Finite Group Databasebeta

← #30607255

Subgroups — #30607255

Order 15616.

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

normal subgroups only

This group has 642 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#21461650C212yesmaximal
2#21460914C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#21448410C211yesmaximal, characteristic
5C2×#14519836—22no—
6C2×#14519836C2214yes—
7#16840705—22no—
8C22×#11758226C2211yescharacteristic
9C2×#14519734C411yescharacteristic, complemented
10C2×C4⋊C4×D61C411yescharacteristic, complemented
11#16839669C411yescharacteristic, complemented
12#16839664C2211yescharacteristic
13#16839663C2211yescharacteristic
14#16839640—22no—
15#16839621—22no—
16C61×#391C411yescharacteristic, complemented
17C2×C22⋊#6923—24no—
18#14519836—28no—
19#14519836—44no—
20(C2×#6923)⋊C4—44no—
21C2×#11758226—24no—
22C2×#11758226—42no—
23C2×#11758226—42no—
24C22×D244D412yes—
25C22×D244C2311yescharacteristic
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29#14519734—42no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—21no—
33C4⋊C4×D61—42no—
34C2×#11758198—21no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×C4×Dic61C2×C411yescharacteristic
38#14519684—22no—
39C22×Q8×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C42×C61C2×C411yescharacteristic
42#95×C61—22no—
43C23×D61C22⋊C411yescharacteristic
44C23×D61C2×D412yes—
45C22×#6923—44no—
46C2×Q8×C61—22no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—21no—
54C22×Dic61—21no—
55C22×Dic61C22⋊C411yescharacteristic
56#11758226—416no—
57C2×D244—24no—
58C2×D244—24no—
59C2×D244—42no—
60C2×D244—41no—
61C2×D244—41no—
62C2×D244#4114yes—
63C2×C4×D61—22no—
64C2×C4×D61—22no—
65C2×C4×D61—22no—
66C2×C4×D61—24no—
67C2×C4×D61—24no—
68C2×C4×D61—42no—
69C2×C4×D61—42no—
70C2×C4×D61—82no—
71C22⋊#6923—44no—
72C22⋊#6923—44no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75#11758198—42no—
76#11758198—82no—
77#11758198—82no—
78C4⋊Dic61—41no—
79#11758196—42no—
80#11758196—42no—
81C4×Dic61—41no—
82C23×C61C2×C22⋊C411yescharacteristic
83C22×D61—21no—
84C22×D61—21no—
85C22×D61—22no—
86C22×D61—22no—
87C22×D61—41no—
88C22×D61—41no—
89C22×D61—42no—
90C22×D61—42no—
91C22×D61#9912yes—
92C22×D61#12314yes—
93C2×#6923—84no—
94C2×#6923—84no—
95C2×#6923—84no—
96Q8×C61—44no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61#11712yes—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—41no—
111C2×Dic61—42no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61#9912yes—
116D244—44no—
117D244—48no—
118C4×D61—28no—
119C4×D61—48no—
120C4×D61—84no—
121#12859—611nomaximal, Sylow
122C22×C61—21no—
123C22×C61—21no—
124C22×C61#38111yescharacteristic
125C22×C61C2×#9911yescharacteristic
126C22×C61C2×#9911yescharacteristic
127C2×D61—22no—
128C2×D61—24no—
129C2×D61—42no—
130C2×D61—41no—
131C2×D61—42no—
132C2×D61—42no—
133C2×D61—41no—
134C2×D61—42no—
135C2×D61—44no—
136C2×D61—82no—
137#6923—164no—
138C4×C61—22no—
139C4×C61—42no—
140C4×C61—81no—
141C4×C61—81no—
142Dic61—22no—
143Dic61—81no—
144Dic61—81no—
145#2311—611no—
146#1782—612no—
147#1393—612no—
148#1393—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61#181911yescharacteristic
