Finite Group Databasebeta

← #29577463

Subgroups — #29577463

Order 15616.

644 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 111 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#21444334C212yesmaximal
3#21444149C211yesmaximal, characteristic, complemented
4#21443705C212yesmaximal, complemented
5C61×#2274C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7#16839726C2212yes—
8#16839701C412yescomplemented
9#16839679C412yescomplemented
10#16839652—22no—
11#16839641—21no—
12#16839641—21no—
13C4×C22⋊Dic61C2211yescharacteristic
14#16839495—24no—
15#16839488—22no—
16C2×#117×C61—21no—
17C22×C4⋊C4×C61C2211yescharacteristic
18C4×C4⋊C4×C61—24no—
19C4×C22⋊C4×C61—21no—
20C4×C22⋊C4×C61C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C2×#11758194—22no—
24#14519751—22no—
25#14519751—22no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C22⋊Dic61—21no—
28C2×#11758198—22no—
29C2×#11758198—21no—
30C2×#11758198—22no—
31C2×C4×Dic61—21no—
32C22⋊#92418—22no—
33C22⋊#92418—22no—
34#14519684—22no—
35#14519684—22no—
36#14519684C2×C412yes—
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61C2311yescharacteristic
40#117×C61—28no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C4⋊C4×C61D412yes—
46C2×C22⋊C4×C61—22no—
47C2×C42×C61—22no—
48C2×C42×C61—22no—
49C2×C42×C61—22no—
50#95×C61—22no—
51#95×C61D414yes—
52C24×C61C2×D411yescharacteristic
53C23×D61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—24no—
57C22×C4×C61—21no—
58C22×C4×C61—24no—
59C22×C4×C61—21no—
60C22×C4×C61—81no—
61C22×C4×C61—81no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D412yes—
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61#4112yes—
68C2×C22⋊D61—44no—
69C2×C22⋊D61—41no—
70C2×C22⋊D61—41no—
71C22×Dic61—21no—
72C22×Dic61—22no—
73C2×D244—22no—
74C2×D244—81no—
75C4⋊C4×C61—28no—
76C4⋊C4×C61—24no—
77C4⋊C4×C61—24no—
78C22⋊C4×C61—28no—
79C42×C61—28no—
80C42×C61—82no—
81C42×C61—82no—
82C22⋊Dic61—42no—
83#11758198—44no—
84#11758198—42no—
85#11758198—82no—
86C4⋊Dic61—22no—
87C4×Dic61—81no—
88#11758194—82no—
89C2×#92418—42no—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61—41no—
93C23×C61C2×C22⋊C411yescharacteristic
94C23×C61#12011yescharacteristic
95C23×C61#12311yescharacteristic
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—41no—
99C22×D61—42no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—24no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—28no—
106C2×C4×C61—22no—
107C2×C4×C61—42no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—41no—
111C2×C4×C61—44no—
112C2×C4×C61—42no—
113C2×C4×C61—44no—
114C2×C4×C61—42no—
115C2×C4×C61—81no—
116C2×C4×C61#10414yes—
117C2×C4×C61C4×D412yes—
118C2×C4×C61#12112yes—
119C22⋊D61—82no—
120C22⋊D61—82no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—41no—
125C2×Dic61—42no—
126D244—82no—
127#92418—82no—
128#9410—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—44no—
137C22×C61#38611yescharacteristic
138C22×C61C2×#10412yes—
139C22×C61#45412yes—
140C2×D61—84no—
141C2×D61—81no—
142C2×D61—81no—
143C2×D61—81no—
144C4×C61—24no—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—48no—
148C4×C61—42no—
149Dic61—82no—
150Dic61—81no—
151#2591—611no—
152#2274D6111yescharacteristic, complemented
153#1507—612no—
154#1463—611no—
155#1297—612no—
156C2×C61—22no—
157C2×C61—22no—
158C2×C61—41no—
159C2×C61#188811yescharacteristic
160C2×C61#189611yescharacteristic
161C2×C61#191311yescharacteristic
162D61—161no—
163C2×#121—611no—
164C2×#117—21no—
165C22×C4⋊C4C2×D6111yescharacteristic
166#407—612no—
167#400—612no—
168#394—612no—
169#388—1222no—
170#386—1221no—
171#386—1221no—
172C4×C4⋊C4—24no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—611no—
175C4×C22⋊C4C2×D6112yes—
176#327—1224no—
177#324—1222no—
178C61#941011yescharacteristic, complemented, Sylow
179C22×D4—611no—
180C22×D4—1221no—
181C23×C4C22⋊D6111yescharacteristic
182C23×C4C22⋊D6111yescharacteristic
183C23×C4C22×D6111yescharacteristic
184C2×M16—1222no—
185#121—1222no—
186#121—1222no—
187#117—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4C4×D6112yes—
193C2×C4⋊C4D24412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1222no—
196C2×C22⋊C4—1221no—
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1221no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—1221no—
203C22⋊C8—1222no—
204C22⋊C8—1222no—
205#95—22no—
206#95—612no—
207#95—1222no—
208#95—1222no—
209#95C22⋊D6114yes—
210C24—1221no—
211C24C2×C22⋊D6111yescharacteristic
212C2×D4—1222no—
213C2×D4—4881no—
214C2×D4—2444no—
215C2×D4—2441no—
216C2×D4—2441no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—1222no—
226C22×C4—1221no—
227C22×C4#1175819812yes—
228C22×C4#1175819812yes—
229C22×C4C2×C4×D6111yescharacteristic
230C22×C4C2×D24411yescharacteristic
231C22×C4#1175822212yes—
232C22×C4C2×C22⋊D6112yes—
233M16—4882no—
234C2×C8—2442no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—1222no—
239C22⋊C4—28no—
240C22⋊C4—4882no—
241C22⋊C4—2442no—
242C22⋊C4—2444no—
243C22⋊C4—2442no—
244C42—28no—
245C42—82no—
246C42—82no—
247C42—4881no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—2442no—
252C23—2442no—
253C23—2442no—
254C23—2441no—
255C23C2×#1175819811yescharacteristic
256C23#1451974611yescharacteristic
257C23#1451976011yescharacteristic
258D4—4882no—
259D4—4882no—
260D4—4882no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2441no—
282C2×C4#1451970514yes—
283C2×C4C4×C22⋊D6112yes—
284C2×C4#1451975112yes—
285C8—4882no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—4884no—
295C22—4881no—
296C22—4881no—
297C22—4881no—
298C22#1683964111yescharacteristic
299C22C2×#1451970512yes—
300C22#1683993512yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—48no—
305C4—42no—
306C4—4882no—
307C4—4881no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—9761no—
312C2#2144622211yescharacteristic
313C2#2144626211yescharacteristic
314C2#2144636111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2274C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#2274D61
hypocenterC61#9410
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #21444149C2semidirect
4. #21443705C2semidirect
5. C61×#2274C2semidirect
8. #16839701C4semidirect
9. #16839679C4semidirect
152. #2274D61semidirect
178. C61#9410semidirect

Maximal subgroups

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

subgroupindex
#214512992
#214443342
#214441492
#214437052
C61×#22742
#941061