Finite Group Databasebeta

← #29573517

Subgroups — #29573517

Order 15616.

786 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 87 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#21448400C211yesmaximal, characteristic
2#21445733C212yesmaximal
3#21443723C212yesmaximal, complemented
4#21443723C211yesmaximal, characteristic, complemented
5C61×#1304C211yesmaximal, characteristic, complemented
6C23×C22⋊D61C2211yescharacteristic
7C2×#14519733C412yescomplemented
8C2×#14519691—24no—
9#16839662—24no—
10#16839618C2212yes—
11#16839618C2211yescharacteristic
12#16839617C412yescomplemented
13C61×#379C2212yes—
14C61×#379C2211yescharacteristic
15C24×D61C2×C411yescharacteristic
16C22×C22⋊D61—22no—
17C22×C22⋊D61—21no—
18C22×C22⋊D61—21no—
19C22×C22⋊D61—44no—
20C22×C22⋊D61D414yes—
21C23×Dic61C2×C411yescharacteristic
22#14519733—44no—
23C2×#11758196—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61C2×C412yes—
29C2×#11758198—24no—
30C2×C4×Dic61—22no—
31#14519691—48no—
32C25×C61C2311yescharacteristic
33C2×C22⋊C4×C61—22no—
34C2×C22⋊C4×C61—22no—
35C2×C22⋊C4×C61—22no—
36C2×C22⋊C4×C61—22no—
37C2×C22⋊C4×C61C2×C412yes—
38C24×C61—42no—
39C24×C61—82no—
40C24×C61C22×C411yescharacteristic
41C24×C61C2×D412yes—
42C24×C61#4112yes—
43C24×C61#4111yescharacteristic
44C24×C61#4111yescharacteristic
45C23×D61—21no—
46C23×D61—21no—
47C23×D61—42no—
48C23×D61—82no—
49C23×D61C22⋊C412yes—
50C22×C4×C61—22no—
51C22×C4×C61—42no—
52C2×C22⋊D61—24no—
53C2×C22⋊D61—24no—
54C2×C22⋊D61—28no—
55C2×C22⋊D61—44no—
56C2×C22⋊D61—44no—
57C2×C22⋊D61—48no—
58C2×C22⋊D61—44no—
59C22×Dic61—21no—
60C22×Dic61—21no—
61C22×Dic61—24no—
62C22×Dic61—22no—
63C22×Dic61—42no—
64C22×Dic61C22⋊C412yes—
65C22⋊C4×C61—42no—
66C22⋊C4×C61—44no—
67C22⋊C4×C61—82no—
68C22⋊C4×C61—82no—
69C22⋊C4×C61—82no—
70C22⋊Dic61—42no—
71C22⋊Dic61—44no—
72C22⋊Dic61—82no—
73C22⋊Dic61—82no—
74C22⋊Dic61—82no—
75#11758198—84no—
76#11758196—44no—
77C4×Dic61—44no—
78C23×C61—22no—
79C23×C61—24no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—42no—
83C23×C61—42no—
84C23×C61—42no—
85C23×C61—42no—
86C23×C61—42no—
87C23×C61—41no—
88C23×C61—42no—
89C23×C61—41no—
90C23×C61—88no—
91C23×C61C2×C22⋊C411yescharacteristic
92C23×C61#11711yescharacteristic
93C23×C61#11711yescharacteristic
94C23×C61#12312yes—
95C23×C61#12312yes—
96C22×D61—22no—
97C22×D61—24no—
98C22×D61—42no—
99C22×D61—42no—
100C22×D61—44no—
101C22×D61—82no—
102C22×D61—82no—
103C22×D61—88no—
104C2×C4×C61—44no—
105C2×C4×C61—42no—
106C2×C4×C61—82no—
107C2×C4×C61—82no—
108C2×C4×C61—82no—
109C22⋊D61—48no—
110C22⋊D61—48no—
111C2×Dic61—24no—
112C2×Dic61—24no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—44no—
116C2×Dic61—42no—
117C2×Dic61—82no—
118C2×Dic61—82no—
119C2×Dic61—82no—
120C2×Dic61#9914yes—
121#8748—611nomaximal, Sylow
122C22×C61—22no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—22no—
128C22×C61—22no—
129C22×C61—22no—
130C22×C61—22no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—44no—
134C22×C61—84no—
135C22×C61—82no—
136C22×C61—84no—
137C22×C61—82no—
138C22×C61#38111yescharacteristic
139C22×C61C2×#9912yes—
140C2×D61—42no—
141C2×D61—44no—
142C2×D61—84no—
143C2×D61—84no—
144C2×D61—84no—
145C4×C61—82no—
146C4×C61—162no—
147Dic61—24no—
148Dic61—82no—
149Dic61—162no—
150#2309—611no—
151#1781—612no—
152#1304—612no—
