Finite Group Databasebeta

← #29679753

Subgroups — #29679753

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×#16840235C211yesmaximal, characteristic
2#21448395C212yesmaximal
3#21445895C214yesmaximal, complemented
4#21445893C211yesmaximal, characteristic, complemented
5#21445891C212yesmaximal, complemented
6#342×D61C214yesmaximal, complemented
7C61×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D61C2211yescharacteristic
9#16840235—28no—
10C2×C4×C22⋊D61C414yescomplemented
11C2×#14519737C414yescomplemented
12C2×#14519733C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C2×C22⋊C4×D61C2214yes—
15C2×C22⋊C4×D61C2212yes—
16#16839638—28no—
17#16839623C2214yes—
18#16839618C2212yes—
19#16839529C2214yes—
20#16839528C2218yes—
21C22×C22⋊C4×C61C2211yescharacteristic
22C61×#379C2212yes—
23C61×#342C2214yes—
24C24×D61C2311yescharacteristic
25C22×C22⋊D61—28no—
26C22×C22⋊D61C2×C412yes—
27C22×C22⋊D61C2×C412yes—
28C22×C22⋊D61D418yes—
29C23×Dic61C2×C412yescomplemented
30C23×Dic61C2311yescharacteristic
31C22×C4×D61—24no—
32C22×C4×D61C2×C414yes—
33C4×C22⋊D61—28no—
34C4×C22⋊D61—28no—
35#14519737—216no—
36#14519733—28no—
37C2×#11758196C2×C414yescomplemented
38C2×C22⋊Dic61—24no—
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61C2×C412yescomplemented
41C2×C22⋊Dic61C2314yes—
42C2×C22⋊Dic61C2312yes—
43C2×#11758198—24no—
44C2×#11758198C2×C414yes—
45C22⋊C4×D61—28no—
46C22⋊C4×D61—216no—
47C2×C4×Dic61C2×C414yescomplemented
48#14519684—28no—
49C25×C61C2311yescharacteristic
50C23×C4×C61C2×C412yescomplemented
51C2×C22⋊C4×C61—24no—
52C2×C22⋊C4×C61—24no—
53C2×C22⋊C4×C61C2×C412yescomplemented
54C2×C22⋊C4×C61C2314yes—
55C2×C22⋊C4×C61C2312yes—
56C24×C61—24no—
57C24×C61—82no—
58C24×C61C22×C411yescharacteristic
59C24×C61C22×C411yescharacteristic
60C24×C61C2×D414yes—
61C24×C61C2411yescharacteristic
62C23×D61—82no—
63C23×D61C22×C412yes—
64C23×D61C2×D414yes—
65C23×D61#4118yes—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—24no—
69C22×C4×C61—82no—
70C22×C4×C61C22×C414yes—
71C2×C22⋊D61—28no—
72C2×C22⋊D61—28no—
73C2×C22⋊D61—216no—
74C2×C22⋊D61—216no—
75C2×C22⋊D61C22⋊C418yes—
76C2×C22⋊D61C22⋊C418yes—
77C22×Dic61—24no—
78C22×Dic61—24no—
79C22×Dic61—24no—
80C22×Dic61—82no—
81C22×Dic61C22×C412yes—
82C22×Dic61C22×C414yes—
83C22×Dic61C2×D414yes—
84C2×C4×D61—28no—
85C2×C4×D61—48no—
86C2×C4×D61—88no—
87C2×C4×D61—84no—
88C22⋊C4×C61—24no—
89C22⋊C4×C61—24no—
90C22⋊C4×C61—28no—
91C22⋊C4×C61—84no—
92C22⋊Dic61—24no—
93C22⋊Dic61—24no—
94C22⋊Dic61—28no—
95C22⋊Dic61—84no—
96#11758198—28no—
97#11758198—48no—
98#11758196—28no—
99C4×Dic61—28no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—28no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—44no—
107C23×C61—44no—
108C23×C61—88no—
109C23×C61C2×C22⋊C412yes—
110C23×C61C2×C22⋊C412yes—
111C23×C61C23×C411yescharacteristic
112C23×C61C22×D412yes—
113C23×C61C2×#4114yes—
114C22×D61—44no—
115C22×D61—48no—
116C22×D61—88no—
117C22×D61C2×C22⋊C414yes—
118C22×D61#11718yes—
119C22×D61#12318yes—
120C22×D61#12318yes—
121C2×C4×C61—24no—
122C2×C4×C61—44no—
123C2×C4×C61—48no—
124C2×C4×C61—44no—
125C2×C4×C61—82no—
126C2×C4×C61—82no—
127C22⋊D61—216no—
128C22⋊D61—216no—
129C2×Dic61—24no—
130C2×Dic61—28no—
131C2×Dic61—24no—
132C2×Dic61—44no—
133C2×Dic61—48no—
134C2×Dic61—44no—
135C2×Dic61—84no—
136C2×Dic61C2×C22⋊C414yes—
137C4×D61—88no—
138C4×D61—816no—
139#20689—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—24no—
142C22×C61—24no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—44no—
148C22×C61—44no—
149C22×C61—44no—
150C22×C61—88no—
151C22×C61C22×C22⋊C411yescharacteristic
152C22×C61C2×#11712yes—
153C22×C61C2×#12314yes—
154C2×D61—28no—
155C2×D61—48no—
156C2×D61—48no—
157C2×D61—48no—
158C2×D61#38118yes—
159C4×C61—44no—
160C4×C61—84no—
161Dic61—24no—
162Dic61—44no—
163Dic61—84no—
164C2×#518—611no—
165#2308—612no—
166#1820—614no—
167#1819—612no—
168#1819—611no—
