Finite Group Databasebeta

← #30601144

Subgroups — #30601144

Order 15616.

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462715C211yesmaximal, characteristic
2#21461348C212yesmaximal
3#21461082C211yesmaximal, characteristic
4#21460670C212yesmaximal
5#21451036C211yesmaximal, characteristic
6#16840813—24no—
7#16840782—21no—
8#16840782C2212yes—
9#16840782C2212yes—
10#16840778—21no—
11#16840621—24no—
12#16840607—21no—
13#16840607—21no—
14C2×#14519776C2211yescharacteristic
15C2×Q8×Dic61C2211yescharacteristic
16#16839704—21no—
17#16839704—21no—
18#16839669C411yescharacteristic
19#16839666—21no—
20#16839666—21no—
21#16839628C411yescharacteristic
22#16839627C411yescharacteristic
23C4×#11758196C2211yescharacteristic
24C61×#391C411yescharacteristic
25#14519844—22no—
26#14519844—22no—
27#14519844D414yes—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33#14519832—22no—
34C4×#92420—22no—
35C22×Dic122—21no—
36#14519776—22no—
37#14519776—22no—
38Q8×Dic61—22no—
39Q8×Dic61—22no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61D411yescharacteristic
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C2×Q8×C61—22no—
63C2×Q8×C61—22no—
64C2×Q8×C61—41no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—44no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×#92420—24no—
80C2×#92420—42no—
81C4⋊C4×C61—22no—
82C42×C61—41no—
83C4⋊Dic61—22no—
84C4⋊Dic61—22no—
85C4⋊Dic61—22no—
86C4⋊Dic61—22no—
87#11758196—42no—
88#11758196—42no—
89C4×Dic61—21no—
90C4×Dic61—21no—
91C4×Dic61—21no—
92C4×Dic61—21no—
93C4×Dic61—41no—
94C4×Dic61C2×D412yes—
95C4×Dic61#4112yes—
96C23×C61C2×C22⋊C411yescharacteristic
97Q8×C61—44no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61#10112yes—
111C2×C4×C61C4×D412yes—
112C2×Dic61—22no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61#12011yescharacteristic
124C2×Dic61#12112yes—
125C2×Dic61#12311yescharacteristic
126Dic122—82no—
127Dic122—82no—
128#92420—44no—
129#92420—82no—
130#9456—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—21no—
133C22×C61#38611yescharacteristic
134C22×C61C2×#10111yescharacteristic
135C22×C61#41311yescharacteristic
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—41no—
139C4×C61—81no—
140C4×C61—81no—
141Dic61—21no—
142Dic61—22no—
143Dic61—41no—
144Dic61—41no—
145Dic61—81no—
146Dic61—81no—
147Dic61#45412yes—
148#2983—611no—
149#2568—611no—
150#1556—612no—
151#1467—611no—
152#1285—612no—
153C2×C61—21no—
154C2×C61—21no—
155C2×C61#188811yescharacteristic
156C2×C61#190111yescharacteristic
157C2×C61#190911yescharacteristic
158C2×C4⋊Q8—611no—
159C2×C4×Q8—611no—
160#446—1224no—
161#432—612no—
162#432—612no—
163#432—1221no—
164#431—1221no—
165#400—1221no—
166#400—1221no—
167#391—611no—
168#391—1221no—
169#391—1221no—
170#391#692311yescharacteristic
171#382—611no—
172#382—611no—
173C4×C4⋊C4—611no—
174#332—1224no—
175#324—1221no—
176#324—1221no—
177C61#945611yescharacteristic, complemented, Sylow
178C22×Q8—1221no—
179C22×Q8C2×#692311yescharacteristic
180C4⋊Q8—1222no—
181C4⋊Q8—1222no—
182C4×Q8—1222no—
183C4×Q8—1222no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—611no—
187C2×C4⋊C4—611no—
188C2×C4⋊C4—1221no—
189C2×C4⋊C4—1221no—
190C2×C4⋊C4C2×#692311yescharacteristic
191C2×C42—611no—
192C2×C42—611no—
193C2×C42—611no—
194C2×C42—611no—
195C2×C42C2×#692311yescharacteristic
196C4⋊C8—614no—
197C4⋊C8—1222no—
198C4⋊C8—1222no—
199C4⋊C8—1222no—
200C22⋊C8—1222no—
201C22⋊C8—1222no—
202#97—1222no—
203#97—1222no—
204#97—1222no—
205C4×C8—1222no—
206#95—21no—
207#95—21no—
208#95—1221no—
209#95—1221no—
210#95—1221no—
211#95—1221no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—2441no—
219C2×Q8—2441no—
220C2×Q8—2444no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4C22⋊#692311yescharacteristic
230C22×C4C22⋊#692311yescharacteristic
231C22×C4C22×#692311yescharacteristic
232C2×C8—1224no—
233C2×C8—2442no—
234C4⋊C4—22no—
235C4⋊C4—1222no—
236C4⋊C4—1222no—
237C4⋊C4—1222no—
238C4⋊C4—1222no—
239C4⋊C4—2442no—
240C4⋊C4—2442no—
241C42—41no—
242C42—612no—
243C42—612no—
244C42—1221no—
245C42—1221no—
246C42—1221no—
247C42—1221no—
248C42—2441no—
249C23C2×C22⋊#692311yescharacteristic
250Q8—44no—
251Q8—4882no—
252Q8—4882no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—611no—
266C2×C4—612no—
267C2×C4—611no—
268C2×C4—1222no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4—2442no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4#1451983812yes—
280C2×C4D4×#692312yes—
281C8—4882no—
282C8—2444no—
283C22—21no—
284C22—21no—
285C22#1684071011yescharacteristic
286C22C2×#1451983811yescharacteristic
287C22#1684074911yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—612no—
294C4—1221no—
295C4—1222no—
296C4—4881no—
297C4—4881no—
298C4—2441no—
299C4—2441no—
300C2—21no—
301C2—21no—
302C2#2146182711yescharacteristic
303C2#2146184611yescharacteristic
304C2#2146186111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9456
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
177. C61#9456semidirect

Maximal subgroups

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

subgroupindex
#214627152
#214613482
#214610822
#214606702
#214510362
#945661