Finite Group Databasebeta

← #29589425

Subgroups — #29589425

Order 15616.

418 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 77 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#21448440C211yesmaximal, characteristic
2#21444152C212yesmaximal
3#21443765C211yesmaximal, characteristic, complemented
4#21443764C212yesmaximal, complemented
5C61×#1325C211yesmaximal, characteristic, complemented
6C2×#14519735C2211yescharacteristic
7C2×#14519734C411yescharacteristic
8C2×C4⋊C4×D61C411yescharacteristic
9#16839816—22no—
10#16839657—22no—
11#16839633—22no—
12#16839631C2211yescharacteristic
13#16839628C411yescharacteristic
14#16839627C411yescharacteristic
15#16839488C2212yes—
16#16839485—22no—
17C61×#384C2211yescharacteristic
18C61×#324C2212yes—
19C22×C4×D61—21no—
20C22×C4×D61C2×C411yescharacteristic
21#14519735—42no—
22#14519734—22no—
23#14519734—22no—
24C2×C4⋊Dic61C2×C411yescharacteristic
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×#11758196—21no—
27C2×#11758196—21no—
28C2×#11758196—22no—
29C4⋊C4×D61—42no—
30C2×#11758198—21no—
31C2×#11758198D412yes—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61D412yes—
34#14519706—42no—
35C22⋊#92418—22no—
36#14519687—42no—
37#14519684—22no—
38#14519684—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C22⋊C8×C61—22no—
44C23×D61C22⋊C411yescharacteristic
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61C22×C411yescharacteristic
48C22×C4×C61C2×D412yes—
49C22×Dic61—21no—
50C22×Dic61—21no—
51C22×Dic61C22⋊C411yescharacteristic
52C22×Dic61C22⋊C412yes—
53C2×C4×D61—22no—
54C2×C4×D61—21no—
55C2×C4×D61—21no—
56C2×C4×D61—22no—
57C2×C4×D61—42no—
58C2×C4×D61—82no—
59C2×C8×C61—42no—
60C4⋊C4×C61—41no—
61C4⋊C4×C61—41no—
62C4⋊C4×C61—82no—
63C42×C61—21no—
64C42×C61M1612yes—
65#11758198—24no—
66#11758198—42no—
67C4⋊Dic61—41no—
68C4⋊Dic61—41no—
69#11758196—24no—
70#11758196—42no—
71#11758196—82no—
72C4×Dic61—21no—
73C4×Dic61—22no—
74C4×Dic61—22no—
75C4×Dic61M1612yes—
76C2×#92418—42no—
77C23×C61C2×C22⋊C411yescharacteristic
78C22×D61—21no—
79C22×D61—21no—
80C22×D61—41no—
81C22×D61—41no—
82C22×D61#10112yes—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—22no—
86C2×C4×C61—22no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—41no—
92C2×C4×C61C2×M1611yescharacteristic
93C2×C4×C61C2×M1611yescharacteristic
94C2×Dic61—21no—
95C2×Dic61—21no—
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—41no—
99C2×Dic61—41no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—41no—
103C2×Dic61#9912yes—
104C4×D61—24no—
105C4×D61—24no—
106C4×D61—84no—
107C8×C61—82no—
108#92418—82no—
109#11516—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61#40711yescharacteristic
113C22×C61C2×#9911yescharacteristic
114C22×C61C2×#10111yescharacteristic
115C2×D61—22no—
116C2×D61—42no—
117C2×D61—41no—
118C2×D61—41no—
119C2×D61—42no—
120C4×C61—21no—
121C4×C61—21no—
122C4×C61—42no—
123C4×C61—42no—
124C4×C61—81no—
125Dic61—22no—
126Dic61—42no—
127Dic61—42no—
128Dic61—81no—
129#2317—611no—
130#1464—612no—
131#1325—611no—
132#1325—612no—
133#1325D6111yescharacteristic, complemented
134C2×C61—21no—
135C2×C61—21no—
136C2×C61#146311yescharacteristic
137C2×C61#212811yescharacteristic
138C2×C61#213511yescharacteristic
139D61—42no—
140C2×#117—611no—
141C2×#117—611no—
142C22×C4⋊C4—611no—
143#432—1222no—
144#389—1222no—
145#385—1222no—
146#384—611no—
147#384C2×D6111yescharacteristic
148#382—611no—
149#382—611no—
150#324—612no—
151#324C2×D6112yes—
152#323—1222no—
153C61#1151611yescharacteristic, complemented, Sylow
154C23×C4—611no—
155C23×C4—1221no—
156#117—1222no—
157#117—1222no—
158#117—2442no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—611no—
161C2×C4⋊C4—611no—
162C2×C4⋊C4—1221no—
163C2×C4⋊C4—1222no—
164C2×C4⋊C4—1221no—
165C2×C4⋊C4—2442no—
166C2×C4⋊C4C4×D6111yescharacteristic
167C2×C4⋊C4C22×D6111yescharacteristic
168C2×C22⋊C4—612no—
169C2×C22⋊C4—1221no—
170C2×C42—611no—
171C2×C42—612no—
172C2×C42C4×D6111yescharacteristic
173C4⋊C8—2442no—
174C22⋊C8—22no—
175C22⋊C8—1222no—
176#97—2442no—
177#95—1222no—
178#95—1222no—
179C24—611no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—611no—
183C22×C4—612no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4—1221no—
187C22×C4—1221no—
188C22×C4—1221no—
189C22×C4—1221no—
190C22×C4—4882no—
191C22×C4—2442no—
192C22×C4C2×C4×D6111yescharacteristic
193C22×C4D4×D6112yes—
194C2×C8—42no—
195C2×C8—2442no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—1224no—
200C4⋊C4—4882no—
201C4⋊C4—2441no—
202C4⋊C4—2441no—
203C4⋊C4—2442no—
204C22⋊C4—1224no—
205C22⋊C4—2442no—
206C42—21no—
207C42—612no—
208C42—1222no—
209C42—1222no—
210C42—1221no—
211C42#1175818912yes—
212C23—612no—
213C23—1221no—
214C23—1221no—
215C23—2441no—
216C23—2441no—
217C23C22⋊C4×D6111yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—612no—
228C2×C4—1224no—
229C2×C4—1224no—
230C2×C4—1221no—
231C2×C4—1221no—
232C2×C4—1222no—
233C2×C4—1222no—
234C2×C4—4884no—
235C2×C4—2442no—
236C2×C4—2441no—
237C2×C4—2441no—
238C2×C4—2441no—
239C2×C4—2442no—
240C2×C4M16×D6111yescharacteristic
241C2×C4C2×#1175818911yescharacteristic
242C8—82no—
243C8—4882no—
244C22—21no—
245C22—21no—
246C22—1222no—
247C22—2442no—
248C22—2442no—
249C22—2441no—
250C22—2441no—
251C22#1683972411yescharacteristic
252C22#99×D6111yescharacteristic
253C22#101×D6111yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—1222no—
260C4—4881no—
261C4—2442no—
262C4—2442no—
263C2—21no—
264C2—21no—
265C2—2442no—
266C2#2144414711yescharacteristic
267C2#2144748911yescharacteristic
268C2#2144751211yescharacteristic
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#11516
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
133. #1325D61semidirect
153. C61#11516semidirect

Maximal subgroups

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

subgroupindex
#214484402
#214441522
#214437652
#214437642
C61×#13252
#1151661