Finite Group Databasebeta

← #29577481

Subgroups — #29577481

Order 15616.

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

normal subgroups only

This group has 552 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#21451299C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic, complemented
3#21444336C212yesmaximal
4#21443704C212yesmaximal, complemented
5C61×#1463C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7#16839701C412yescomplemented
8#16839679C412yescomplemented
9#16839652—22no—
10#16839648—24no—
11#16839641—21no—
12#16839641—21no—
13#16839625—21no—
14#16839622—21no—
15C4×C22⋊Dic61C2211yescharacteristic
16C4×C22⋊Dic61C2212yes—
17#16839497—24no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C61×#407C2212yes—
20C61×#324—22no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23#14519751—22no—
24#14519751—22no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—21no—
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×C4×Dic61—22no—
33C2×C4×Dic61—21no—
34#14519684—22no—
35#14519684—24no—
36#14519684—22no—
37#14519684—22no—
38#14519684C2×C412yes—
39#14519684D414yes—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C2×M16×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61C2×C412yes—
47C2×C4⋊C4×C61D412yes—
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C24×C61C2×D411yescharacteristic
51C23×D61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—81no—
58C22×C4×C61—81no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D412yes—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61#4112yes—
65C2×C22⋊D61—44no—
66C2×C22⋊D61—41no—
67C2×C22⋊D61—41no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—24no—
71C2×D244—22no—
72C2×D244—81no—
73M16×C61—82no—
74C2×C8×C61—42no—
75C4⋊C4×C61—24no—
76C4⋊C4×C61—24no—
77C22⋊Dic61—44no—
78C22⋊Dic61—42no—
79#11758198—44no—
80#11758198—42no—
81#11758198—82no—
82C4⋊Dic61—22no—
83C4⋊Dic61—82no—
84C4×Dic61—82no—
85C4×Dic61—81no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—41no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61#12011yescharacteristic
91C23×C61#12311yescharacteristic
92C22×D61—42no—
93C22×D61—42no—
94C22×D61—41no—
95C22×D61—42no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—81no—
109C2×C4×C61C4×D412yes—
110C2×C4×C61#12112yes—
111C22⋊D61—82no—
112C22⋊D61—82no—
113C2×Dic61—44no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—44no—
117C2×Dic61—42no—
118C2×Dic61—41no—
119C2×Dic61—42no—
120C2×Dic61—44no—
121D244—82no—
122C8×C61—82no—
123#9414—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—44no—
132C22×C61#38611yescharacteristic
133C22×C61#42112yes—
134C22×C61#45412yes—
135C2×D61—84no—
136C2×D61—81no—
137C2×D61—81no—
138C2×D61—81no—
139C4×C61—22no—
140C4×C61—22no—
141C4×C61—42no—
142C4×C61—42no—
143Dic61—84no—
144Dic61—82no—
145Dic61—81no—
146#2591—611no—
147#2313—611no—
148#1507—612no—
149#1463D6111yescharacteristic, complemented
150#1297—612no—
151C2×C61—22no—
152C2×C61—22no—
153C2×C61—41no—
154C2×C61#188811yescharacteristic
155C2×C61#189711yescharacteristic
156C2×C61#191311yescharacteristic
157D61—161no—
158C2×#121—611no—
159C22×C4⋊C4C2×D6111yescharacteristic
160#407C2×D6112yes—
161#400—612no—
162#394—612no—
163#388—1222no—
164#387—1224no—
165#386—1221no—
166#386—1221no—
167#381—1221no—
168#380—1221no—
169C4×C22⋊C4—612no—
170C4×C22⋊C4—611no—
171#327—1224no—
172#324—22no—
173C61#941411yescharacteristic, complemented, Sylow
174C22×D4—611no—
175C22×D4—1221no—
176C23×C4D24411yescharacteristic
177C23×C4D24411yescharacteristic
178C23×C4C22×D6111yescharacteristic
179C2×M16—22no—
180#121—1222no—
181#121—1222no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—1222no—
186C2×C4⋊C4C4×D6112yes—
187C2×C4⋊C4C22⋊D6112yes—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1221no—
191C2×C22⋊C4—1222no—
192C2×C22⋊C4—1221no—
193C2×C42—1222no—
194C2×C42—1221no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—612no—
198#95—614no—
199#95—1222no—
200#95—1224no—
201#95—1222no—
202#95—1222no—
203C24—1221no—
204C24C2×D24411yescharacteristic
205C2×D4—1222no—
206C2×D4—4881no—
207C2×D4—2444no—
208C2×D4—2441no—
209C2×D4—2441no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—1222no—
218C22×C4—1221no—
219C22×C4—1224no—
220C22×C4#1175819812yes—
221C22×C4#1175819812yes—
222C22×C4C2×C4×D6111yescharacteristic
223C22×C4D4×D6112yes—
224C22×C4#1175822412yes—
225C22×C4C2×C22⋊D6111yescharacteristic
226M16—82no—
227C2×C8—42no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—1222no—
231C4⋊C4—4882no—
232C22⋊C4—4882no—
233C22⋊C4—2444no—
234C22⋊C4—2442no—
235C22⋊C4—2444no—
236C22⋊C4—2442no—
237C42—4882no—
238C42—4881no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—2442no—
243C23—2442no—
244C23—2442no—
245C23—2441no—
246C23C2×#1175819811yescharacteristic
247C23#1451974711yescharacteristic
248C23#1451976111yescharacteristic
249D4—4882no—
250D4—4882no—
251D4—4882no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—2442no—
266C2×C4—2444no—
267C2×C4—2442no—
268C2×C4—2444no—
269C2×C4—2442no—
270C2×C4—2444no—
271C2×C4—2442no—
272C2×C4—2441no—
273C2×C4#1451973712yes—
274C2×C4#1451974912yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—4884no—
285C22—4881no—
286C22—4881no—
287C22—4881no—
288C22#1683963911yescharacteristic
289C22#1683977812yes—
290C22#1683993712yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—4884no—
296C4—4882no—
297C4—4881no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—9761no—
302C2#2144622011yescharacteristic
303C2#2144626511yescharacteristic
304C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1463C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1463D61
hypocenterC61#9414
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #21448422C2semidirect
4. #21443704C2semidirect
5. C61×#1463C2semidirect
7. #16839701C4semidirect
8. #16839679C4semidirect
149. #1463D61semidirect
173. C61#9414semidirect

Maximal subgroups

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

subgroupindex
#214512992
#214484222
#214443362
#214437042
C61×#14632
#941461