152C2×C61#212811yescharacteristic
153C2×C61#212911yescharacteristic
154D61—44no—
155D61—42no—
156C22×#41—611no—
157C2×#117—611no—
158C22×C4⋊C4—611no—
159C2×#99—614no—
160C2×#99—1222no—
161#391—611no—
162#391#692311yescharacteristic, complemented
163#390—611no—
164#390—611no—
165#386—1222no—
166#381—1222no—
167#380—1222no—
168C61#1285911yescharacteristic, complemented, Sylow
169C2×#41—1224no—
170C2×#41—2442no—
171C2×#41—2442no—
172C22×Q8C2×#692311yescharacteristic
173C22×D4—612no—
174C22×D4—611no—
175C23×C4—611no—
176C23×C4—612no—
177C23×C4—1221no—
178#117—2442no—
179C2×C4⋊C4—611no—
180C2×C4⋊C4—1221no—
181C2×C4⋊C4—1221no—
182C2×C4⋊C4—2442no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C22⋊C4—1222no—
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1221no—
187C2×C22⋊C4—1224no—
188C2×C42—611no—
189C2×C42C2×#692311yescharacteristic
190#99—1228no—
191#99—2444no—
192#95—22no—
193#95—1222no—
194#95—2444no—
195C24—612no—
196C24—611no—
197#41—24416no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×D4—614no—
202C2×D4—1224no—
203C2×D4—1224no—
204C2×D4—2441no—
205C2×D4—2441no—
206C2×D4—2442no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—611no—
210C22×C4—1222no—
211C22×C4—1224no—
212C22×C4—1222no—
213C22×C4—1224no—
214C22×C4—1222no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—4882no—
218C22×C4—2442no—
219C22×C4—2442no—
220C22×C4—2444no—
221C22×C4C22⋊#692312yes—
222C22×C4C22×#692311yescharacteristic
223C4⋊C4—41no—
224C4⋊C4—2441no—
225C4⋊C4—2442no—
226C4⋊C4—2442no—
227C22⋊C4—4882no—
228C22⋊C4—4882no—
229C22⋊C4—2442no—
230C22⋊C4—2444no—
231C22⋊C4—2444no—
232C42—41no—
233C42—2441no—
234C23—614no—
235C23—612no—
236C23—1222no—
237C23—1221no—
238C23—1221no—
239C23—1222no—
240C23—2442no—
241C23—2442no—
242C23—2441no—
243C23—2441no—
244C23C2×C22⋊#692311yescharacteristic
245Q8—44no—
246D4—2444no—
247D4—2448no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—612no—
259C2×C4—1228no—
260C2×C4—1221no—
261C2×C4—1221no—
262C2×C4—4884no—
263C2×C4—4884no—
264C2×C4—4884no—
265C2×C4—4884no—
266C2×C4—2448no—
267C2×C4—2441no—
268C2×C4—2442no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4#1451985612yes—
273C22—21no—
274C22—21no—
275C22—1222no—
276C22—1224no—
277C22—4882no—
278C22—2442no—
279C22—2444no—
280C22—2442no—
281C22—2441no—
282C22—2442no—
283C22—2442no—
284C22—2441no—
285C22#1684070511yescharacteristic
286C22D61⋊#9911yescharacteristic
287C22D61⋊#9911yescharacteristic
288C4—22no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—1222no—
293C4—9764no—
294C4—4881no—
295C4—4881no—
296C2—21no—
297C2—21no—
298C2—2444no—
299C2—2442no—
300C2#2146171511yescharacteristic
301C2#2146209911yescharacteristic
302C2#2146210211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×D244C23
hypercenter#391#6923
hypocenterC61#12859
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
3. #21460913C2semidirect
9. C2×#14519734C4semidirect
10. C2×C4⋊C4×D61C4semidirect
11. #16839669C4semidirect
16. C61×#391C4semidirect
162. #391#6923semidirect
168. C61#12859semidirect

Maximal subgroups

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

subgroupindex
#214616502
#214609142
#214609132
#214484102
#1285961