Finite Group Databasebeta

← #29679756

Subgroups — #29679756

Order 15616.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 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
1C2×#16840236C211yesmaximal, characteristic
2#21448396C212yesmaximal
3#21445896C211yesmaximal, characteristic, complemented
4#21445894C212yesmaximal, complemented
5#21445892C214yesmaximal, complemented
6#342×D61C214yesmaximal, complemented
7C61×#1820C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9#16840236—28no—
10C2×C4×D244C414yescomplemented
11C2×#14519739C414yescomplemented
12C2×#14519736C2211yescharacteristic
13C2×C22⋊C4×D61C2214yes—
14C2×C22⋊C4×D61C2212yes—
15C2×C22⋊C4×D61C2212yes—
16#16839640—28no—
17#16839625C2212yes—
18#16839617C2214yes—
19#16839529C2214yes—
20#16839528C2218yes—
21C2×#117×C61C2211yescharacteristic
22C61×#381C2212yes—
23C61×#342C2214yes—
24C24×D61C2311yescharacteristic
25C24×D61C2311yescharacteristic
26C22×D244—28no—
27C22×D244C2×C412yes—
28C22×D244C2×C412yes—
29C22×D244D418yes—
30C22×C4×D61—24no—
31C22×C4×D61C2×C414yes—
32C22×C4×D61C2×C414yescomplemented
33C4×D244—216no—
34#14519739—216no—
35#14519736—24no—
36C2×C4⋊Dic61C2×C412yescomplemented
37C2×C22⋊Dic61C2314yes—
38C2×C22⋊Dic61C2312yes—
39C2×#11758198—24no—
40C2×#11758198—28no—
41C2×#11758198C2×C414yes—
42C2×#11758198C2×C414yescomplemented
43C22⋊C4×D61—28no—
44C22⋊C4×D61—28no—
45C22⋊C4×D61—216no—
46C2×C4×Dic61C2×C412yescomplemented
47#14519684—24no—
48C23×C4×C61C2311yescharacteristic
49#117×C61—24no—
50C2×C4⋊C4×C61C2×C412yescomplemented
51C2×C22⋊C4×C61C2314yes—
52C2×C22⋊C4×C61C2312yes—
53C2×C42×C61C2×C412yescomplemented
54#95×C61—24no—
55C24×C61C2411yescharacteristic
56C23×D61—24no—
57C23×D61—82no—
58C23×D61—82no—
59C23×D61C22×C412yes—
60C23×D61C22×C412yes—
61C23×D61C2×D414yes—
62C23×D61C2×D414yes—
63C23×D61#4118yes—
64C22×C4×C61—24no—
65C22×C4×C61—24no—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C22×C414yes—
69C22×C4×C61C2×D414yes—
70C22×Dic61—24no—
71C22×Dic61C22×C414yes—
72C2×D244—28no—
73C2×D244—28no—
74C2×D244—216no—
75C2×D244—216no—
76C2×D244C22⋊C418yes—
77C2×D244C22⋊C418yes—
78C2×C4×D61—28no—
79C2×C4×D61—28no—
80C2×C4×D61—48no—
81C2×C4×D61—88no—
82C2×C4×D61—84no—
83C2×C4×D61—84no—
84C4⋊C4×C61—24no—
85C22⋊C4×C61—24no—
86C22⋊C4×C61—28no—
87C42×C61—24no—
88C22⋊Dic61—24no—
89C22⋊Dic61—28no—
90#11758198—28no—
91#11758198—28no—
92#11758198—48no—
93#11758198—88no—
94C4⋊Dic61—24no—
95C4×Dic61—24no—
96C23×C61—42no—
97C23×C61C23×C411yescharacteristic
98C23×C61C22×D412yes—
99C23×C61C2×#4114yes—
100C22×D61—24no—
101C22×D61—28no—
102C22×D61—44no—
103C22×D61—44no—
104C22×D61—48no—
105C22×D61—48no—
106C22×D61—88no—
107C22×D61—88no—
108C22×D61C2×C22⋊C414yes—
109C22×D61C2×C22⋊C414yes—
110C22×D61#11718yes—
111C22×D61#12318yes—
112C22×D61#12318yes—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—28no—
116C2×C4×C61—24no—
117C2×C4×C61—44no—
118C2×C4×C61—48no—
119C2×C4×C61—44no—
120C2×C4×C61C2×C22⋊C412yes—
121C2×C4×C61C2×C22⋊C412yes—
122C2×Dic61—24no—
123C2×Dic61—44no—
124C2×Dic61—48no—
125C2×Dic61—44no—
126D244—216no—
127D244—216no—
128C4×D61—88no—
129C4×D61—88no—
130C4×D61—816no—
131#20689—611nomaximal, Sylow
132C22×C61—22no—
133C22×C61—24no—
134C22×C61—44no—
135C22×C61C22×C22⋊C411yescharacteristic
136C22×C61C2×#11712yes—
137C22×C61C2×#12314yes—
138C2×D61—28no—
139C2×D61—28no—
140C2×D61—24no—
141C2×D61—48no—
142C2×D61—48no—
143C2×D61—48no—
144C2×D61—48no—
145C2×D61—88no—
146C2×D61#38118yes—
147C4×C61—22no—
148C4×C61—22no—
149C4×C61—44no—
150C4×C61—84no—
151Dic61—44no—
152Dic61—84no—
153C2×#518—611no—
154#2308—612no—
155#1820—612no—
156#1820—611no—
157#1820D6111yescharacteristic, complemented
158#1819—614no—
