Finite Group Databasebeta

← #29577735

Subgroups — #29577735

Order 15616.

512 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450300C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic
3#21444353C212yesmaximal
4#21443673C212yesmaximal
5C61×#1821C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7#16839700—22no—
8#16839667C412yescomplemented
9#16839648—24no—
10#16839627C412yescomplemented
11#16839625—21no—
12#16839625—21no—
13#16839625C2212yes—
14#16839622—21no—
15C4×C22⋊Dic61—21no—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839500—24no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C61×#407C2212yes—
20C61×#343—22no—
21C22×Dic122C2×C411yescharacteristic
22#14519756—22no—
23#14519756—22no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×#11758196—21no—
27C2×C22⋊Dic61—21no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—21no—
30C2×C4×Dic61—22no—
31C2×C4×Dic61C2×C412yes—
32#14519684—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—21no—
39#14519684—22no—
40#14519684D414yes—
41C23×C4×C61D411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61—22no—
47C2×C4⋊C4×C61C2×C412yes—
48C2×C4⋊C4×C61D412yes—
49C22⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61C2×D411yescharacteristic
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D412yes—
64C22×C4×C61#4112yes—
65C22×Dic61—21no—
66C22×Dic61—22no—
67C22×Dic61—21no—
68C22×Dic61—24no—
69C2×Dic122—22no—
70C2×Dic122—42no—
71M16×C61—44no—
72C2×C8×C61—42no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—24no—
75C22⋊Dic61—44no—
76C22⋊Dic61—42no—
77C22⋊Dic61—41no—
78C22⋊Dic61—41no—
79C4⋊Dic61—22no—
80C4⋊Dic61—82no—
81#11758196—42no—
82C4×Dic61—42no—
83C4×Dic61—82no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—41no—
87C23×C61C2×C22⋊C411yescharacteristic
88C23×C61#12011yescharacteristic
89C23×C61#12311yescharacteristic
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—81no—
101C2×C4×C61—81no—
102C2×C4×C61C4×D412yes—
103C2×C4×C61#12112yes—
104C2×Dic61—42no—
105C2×Dic61—44no—
106C2×Dic61—42no—
107C2×Dic61—44no—
108C2×Dic61—42no—
109C2×Dic61—41no—
110C2×Dic61—41no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—44no—
114Dic122—82no—
115C8×C61—82no—
116#9460—611nomaximal, Sylow
117C22×C61—21no—
118C22×C61—21no—
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—44no—
125C22×C61#38611yescharacteristic
126C22×C61#41012yes—
127C22×C61#45512yes—
128C4×C61—22no—
129C4×C61—22no—
130C4×C61—42no—
131C4×C61—42no—
132Dic61—84no—
133Dic61—82no—
134Dic61—81no—
135Dic61—81no—
136#2503—611no—
137#2313—611no—
138#1821D6111yescharacteristic
139#1511—612no—
140#1284—612no—
141C2×C61—22no—
142C2×C61—22no—
143C2×C61—41no—
144C2×C61#188511yescharacteristic
145C2×C61#189911yescharacteristic
146C2×C61#191211yescharacteristic
147C2×C22⋊Q8—611no—
148C22×C4⋊C4C2×D6111yescharacteristic
149#407C2×D6112yes—
150#400—1222no—
151#391—612no—
152#387—1224no—
153#382—612no—
154#381—612no—
155#381—1221no—
156#381—1221no—
157#380—1221no—
158C4×C22⋊C4—611no—
159C4×C22⋊C4—1221no—
160#343—22no—
161#328—1224no—
162C61#946011yescharacteristic, complemented, Sylow
163C22×Q8—611no—
164C23×C4D24411yescharacteristic
165C23×C4D24411yescharacteristic
166C23×C4C22×D6111yescharacteristic
167C2×M16—22no—
168C22⋊Q8—1222no—
169C22⋊Q8—1222no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—611no—
173C2×C4⋊C4—1222no—
174C2×C4⋊C4—1221no—
175C2×C4⋊C4C4×D6112yes—
176C2×C4⋊C4C22⋊D6112yes—
177C2×C22⋊C4—1221no—
178C2×C22⋊C4—1221no—
179C2×C22⋊C4—1222no—
180C2×C42—612no—
181C2×C42—1222no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184#95—614no—
185#95—1222no—
186#95—1222no—
187#95—1222no—
188#95—1222no—
189#95—1222no—
190#95—1222no—
191#95—1222no—
192#95—1221no—
193C24C2×D24411yescharacteristic
194C2×Q8—1222no—
195C2×Q8—2442no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—1222no—
204C22×C4—1221no—
205C22×C4—1221no—
206C22×C4—1224no—
207C22×C4#1175819812yes—
208C22×C4#1175819812yes—
209C22×C4C2×C4×D6111yescharacteristic
210C22×C4D4×D6112yes—
211C22×C4#1175822412yes—
212C22×C4C2×C22⋊D6111yescharacteristic
213M16—44no—
214C2×C8—42no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—1222no—
218C4⋊C4—4882no—
219C4⋊C4—2442no—
220C22⋊C4—2442no—
221C22⋊C4—2444no—
222C22⋊C4—2441no—
223C22⋊C4—2441no—
224C42—4882no—
225C42—2442no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×#1175819811yescharacteristic
230C23#1451974711yescharacteristic
231C23#1451976111yescharacteristic
232Q8—4882no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—2442no—
246C2×C4—2444no—
247C2×C4—2442no—
248C2×C4—2444no—
249C2×C4—2442no—
250C2×C4—2444no—
251C2×C4—2442no—
252C2×C4—2442no—
253C2×C4—2441no—
254C2×C4—2441no—
255C2×C4#1451973712yes—
256C2×C4#1451974912yes—
257C8—82no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1683963911yescharacteristic
267C22#1683973512yes—
268C22#1683994212yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—4884no—
274C4—4882no—
275C4—4881no—
276C4—4881no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2144620511yescharacteristic
281C2#2144627511yescharacteristic
282C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1821D61
hypocenterC61#9460
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
8. #16839667C4semidirect
10. #16839627C4semidirect
162. C61#9460semidirect

Maximal subgroups

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

subgroupindex
#214503002
#214484222
#214443532
#214436732
C61×#18212
#946061