Finite Group Databasebeta

← #29589411

Subgroups — #29589411

Order 15616.

578 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 81 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#384×D61C211yesmaximal, characteristic
2#21444147C212yesmaximal
3#21443765C211yesmaximal, characteristic, complemented
4#21443764C212yesmaximal, complemented
5C61×#1325C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D61—22no—
7C2×C4⋊C4×D61C411yescharacteristic
8C2×C4⋊C4×D61C2211yescharacteristic
9C2×C42×D61C411yescharacteristic
10#16839724—22no—
11#16839632C411yescharacteristic
12#16839631—22no—
13#16839631C411yescharacteristic
14#16839631C2211yescharacteristic
15#16839488C2212yes—
16#16839487—22no—
17C61×#384C2211yescharacteristic
18C61×#324C2212yes—
19C22×C4×D61—21no—
20C22×C4×D61—21no—
21C22×C4×D61C2×C411yescharacteristic
22C22×C4×D61D412yes—
23C2×#11758189—42no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×#11758196—22no—
28C2×#11758196—22no—
29C2×#11758196—21no—
30C2×#11758196D412yes—
31C4⋊C4×D61—41no—
32C4⋊C4×D61—41no—
33C4⋊C4×D61—41no—
34C4⋊C4×D61—41no—
35C4⋊C4×D61—48no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C42×D61—22no—
39C42×D61—22no—
40C22⋊#92418—22no—
41#14519689—42no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2311yescharacteristic
45C2×C42×C61C2×C411yescharacteristic
46C22⋊C8×C61—22no—
47C23×D61C22⋊C411yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22×C411yescharacteristic
51C22×C4×C61C2×D412yes—
52C22×Dic61—21no—
53C22×Dic61—21no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C22⋊C412yes—
56C2×C4×D61—24no—
57C2×C4×D61—24no—
58C2×C4×D61—24no—
59C2×C4×D61—21no—
60C2×C4×D61—21no—
61C2×C4×D61—21no—
62C2×C4×D61—21no—
63C2×C4×D61—22no—
64C2×C4×D61—22no—
65C2×C4×D61—22no—
66C2×C4×D61—44no—
67C2×C4×D61—44no—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C8×C61—42no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—44no—
73C4⋊C4×C61—41no—
74C42×C61—21no—
75C42×C61M1612yescomplemented
76C4⋊Dic61—41no—
77C4⋊Dic61—44no—
78C4⋊Dic61—41no—
79#11758196—42no—
80#11758196—44no—
81#11758196—44no—
82#11758196—42no—
83C4×Dic61—24no—
84C4×Dic61—21no—
85C4×Dic61M1612yescomplemented
86C2×#92418—42no—
87#11758189—84no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—21no—
90C22×D61—21no—
91C22×D61—22no—
92C22×D61—22no—
93C22×D61#9911yescharacteristic
94C22×D61#9911yescharacteristic
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61C2×M1611yescharacteristic
105C2×C4×C61C2×M1611yescharacteristic
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—21no—
109C2×Dic61—22no—
110C2×Dic61—21no—
111C2×Dic61—22no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61#10112yes—
116C4×D61—216no—
117C4×D61—22no—
118C4×D61—22no—
119C4×D61—22no—
120C4×D61—22no—
121C4×D61—44no—
122C4×D61—82no—
123C4×D61—82no—
124C4×D61—82no—
125C8×C61—82no—
126#92418—82no—
127#11512—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—21no—
130C22×C61#40711yescharacteristic
131C22×C61C2×#9911yescharacteristic
132C22×C61C2×#10111yescharacteristic
133C2×D61—24no—
134C2×D61—24no—
135C2×D61—21no—
136C2×D61—21no—
137C2×D61—21no—
138C2×D61—21no—
139C2×D61#35312yes—
140C2×D61#35612yes—
141C4×C61—24no—
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—42no—
145C4×C61—81no—
146Dic61—24no—
147Dic61—22no—
148Dic61—42no—
149Dic61—81no—
150C2×#384—611no—
151#1463—612no—
152#1325—611no—
153#1325—612no—
