C4#1684074111yescharacteristic
298C2—21no—
299C2—21no—
300C2—42no—
301C2—2442no—
302C2#2146167911yescharacteristic
303C2#2146169811yescharacteristic
304C2#2146170311yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×C8×C61C4
socleC22×C61#380
prime radicalC22×C22⋊D61Q8
cosocleC22×C4×D61C23
hypercenterC23×C8#6923
hypocenterC61#8611
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #21461698C2semidirect
3. #21460691C2semidirect
90. C23×C61#108semidirect
112. C2×Dic61#108semidirect
171. C61#8611semidirect
248. C23#14519847semidirect

Maximal subgroups

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

subgroupindex
#214616982
#214610742
#214606912
C2×C8×C22⋊D612
#861161