169#1819D6111yescharacteristic, complemented
170C2×#342—614no—
171C2×C61—24no—
172C2×C61—22no—
173C2×C61—42no—
174C2×C61—42no—
175C2×C61C2×#38111yescharacteristic
176C2×C61#288312yes—
177D61—28no—
178C23×D4—611no—
179#518—1228no—
180C2×C4×D4—614no—
181C2×C4×D4—614no—
182C2×#117—612no—
183C22×C22⋊C4—611no—
184C22×C22⋊C4—612no—
185C22×C22⋊C4—614no—
186C22×C22⋊C4C2×D6111yescharacteristic
187#386—1228no—
188#381—614no—
189#379—612no—
190#379C2×D6112yes—
191#342—618no—
192#342—614no—
193#342C2×D6114yes—
194C61#2068911yescharacteristic, complemented, Sylow
195C25—611no—
196C25C22×D6111yescharacteristic
197C22×D4—612no—
198C22×D4—612no—
199C22×D4—618no—
200C22×D4—1228no—
201C23×C4—611no—
202C23×C4—612no—
203C23×C4—614no—
204C23×C4—1224no—
205C23×C4C4×D6112yescomplemented
206C4×D4—12216no—
207C4×D4—1228no—
208C4×D4—1228no—
209#117—1228no—
210C2×C4⋊C4—614no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—612no—
214C2×C22⋊C4—612no—
215C2×C22⋊C4—614no—
216C2×C22⋊C4—614no—
217C2×C22⋊C4—1228no—
218C2×C22⋊C4—12216no—
219C2×C22⋊C4—1224no—
220C2×C22⋊C4—1224no—
221C2×C22⋊C4—1224no—
222C2×C22⋊C4C4×D6112yescomplemented
223C2×C22⋊C4C22×D6114yes—
224C2×C22⋊C4C22×D6112yes—
225C2×C42—614no—
226#95—1228no—
227C24—24no—
228C24—82no—
229C24—618no—
230C24—612no—
231C24—614no—
232C24—4882no—
233C24C2×C4×D6111yescharacteristic
234C24C2×C4×D6111yescharacteristic
235C24D4×D6114yes—
236C24C23×D6111yescharacteristic
237C2×D4—618no—
238C2×D4—618no—
239C2×D4—1228no—
240C2×D4—1228no—
241C2×D4—12216no—
242C2×D4—12216no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—82no—
247C22×C4—612no—
248C22×C4—614no—
249C22×C4—614no—
250C22×C4—1228no—
251C22×C4—1224no—
252C22×C4—1224no—
253C22×C4—1224no—
254C22×C4—4888no—
255C22×C4—4884no—
256C22×C4—4882no—
257C22×C4—2448no—
258C22×C4C2×C4×D6114yes—
259C4⋊C4—1228no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—28no—
263C22⋊C4—84no—
264C22⋊C4—1228no—
265C22⋊C4—1224no—
266C22⋊C4—1224no—
267C22⋊C4—1228no—
268C22⋊C4—4884no—
269C22⋊C4—2448no—
270C42—1228no—
271C23—22no—
272C23—22no—
273C23—28no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—88no—
280C23—614no—
281C23—618no—
282C23—618no—
283C23—618no—
284C23—4888no—
285C23—2444no—
286C23—2448no—
287C23C22⋊C4×D6112yes—
288C23C22⋊C4×D6112yes—
289C23C22×C4×D6111yescharacteristic
290C23C2×D4×D6112yes—
291C23#41×D6114yes—
292D4—12216no—
293D4—12216no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—614no—
301C2×C4—1224no—
302C2×C4—1228no—
303C2×C4—1224no—
304C2×C4—4888no—
305C2×C4—48816no—
306C2×C4—4884no—
307C2×C4—2448no—
308C2×C4—2444no—
309C2×C4—2444no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—24no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—88no—
321C22—618no—
322C22—1228no—
323C22—2448no—
324C22—2448no—
325C22—2448no—
326C22C2×C22⋊C4×D6111yescharacteristic
327C22#117×D6112yes—
328C22#123×D6114yes—
329C4—44no—
330C4—84no—
331C4—1224no—
332C4—4884no—
333C4—2444no—
334C2—24no—
335C2—22no—
336C2—42no—
337C2—42no—
338C2—1228no—
339C2#381×D6111yescharacteristic
340C2#2145427712yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1819C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C22
cosocleC24×C61C24
hypercenter#1819D61
hypocenterC61#20689
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #21445895C2semidirect
4. #21445893C2semidirect
5. #21445891C2semidirect
6. #342×D61C2semidirect
7. C61×#1819C2semidirect
10. C2×C4×C22⋊D61C4semidirect
11. C2×#14519737C4semidirect
29. C23×Dic61C2×C4semidirect
37. C2×#11758196C2×C4semidirect
40. C2×C22⋊Dic61C2×C4semidirect
47. C2×C4×Dic61C2×C4semidirect
50. C23×C4×C61C2×C4semidirect
53. C2×C22⋊C4×C61C2×C4semidirect
169. #1819D61semidirect
194. C61#20689semidirect
205. C23×C4C4×D61semidirect
222. C2×C22⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402352
#214483952
#214458952
#214458932
#214458912
#342×D612
C61×#18192
#2068961