159C2×#342—614no—
160C2×C61—24no—
161C2×C61—22no—
162C2×C61C2×#38111yescharacteristic
163C2×C61#288312yes—
164D61—28no—
165D61—44no—
166C23×D4—611no—
167#518—1228no—
168C2×C4×D4—614no—
169C2×C4×D4—614no—
170C2×#117—611no—
171C2×#117C2×D6111yescharacteristic
172C22×C22⋊C4—612no—
173C22×C22⋊C4—612no—
174C22×C22⋊C4—614no—
175#386—1228no—
176#381—612no—
177#381C2×D6112yes—
178#379—614no—
179#342—618no—
180#342—614no—
181#342C2×D6114yes—
182C61#2068911yescharacteristic, complemented, Sylow
183C25—611no—
184C25—611no—
185C22×D4—612no—
186C22×D4—612no—
187C22×D4—618no—
188C22×D4—1228no—
189C23×C4—614no—
190C23×C4—614no—
191C23×C4—1224no—
192C23×C4C22×D6111yescharacteristic
193C4×D4—12216no—
194C4×D4—12216no—
195#117—24no—
196#117—1224no—
197C2×C4⋊C4—612no—
198C2×C4⋊C4C4×D6112yescomplemented
199C2×C22⋊C4—612no—
200C2×C22⋊C4—614no—
201C2×C22⋊C4—614no—
202C2×C22⋊C4—614no—
203C2×C22⋊C4—1228no—
204C2×C22⋊C4—1228no—
205C2×C22⋊C4—12216no—
206C2×C22⋊C4—1228no—
207C2×C22⋊C4—1224no—
208C2×C22⋊C4C22×D6114yes—
209C2×C22⋊C4C22×D6112yes—
210C2×C42—612no—
211C2×C42C4×D6112yescomplemented
212#95—24no—
213#95—1224no—
214C24—618no—
215C24—612no—
216C24—614no—
217C24—612no—
218C24—614no—
219C24—1224no—
220C24—4882no—
221C24—4882no—
222C24C23×D6111yescharacteristic
223C2×D4—618no—
224C2×D4—618no—
225C2×D4—1228no—
226C2×D4—1228no—
227C2×D4—12216no—
228C2×D4—12216no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—614no—
232C22×C4—1228no—
233C22×C4—1228no—
234C22×C4—1224no—
235C22×C4—4888no—
236C22×C4—4884no—
237C22×C4—4884no—
238C22×C4—2448no—
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×C4×D6111yescharacteristic
241C22×C4C2×C4×D6114yes—
242C22×C4D4×D6114yes—
243C4⋊C4—24no—
244C4⋊C4—1224no—
245C22⋊C4—24no—
246C22⋊C4—28no—
247C22⋊C4—1228no—
248C22⋊C4—1228no—
249C22⋊C4—1224no—
250C22⋊C4—1228no—
251C22⋊C4—4888no—
252C22⋊C4—2448no—
253C42—24no—
254C42—1224no—
255C23—42no—
256C23—614no—
257C23—618no—
258C23—618no—
259C23—618no—
260C23—614no—
261C23—1228no—
262C23—1224no—
263C23—4888no—
264C23—4888no—
265C23—2448no—
266C23—2444no—
267C23—2444no—
268C23—2448no—
269C23C22×C4×D6111yescharacteristic
270C23C2×D4×D6112yes—
271C23#41×D6114yes—
272D4—12216no—
273D4—12216no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—1224no—
282C2×C4—4888no—
283C2×C4—4888no—
284C2×C4—48816no—
285C2×C4—2448no—
286C2×C4—2444no—
287C2×C4—2444no—
288C2×C4C22⋊C4×D6112yes—
289C2×C4C22⋊C4×D6112yes—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—618no—
294C22—1228no—
295C22—1224no—
296C22—1228no—
297C22—4888no—
298C22—2448no—
299C22—2448no—
300C22—2448no—
301C22—2448no—
302C22C2×C22⋊C4×D6111yescharacteristic
303C22#117×D6112yes—
304C22#123×D6114yes—
305C4—22no—
306C4—22no—
307C4—44no—
308C4—84no—
309C4—4884no—
310C4—2444no—
311C2—24no—
312C2—22no—
313C2—1228no—
314C2—2444no—
315C2#381×D6111yescharacteristic
316C2#2145427812yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1820C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D244C22
cosocleC24×C61C24
hypercenter#1820D61
hypocenterC61#20689
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
3. #21445896C2semidirect
4. #21445894C2semidirect
5. #21445892C2semidirect
6. #342×D61C2semidirect
7. C61×#1820C2semidirect
10. C2×C4×D244C4semidirect
11. C2×#14519739C4semidirect
32. C22×C4×D61C2×C4semidirect
36. C2×C4⋊Dic61C2×C4semidirect
42. C2×#11758198C2×C4semidirect
46. C2×C4×Dic61C2×C4semidirect
50. C2×C4⋊C4×C61C2×C4semidirect
53. C2×C42×C61C2×C4semidirect
157. #1820D61semidirect
182. C61#20689semidirect
198. C2×C4⋊C4C4×D61semidirect
211. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402362
#214483962
#214458962
#214458942
#214458922
#342×D612
C61×#18202
#2068961