153#1304—611no—
154#1304D6111yescharacteristic, complemented
155C2×C61—22no—
156C2×C61—42no—
157C2×C61—42no—
158C2×C61—41no—
159C2×C61—41no—
160C2×C61#181911yescharacteristic
161C2×C61#182812yes—
162D61—82no—
163C23×D4—611no—
164C2×#117—612no—
165C2×#99—1224no—
166#390—1224no—
167#379—611no—
168#379—612no—
169#379—612no—
170#379C2×D6112yes—
171#379C2×D6111yescharacteristic
172C61#874811yescharacteristic, complemented, Sylow
173C25—611no—
174C25C22×D6111yescharacteristic
175C22×D4—614no—
176C22×D4—1222no—
177C22×D4—1221no—
178C22×D4—1221no—
179C22×D4—2444no—
180C23×C4—611no—
181#117—2444no—
182C2×C4⋊C4—1222no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—612no—
188C2×C22⋊C4—1224no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1222no—
191C2×C22⋊C4—1222no—
192C2×C22⋊C4—1222no—
193C2×C22⋊C4C4×D6112yes—
194C2×C42—1222no—
195#99—2448no—
196C24—42no—
197C24—82no—
198C24—612no—
199C24—1221no—
200C24—1221no—
201C24—4882no—
202C24—2442no—
203C24C2×C4×D6111yescharacteristic
204C24#1175822212yes—
205C24D4×D6112yes—
206C24#1175822411yescharacteristic
207C24#1175822611yescharacteristic
208C2×D4—1228no—
209C2×D4—1224no—
210C2×D4—1224no—
211C2×D4—2444no—
212C2×D4—2448no—
213C2×D4—2444no—
214C2×D4—2444no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—612no—
218C22×C4—1224no—
219C22×C4—1221no—
220C22×C4—1221no—
221C22×C4—1222no—
222C22×C4—2442no—
223C4⋊C4—2444no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—4884no—
230C22⋊C4—4882no—
231C22⋊C4—4882no—
232C22⋊C4—4882no—
233C22⋊C4—2444no—
234C22⋊C4—2442no—
235C42—2444no—
236C23—22no—
237C23—24no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—88no—
249C23—1224no—
250C23—1222no—
251C23—4882no—
252C23—4882no—
253C23—4888no—
254C23—2444no—
255C23—2442no—
256C23—2442no—
257C23C22⋊C4×D6111yescharacteristic
258C23#1451973411yescharacteristic
259C23#1451973511yescharacteristic
260C23#1451975912yes—
261C23#1451976212yes—
262D4—2448no—
263D4—2448no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—614no—
270C2×C4—1224no—
271C2×C4—1224no—
272C2×C4—1222no—
273C2×C4—1222no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—4882no—
277C2×C4—2444no—
278C2×C4—2442no—
279C22—22no—
280C22—22no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—44no—
291C22—84no—
292C22—82no—
293C22—84no—
294C22—82no—
295C22—4884no—
296C22—4884no—
297C22—4884no—
298C22—2442no—
299C22—2444no—
300C22#1683962411yescharacteristic
301C22#99×D6112yes—
302C4—82no—
303C4—162no—
304C4—1224no—
305C4—9762no—
306C4—4882no—
307C2—22no—
308C2—42no—
309C2—42no—
310C2—41no—
311C2—41no—
312C2—4882no—
313C2#2144589211yescharacteristic
314C2#2144592612yes—
subgroupG/H
centerC22#16839624
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#1304C2
socleC22×C61#381
prime radicalC23×C22⋊D61C22
cosocleC25×C61C23
hypercenter#1304D61
hypocenterC61#8748
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1304

Complements

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

NH ≅ G/Ntype
3. #21443723C2semidirect
4. #21443723C2semidirect
5. C61×#1304C2semidirect
7. C2×#14519733C4semidirect
12. #16839617C4semidirect
154. #1304D61semidirect
172. C61#8748semidirect

Maximal subgroups

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

subgroupindex
#214484002
#214457332
#214437232
C61×#13042
#874861