154#1325D6111yescharacteristic, complemented
155C2×C61—21no—
156C2×C61—21no—
157C2×C61#146311yescharacteristic
158C2×C61C2×#35311yescharacteristic
159C2×C61C2×#35611yescharacteristic
160D61—22no—
161D61—22no—
162C22×C4⋊C4—611no—
163C22×C4⋊C4—611no—
164C22×C4⋊C4—1222no—
165C22×C42—611no—
166#407—1222no—
167#384—611no—
168#384—611no—
169#384—611no—
170#384—1222no—
171#384C2×D6111yescharacteristic
172#324—612no—
173#324—1222no—
174#324C2×D6112yes—
175C61#1151211yescharacteristic, complemented, Sylow
176C23×C4—611no—
177C23×C4—612no—
178C23×C4—1221no—
179C23×C4—1221no—
180C2×M16—2442no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4—612no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4—1222no—
187C2×C4⋊C4—1222no—
188C2×C4⋊C4—1222no—
189C2×C4⋊C4—2441no—
190C2×C4⋊C4—2441no—
191C2×C4⋊C4—2441no—
192C2×C4⋊C4—2441no—
193C2×C4⋊C4—2448no—
194C2×C4⋊C4C4×D6111yescharacteristic
195C2×C4⋊C4C22×D6111yescharacteristic
196C2×C42—611no—
197C2×C42—1222no—
198C2×C42—1221no—
199C2×C42—1222no—
200C2×C42C4×D6111yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—1222no—
203C22⋊C8—2442no—
204C24—611no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—611no—
208C22×C4—612no—
209C22×C4—1222no—
210C22×C4—1221no—
211C22×C4—1221no—
212C22×C4—1224no—
213C22×C4—1224no—
214C22×C4—1224no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—1221no—
220C22×C4—1221no—
221C22×C4—2441no—
222C22×C4—2441no—
223C22×C4—2444no—
224C22×C4—2444no—
225C22×C4C2×C4×D6111yescharacteristic
226C22×C4D4×D6112yes—
227M16—4884no—
228C2×C8—42no—
229C2×C8—2442no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C4⋊C4—2442no—
234C4⋊C4—2441no—
235C4⋊C4—2441no—
236C4⋊C4—2444no—
237C4⋊C4—2444no—
238C4⋊C4—2444no—
239C4⋊C4—2442no—
240C42—21no—
241C42—612no—
242C42—1224no—
243C42—1221no—
244C42#1175818912yescomplemented
245C23—611no—
246C23—611no—
247C23—1222no—
248C23—1222no—
249C23—1221no—
250C23—1221no—
251C23C22⋊C4×D6111yescharacteristic
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—612no—
262C2×C4—12216no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1222no—
270C2×C4—1221no—
271C2×C4—1221no—
272C2×C4—1222no—
273C2×C4—4882no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—2444no—
277C2×C4—2442no—
278C2×C4—2441no—
279C2×C4—2442no—
280C2×C4M16×D6111yescharacteristic
281C2×C4C2×#1175818911yescharacteristic
282C8—82no—
283C8—4882no—
284C22—21no—
285C22—21no—
286C22—612no—
287C22—612no—
288C22—1224no—
289C22—1224no—
290C22—1221no—
291C22—1221no—
292C22—1221no—
293C22—1221no—
294C22#1683972411yescharacteristic
295C22#99×D6111yescharacteristic
296C22#101×D6111yescharacteristic
297C4—24no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—1224no—
303C4—1222no—
304C4—4881no—
305C4—2442no—
306C2—21no—
307C2—21no—
308C2—1222no—
309C2—1222no—
310C2#2144414711yescharacteristic
311C2#353×D6111yescharacteristic
312C2#356×D6111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1325C2
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1325D61
hypocenterC61#11512
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #21443765C2semidirect
4. #21443764C2semidirect
5. C61×#1325C2semidirect
75. C42×C61M16semidirect
85. C4×Dic61M16semidirect
154. #1325D61semidirect
175. C61#11512semidirect
244. C42#11758189semidirect

Maximal subgroups

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

subgroupindex
#384×D612
#214441472
#214437652
#214437642
C61×#13252
#1151261