Finite Group Databasebeta

← #30604104

Subgroups — #30604104

Order 15616.

574 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461067C212yesmaximal, complemented
2#21460750C212yesmaximal, complemented
3#21460748C212yesmaximal, complemented
4#21448407C211yesmaximal, characteristic
5#16840732—22no—
6#16840607—22no—
7#16840607C2214yes—
8C22×#11758222C411yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16839668C411yescharacteristic
11#16839668C2211yescharacteristic
12#16839664C411yescharacteristic
13#16839664C2211yescharacteristic
14#16839639—22no—
15#16839619—22no—
16C2×#117×C61C411yescharacteristic
17C2×#11758217—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C2×#11758222—22no—
21C2×#11758222—22no—
22C2×#11758222—42no—
23C2×#11758222—42no—
24C22×Dic122C2×C411yescharacteristic
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C23×Dic61D412yes—
28C22×C4×D61—21no—
29C2×C4⋊Dic61—42no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C4⋊Dic61D412yes—
32C2×C4⋊Dic61C2311yescharacteristic
33C2×#11758198—22no—
34C2×#11758198—22no—
35#14519684—22no—
36#14519684—22no—
37C23×C4×C61C2×C411yescharacteristic
38#117×C61—42no—
39C2×C4⋊C4×C61—21no—
40C2×C22⋊C4×C61—21no—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C24×C61C22⋊C411yescharacteristic
44C23×D61—21no—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C2×C22⋊D61—24no—
54C2×C22⋊D61—42no—
55C2×C22⋊D61—42no—
56C22×Dic61—22no—
57C22×Dic61—21no—
58C22×Dic61—22no—
59C22×Dic61—82no—
60C22×Dic61C22⋊C412yes—
61C22×Dic61C2×D412yes—
62#11758222—48no—
63#11758222—48no—
64C2×D244—21no—
65C2×D244—41no—
66C2×D244—41no—
67C2×D244M1612yescomplemented
68C2×C4×D61—24no—
69C2×C4×D61—42no—
70C2×C4×D61—42no—
71C2×Dic122—21no—
72C2×Dic122—41no—
73C2×Dic122—41no—
74C2×Dic122M1612yescomplemented
75#11758217—84no—
76C2×#92420—44no—
77C4⋊C4×C61—42no—
78C22⋊C4×C61—42no—
79C42×C61—41no—
80C42×C61—41no—
81#11758198—82no—
82#11758198—82no—
83C4⋊Dic61—41no—
84C4⋊Dic61—42no—
85C4⋊Dic61—41no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61#9912yes—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—22no—
93C22×D61—42no—
94C22×D61—42no—
95C22×D61—41no—
96C22×D61—41no—
97C2×C4×C61—24no—
98C2×C4×C61—24no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61#10012yes—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C22⋊D61—48no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—44no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—82no—
122D244—42no—
123D244—42no—
124C4×D61—48no—
125Dic122—42no—
126Dic122—42no—
127#92420—84no—
128#11508—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—42no—
136C22×C61#40711yescharacteristic
137C22×C61C2×#9911yescharacteristic
138C22×C61C2×#10011yescharacteristic
139C2×D61—44no—
140C2×D61—42no—
141C2×D61—82no—
142C2×D61—81no—
143C2×D61—81no—
144C4×C61—21no—
145C4×C61—21no—
146C4×C61—22no—
147C4×C61—81no—
148C4×C61—81no—
149Dic61—42no—
150Dic61—42no—
151Dic61—42no—
152#2311—611no—
153#1463—612no—
154#1319—612no—
155#1318—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—42no—
159C2×C61#146211yescharacteristic
160C2×C61#212711yescharacteristic
161C2×C61#213311yescharacteristic
162D61—82no—
163C22×#41—611no—
164C2×#117#692311yescharacteristic
165C22×C4⋊C4—611no—
166#407—1222no—
167#391—611no—
168#391—611no—
169#390—611no—
170#390—611no—
171#386—1222no—
172#380—1222no—
173#324—614no—
174#324—1222no—
175C61#1150811yescharacteristic, complemented, Sylow
176C2×#41—1222no—
177C2×#41—1222no—
178C2×#41—2442no—
179C2×#41—2442no—
180C22×Q8—611no—
181C22×D4—611no—
182C22×D4—1221no—
183C23×C4—612no—
184C23×C4—1221no—
185C23×C4C2×#692311yescharacteristic
186C2×M16—2442no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—612no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—2442no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1222no—
196C2×C42C2×#692311yescharacteristic
197C2×C42C2×#692311yescharacteristic
198C22⋊C8—1224no—
199C22⋊C8—2442no—
200#95—1222no—
201#95—1222no—
202C24—1221no—
203C24C22⋊#692311yescharacteristic
204#41—2448no—
205#41—2448no—
206C2×Q8—612no—
207C2×Q8—1221no—
208C2×Q8—2441no—
209C2×Q8—2441no—
210C2×D4—612no—
211C2×D4—1224no—
212C2×D4—1221no—
213C2×D4—2441no—
214C2×D4—2441no—
215C2×D4—2442no—
216C2×D4—2442no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—612no—
224C22×C4—612no—
225C22×C4—1224no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—1221no—
229C22×C4—4882no—
230C22×C4—2442no—
231C22×C4—2442no—
232C22×C4C22⋊#692311yescharacteristic
233C22×C4C22×#692311yescharacteristic
234M16—4884no—
235C2×C8—2444no—
236C4⋊C4—42no—
237C4⋊C4—2442no—
238C4⋊C4—2441no—
239C4⋊C4—2441no—
240C22⋊C4—42no—
241C22⋊C4—4882no—
242C22⋊C4—4882no—
243C42—41no—
244C42—41no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—1222no—
250C23—2442no—
251C23—2442no—
252C23—2441no—
253C23—2441no—
254C23#1451983612yes—
255C23C2×C22⋊#692311yescharacteristic
256Q8—2442no—
257Q8—2442no—
258D4—2442no—
259D4—2442no—
260D4—2448no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1222no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—4882no—
277C2×C4—2448no—
278C2×C4—2444no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4#1451983712yes—
284C2×C4D61⋊M1611yescharacteristic
285C2×C4D61⋊M1611yescharacteristic
286C8—4884no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—4882no—
295C22—4881no—
296C22—4881no—
297C22—2444no—
298C22—2442no—
299C22#1684073411yescharacteristic
300C22C2×#1451983611yescharacteristic
301C22C2×#1451983711yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—81no—
306C4—81no—
307C4—2442no—
308C4—2442no—
309C4—2442no—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—4882no—
314C2#2146106611yescharacteristic
315C2#2146209611yescharacteristic
316C2#2146211611yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×#117#6923
hypocenterC61#11508
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21461067C2semidirect
2. #21460750C2semidirect
3. #21460748C2semidirect
67. C2×D244M16semidirect
74. C2×Dic122M16semidirect
175. C61#11508semidirect

Maximal subgroups

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

subgroupindex
#214610672
#214607502
#214607482
#214484072
#1150861